Barry D Ganapol
 Professor, AerospaceMechanical Engineering
 Professor, Global Change  GIDP
 Professor, Hydrology / Atmospheric Sciences
 Member of the Graduate Faculty
Contact
 (520) 6214728
 Engineering, Rm. 000204
 Tucson, AZ 85721
 ganapol@cowboy.ame.arizona.edu
Bio
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Interests
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Courses
202122 Courses

Adv Engineering Analysis
AME 500B (Spring 2022) 
Adv Engineering Analysis
AME 500A (Fall 2021) 
Numerical Methods
AME 302 (Fall 2021)
202021 Courses

Intro To Nucrlear Engr
AME 580 (Spring 2021) 
Intro to Nuclear Engr
AME 480 (Spring 2021) 
Numerical Methods
AME 302 (Spring 2021) 
Numerical Methods
AME 302 (Fall 2020) 
Senior Colloquium
AME 495S (Fall 2020)
201920 Courses

Adv Engineering Analysis
AME 500B (Spring 2020) 
Intro To Nucrlear Engr
AME 580 (Spring 2020) 
Intro to Nuclear Engr
AME 480 (Spring 2020) 
Numerical Methods
AME 302 (Spring 2020) 
Adv Engineering Analysis
AME 500A (Fall 2019) 
Senior Colloquium
AME 495S (Fall 2019)
201819 Courses

Adv Engineering Analysis
AME 500A (Spring 2019) 
Intro To Nucrlear Engr
AME 580 (Spring 2019) 
Intro to Nuclear Engr
AME 480 (Spring 2019) 
Num Mth Fld Mech+Ht Trsf
AME 531 (Spring 2019) 
Numerical Methods
AME 302 (Fall 2018) 
Senior Colloquium
AME 495S (Fall 2018)
201718 Courses

Graduate Seminar
AME 696G (Fall 2017) 
Independent Study
AME 499 (Fall 2017) 
Senior Colloquium
AME 495S (Fall 2017)
201617 Courses

Intro To Nucrlear Engr
AME 580 (Spring 2017) 
Intro to Nuclear Engr
AME 480 (Spring 2017) 
Numerical Methods
AME 302 (Spring 2017) 
Graduate Seminar
AME 696G (Fall 2016) 
Senior Colloquium
AME 495S (Fall 2016)
Scholarly Contributions
Journals/Publications
 Ganapol, B. D., Schnurt, S., & Calvin, O. (2020). Consistency of the Chebyshev Rational Approximation Method. Annals of Nuclear Energy, 10.
 Ganapol, B. D. (2019). 1D thermal creep channel flow in the BGK approximation by adding and doubling. ANNALS OF NUCLEAR ENERGY, 134, 441451.
 Molinari, V., Mostacci, D., Pizzio, F., & Ganapol, B. D. (2019). Internal energy and hydrostatic pressure of a quantum relativistic ideal gas. RADIATION EFFECTS AND DEFECTS IN SOLIDS, 174(12), 140147.
 Ganapol, B., Mostacci, D., & Molinari, V. (2018). A Mathematical Realization of Entropy through Neutron Slowing Down. ENTROPY, 20(4).
 , V. M., , B. D., & , D. M. (2017). Density distribution for the molecules of a liquid in a semiinfinite space.More infoThe Sutherland approximation to the van der Waals forces is applied to thederivation of a selfconsistent Vlasovtype field in a liquid filling a halfspace, bordering vacuum. The ensuing Vlasov equation is then derived, andsolved to predict the behavior of the density at and in the vicinity of theliquidvacuum interface. A numerical solution to the Vlasov equation is alsoproduced and the density profile shown and discussed.[Journal_ref: ]
 Ganapol, B. D. (2017). An Accurate Numerical Solution to the Kinetics of Breakable Filament Assembly.More infoProteinaceous aggregation occurs through selfassembly a process notentirely understood. In a recent article [1], an analytical theory for amyloidfibril growth via secondary rather than primary nucleation was presented.Remarkably, with only a single kinetic parameter, the authors were able tounify growth characteristics for a variety of experimental data. In essence,they seem to have uncovered the underlying allometric laws governing theevolution of filament elongation simply from two coupled nonlinear ordinarydifferential equations (ODEs) stemming from a master equation. While this workadds significantly to our understanding of filament selfassembly, it requiredan approximate analytical solution representation. Here, we show that the sameresults are found by purely numerical means once a straightforward and reliablenumerical solution to the set of ODEs has been established.[Journal_ref: ]
 Ganapol, B. D. (2017). Highly Accurate Solutions of the Blasius and FalknerSkan Boundary Layer Equations via Convergence Acceleration.More infoA new highly accurate algorithm for the solution of the FalknerSkan equationof boundary layer theory is presented. The algorithm, based on a Maclaurinseries representation, finds its coefficients from recurrence. In addition,Wynnepsilon convergence acceleration and continuous analytical continuationenable an accurate evaluation. The most accurate skin friction coefficients(shooting angle) to date are presented along with comparisons to past andpresent values found in the literature. The algorithm, coded in FORTRAN, usesneither enhanced precision arithmetic beyond quadruple precision nor computeralgebra to achieve results in a timely fashion. Key Words: FalknerSkan flow; Blasius flow; Wynnepsilon acceleration;Romberg acceleration; Continuous analytical continuation[Journal_ref: ]
 Ganapol, B. D. (2017). Particle Transport in a 3D duct by adding and doubling.More infoParticle transport through a duct by Lambertian reflection from duct walls isagain considered. This popular transport example has been solved by mostnumerical transport methods except notably one the method of doubling. Weshall show that the method of doubling provides every bit as, or more, accuratereflectances and transmittances as the numerical discrete ordinates (NDO) andanalytical discrete ordinates (ADO) methods with less mathematical andnumerical effort.[Journal_ref: ]
 Ganapol, B. D. (2017). The Fourier transform solution for the Green's function of monoenergetic neutron transport theory.More infoNearly 45 years ago, Ken Case published his seminal paper on the singulareigenfunction solution for the Green's function of the monoenergetic neutrontransport equation with isotropic scattering. Previously, the solution had beenobtained by Fourier transform. While it is apparent the two had to beequivalent, a convincing equivalence proof for general anisotropic scatteringremained a challenge until now.[Journal_ref: ]
 Ganapol, B. D. (2017). The Response Matrix Discrete Ordinates Solution to the 1D Radiative Transfer Equation.More infoThe discrete ordinates method (DOM) of solution to the 1D radiative transferequation has been an effective method of solution for nearly 70 years. Duringthat time, the method has experienced numerous improvements as numerical andcomputational techniques have become more powerful and efficient. Here, weagain consider the analytical solution to the discrete radiative transferequation in a homogeneous medium by proposing a new, and consistent, form ofsolution that improves upon previous forms. Aided by a Wynnepsilon convergenceacceleration, its numerical evaluation can achieve extreme accuracy asdemonstrated by comparison with published benchmarks. Finally, we readilyextend the solution to a heterogeneous medium through the star productformulation producing a novel benchmark for closed form HenyeyGreensteinscattering as an example.[Journal_ref: ]
 McCormick, N. J., Siewert, C. E., Ganapol, B. D., & Prinja, A. K. (2017). Paul Frederick Zweifel. PHYSICS TODAY, 70(8), 7373.
 Ganapol, B. D. (2016). Particle Transport in a 3D duct by adding and doubling. Transport Theory and Computational Pysics.
 Molinari, V., Mostacci, D., & Ganapol, B. D. (2016). The Specific Heat of Liquid Helium. JOURNAL OF COMPUTATIONAL AND THEORETICAL TRANSPORT, 45(3), 212218.
 Molinari, V., Mostacci, D., & Ganapol, B. D. (2016). Wave Propagation in an Ideal Gas: First and Second Sound. JOURNAL OF COMPUTATIONAL AND THEORETICAL TRANSPORT, 45(4), 268274.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2016). Application of NonLinear Extrapolations for the Convergence Acceleration of Source Iteration. JOURNAL OF COMPUTATIONAL AND THEORETICAL TRANSPORT, 45(5), 351367.
 Ganapol, B. D. (2015). The response matrix discrete ordinates solution to the 1D radiative transfer equation. JOURNAL OF QUANTITATIVE SPECTROSCOPY & RADIATIVE TRANSFER, 154, 7290.
 Molinari, V., Ganapol, B. D., & Mostacci, D. (2015). Density distribution for the molecules of a liquid in a semiinfinite space. MODERN PHYSICS LETTERS B, 29(21).
 Ganapol, B. D. (2014). A 1D MONOENERGETIC NEUTRON TRANSPORT BENCHMARK IN AN INFINITE MEDIUM. PROCEEDINGS OF THE 22ND INTERNATIONAL CONFERENCE ON NUCLEAR ENGINEERING  2014, VOL 4.
 Ganapol, B. D. (2014). Chandrasekhar Polynomials and the Solution to the Transport Equation in an Infinite Medium. Transport Theory and Statistical Physics, 43(17), 433473.
 Ganapol, B. D. (2013). A highly accurate algorithm for the solution of the point kinetics equations. Annals of Nuclear Energy, 62, 564571.More infoAbstract: Attempts to resolve the point kinetics equations (PKEs) describing nuclear reactor transients have been the subject of numerous articles and texts over the past 50 years. Some very innovative methods, such as the RTS (Reactor Transient Simulation) and CAC (Continuous Analytical Continuation) methods of G.R. Keepin and J. Vigil respectively, have been shown to be exceptionally useful. Recently however, several authors have developed methods they consider accurate without a clear basis for their assertion. In response, this presentation will establish a definitive set of benchmarks to enable those developing PKE methods to truthfully assess the degree of accuracy of their methods. Then, with these benchmarks, two recently published methods, found in this journal will be shown to be less accurate than claimed and a legacy method from 1984 will be confirmed. © 2012 Elsevier Ltd. All rights reserved.
 Ganapol, B., Picca, P., Previti, A., & Mostacci, D. (2013). Benchmarks for the point kinetics equations. International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013, 3, 21192146.More infoAbstract: A new numerical algorithm is presented for the solution to the point kinetics equations (PKEs), whose accurate solution has been sought for over 60 years. The method couples the simplest of finite difference methods, a backward Euler, with Richardsons extrapolation, also called an acceleration. From this coupling, a series of benchmarks have emerged. These include cases from the literature as well as several new ones. The novelty of this presentation lies in the breadth of reactivity insertions considered, covering both prescribed and feedback reactivities, and the extreme 8 to 9 digit accuracy achievable. The benchmarks presented are to provide guidance to those who wish to develop further numerical improvements.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2013). A highly accurate technique for the solution of the nonlinear point kinetics equations. ANNALS OF NUCLEAR ENERGY, 58, 4353.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2013). A highly accurate technique for the solution of the nonlinear point kinetics equations. Annals of Nuclear Energy, 58, 4353.More infoAbstract: A novel methodology for the solution of nonlinear point kinetic (PK) equations is proposed. The technique, based on a piecewise constant approximation (Kinard and Allen, 2004), is enhanced by explicitly accounting for the feedback and the reactivity variation within a time step through an iterative cycle. High accuracy is achieved by introducing a submesh for the numerical evaluation of integrals involved and by correcting the source term to include the nonlinear effect on a finer time scale. The resulting Enhanced Piecewise Constant Approximation (EPCA) is tested on a set of classical linear problems with several types of reactivity insertions (step, linear, sinusoidal, zigzag) and shows extreme accuracy (to 9 digits) even when large time steps are considered (i.e., 100 times the neutron mean life). Nonlinear reactor kinetics is then considered and compared to highly accurate results obtained via convergence acceleration. Its accuracy and the fast convergence make the EPCA algorithm particularly attractive for applications. © 2013 Elsevier Ltd. All rights reserved.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2013). Application of convergence acceleration to the reactor kinetic equations: A comparative study. International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013, 4, 26662677.More infoAbstract: This presentation provides a comparison of two methodologies for the solution of reactor kinetic equations, namely for a standard finite difference and a semianalytical approach. The abovementioned methods are implemented in a convergence acceleration framework to enhance their efficiency and a comparative study is reported to verify whether it is more convenient to use a rudimentary but fast algorithm (finite difference) with respect to the more refined but computationally intense approach of the semianalytical method. Performance on several test cases from the literature are compared.
 Terranova, N., Mostacci, D., & Ganapol, B. D. (2013). Converged accelerated finite difference scheme for the multigroup neutron diffusion equation. International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013, 3, 20882102.More infoAbstract: Computer codes involving neutron transport theory for nuclear engineering applications always require verification to assess improvement. Generally, analytical and semianalytical benchmarks are desirable, since they are capable of high precision solutions to provide accurate standards of comparison. However, these benchmarks often involve relatively simple problems, usually assuming a certain degree of abstract modeling. In the present work, we show how semianalytical equivalent benchmarks can be numerically generated using convergence acceleration. Specifically, we investigate the error behavior of a 1D spatial finite difference scheme for the multigroup (MG) steadystate neutron diffusion equation in plane geometry. Since solutions depending on subsequent discretization can be envisioned as terms of an infinite sequence converging to the true solution, extrapolation methods can accelerate an iterative process to obtain the limit before numerical instability sets in. The obtained results have been compared to the analytical solution to the 1D multigroup diffusion equation when available, using FORTRAN as the computational language. Finally, a slowing down problem has been solved using a cascading source update, showing how a finite difference scheme performs for ultrafine groups (104 groups) in a reasonable computational time using convergence acceleration.
 Ayoobian, N., Hadada, K., & Ganapol, B. D. (2012). Dosimetry parameters calculation of two commercial iodine brachytherapy sources using SMARTEPANTS with EPDL97 library. JOURNAL OF CANCER RESEARCH AND THERAPEUTICS, 8(4), 610618.
 Ayoobian, N., Hadada, K., & Ganapol, B. D. (2012). Dosimetry parameters calculation of two commercial iodine brachytherapy sources using SMARTEPANTS with EPDL97 library. Journal of Cancer Research and Therapeutics, 8(4), 610618.More infoPMID: 23361283;Abstract: Aim: Simulating Many Accumulative Rutherford Trajectories Electron Photon and Neutral Transport Solver (SMARTEPANTS) is a discrete ordinates S N Boltzmann/SpencerLewis solver that was developed during 19881993 by William Filippone and his students. The code calculates particle fluxes, leakage currents as well as energy and charge deposition for coupled electron/photon in xyz geometries both in forward and in adjoin modes. Originally, SMARTEPANTS was designed to utilize CEPXS crosssection library for shielding calculation in satellite electronics. The aim of this study was to adapt SMARTEPANTS to use a new photon crosssection library from Evaluated Photon Data Library, 1997 version (EPDL97) for intravascular brachytherapy 125 Isimulations. Materials and Methods: A MATLAB (MathworkNatick, Massachusetts) program was written to generate an updated multigroupLegendre crosssection from EPDL97. The new library was confirmed by simulating intravascular brachytherapy Best® Model 2301 and Intersource 125 I dosimetry parameters using SMARTEPANTS with different energy groups (g), Legendre moments (L) and discrete ordinate orders (S). Results: The dosimetry parameters for these sources were tabulated and compared with the data given by AAPM TG43 and other reports. The computation time for producing TG43 parameters was about 29.4 min in case of g = 20, L = 7 and S = 16. Conclusion: The good agreement between the results of this study and previous reports and high computational speed suggest that SMARTEPANTS could be extended to a realtime treatment planning system for 125 I brachytherapy treatments.
 Crawford, D. S., Ganapol, B. D., Nigg, D. W., & Wemple, C. A. (2012). A spectral verification of the HELIOS2 lattice physics code. Transactions of the American Nuclear Society, 107, 537540.
 Ganapol, B. D. (2012). An Accurate Numerical Solution to the Kinetics of Breakable Filament Assembly. Transport Theory and Statistical Physics, 41(12), 153174.More infoAbstract: Proteinaceous aggregation occurs through selfassemblya process not entirely understood. In a recent article, Knowles and colleagues (2009) presented an analytical theory for amyloid fibril growth via secondary rather than primary nucleation. Remarkably, with only a single kinetic parameter, the authors were able to unify growth characteristics for a variety of experimental data. In essence, they seem to have uncovered the underlying allometric law governing the evolution of filament elongation simply from two coupled nonlinear ordinary differential equations originally obtained from a master equation. While this work adds significantly to our understanding of filament selfassembly, it required an "approximate" analytical solution representation for the moments of the chain length distribution. If this were always true, the discovery of such scaling laws would be infrequent. Here, we show that the same results are found by purely numerical means. In addition, the numerical method used features a highly accurate solution strategy for the coupled Ordinary Differential Equations (ODEs) based only on a fundamental finite difference scheme and convergence acceleration. Once a reliable numerical solution has been established, a dimensional analysis then provides the scaling laws. © 2012 Copyright Taylor and Francis Group, LLC.
 Ganapol, B. D., Wang, Y., I, R. M., & II, F. G. (2012). Is convergence acceleration an advantage for neutron transport algorithms?. Transactions of the American Nuclear Society, 107, 531534.
 Ganapol, B., Picca, P., Previti, A., & Mostacci, D. (2012). The solution of the point kinetics equations via converged accelerated Taylor series (CATS). International Conference on the Physics of Reactors 2012, PHYSOR 2012: Advances in Reactor Physics, 4, 33913409.More infoAbstract: This paper deals with finding accurate solutions of the point kinetics equations including nonlinear feedback, in a fast, efficient and straightforward way. A truncated Taylor series is coupled to continuous analytical continuation to provide the recurrence relations to solve the ordinary differential equations of point kinetics. Nonlinear (Wynnepsilon) and linear (Romberg) convergence accelerations are employed to provide highly accurate results for the evaluation of Taylor series expansions and extrapolated values of neutron and precursor densities at desired edits. The proposed Converged Accelerated Taylor Series, or CATS, algorithm automatically performs successive mesh refinements until the desired accuracy is obtained, making use of the intermediate results for converged initial values at each interval. Numerical performance is evaluated using case studies available from the literature. Nearly perfect agreement is found with the literature results generally considered most accurate. Benchmark quality results are reported for several cases of interest including step, ramp, zigzag and sinusoidal prescribed insertions and insertions with adiabatic Doppler feedback. A larger than usual (9) number of digits is included to encourage honest benchmarking. The benchmark is then applied to the enhanced piecewise constant algorithm (EPCA) currently being developed by the second author.
 Joseph, R. A., Ganapol, B., & Maldonado, I. (2012). The ultrafinegroup panel method for neutron slowing down. International Conference on Nuclear Engineering, Proceedings, ICONE, 1, 725731.More infoAbstract: The CENTRM1 algorithm has been developed as a part of the SCALE2 code package to process pointwise and groupwise cross section data. While the CENTRM code is the state of the art for cross section generation, there is always a need for further verification and improvement. The theme of this presentation therefore is verification through an independent computational strategy. An entirely independent method of treating a large number of energy groups, on the order of 70,000, called the method of panels, is proposed and tested. Here, we report on the numerical method used to implement the panel procedure and the progress made to date with CENTRM verification. Copyright ©2012 by ASME.
 Joseph, R., Ganapol, B., & Maldonado, I. (2012). THE ULTRAFINEGROUP PANEL METHOD FOR NEUTRON SLOWING DOWN. PROCEEDINGS OF THE 20TH INTERNATIONAL CONFERENCE ON NUCLEAR ENGINEERING AND THE ASME 2012 POWER CONFERENCE  2012, VOL 1, 725731.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2012). An Efficient Multiproblem Strategy for Accurate Solutions of Linear Particle Transport Problems in Spherical Geometry. NUCLEAR SCIENCE AND ENGINEERING, 170(2), 103124.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2012). Derivation of a Physically Based Hybrid Technique for the Solution of SourceDriven TimeDependent Linear Boltzmann Equations. TRANSPORT THEORY AND STATISTICAL PHYSICS, 41(12), 2339.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2012). Derivation of a Physically Based Hybrid Technique for the Solution of SourceDriven TimeDependent Linear Boltzmann Equations. Transport Theory and Statistical Physics, 41(12), 2339.More infoAbstract: The article presents a novel kinetic method for the solution of timedependent sourcedriven linear transport problems. As will be discussed, the method is based on a multigeneration representation of linear Boltzmann equations and is derived from physically grounded considerations. The technique is said to be hybrid since it combines together collisionless transport solutions and factorization techniques. The methodology is tested on systems with different scattering properties and shows very promising performance both in terms of accuracy and computational times. © 2012 Copyright Taylor and Francis Group, LLC.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2012). On Radiative Transfer in Dense Vegetation Canopies. TRANSPORT THEORY AND STATISTICAL PHYSICS, 41(34), 223244.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2012). On Radiative Transfer in Dense Vegetation Canopies. Transport Theory and Statistical Physics, 41(34), 223244.More infoAbstract: This article presents a review of the special features exhibited by the transport of photons in dense vegetation canopies. The conservation of photons for particles moving in canopy media yields a linear Boltzmann Equation similar to conventional participating media. However, because of the turbid medium approximation where the atomic leaves are modeled as a gas cloud of infinitesimal plates with specified orientation, the overall system is not rotationally invariant. The latter affects the mathematical nature of the canopy equation and requires special numerical treatment to generate accurate numerical solutions. Here, the most common solution techniques are reviewed with special emphasis on two highly accurate methodologies, namely the Singular Eigenfunction Expansion, which is the basis of the F N method for canopy transport and the Analytical Discrete Ordinate (ADO) method. It is shown that the conventional ADO method can be easily extended to canopy transport problems by taking advantage of special symmetries exhibited by the intercept function and the area scattering function employed to describe the canopy optical properties. © 2012 Copyright Taylor and Francis Group, LLC.
 Picea, P., Furfaro, R., & Ganapol, B. D. (2012). An efficient multiproblem strategy for accurate solutions of linear particle transport problems in spherical geometry. Nuclear Science and Engineering, 170(2), 103124.More infoAbstract: A novel multiproblem methodology devised to manufacture highly accurate numerical solutions of the linear Boltzmann equation is proposed. As an alternative to classical discretization schemes that focus on a single mesh, the multiproblem approach seeks transport solutions as the limit of a sequence of calculations executed on successively more refined grids. The sequence of approximations serves as a basis for the extrapolation of the solution toward its meshindependent limit. Furthermore, the multiproblem strategy allows an optimization of the computational effort whenever compared to the singlegrid approach. Indeed, the solution obtained on an unrefined mesh is employed as the starting guess for transport calculations on the next grid of the sequence, drastically reducing the number of inner iterations needed on the highly refined mesh. The efficiency of the algorithm may be further improved by combining the source iterations with a convergence acceleration scheme based on nonlinear extrapolation algorithms. To evaluate the performance of the proposed approach, the multiproblem methodology is applied to solve linear transport problems in spherical geometry, which are known to feature special properties whenever compared with the transport of particles in Cartesian geometry. The methodology is implemented by choosing the presumably simplest and most widespread numerical transport algorithm (i.e., discrete ordinates with diamond differences). Results show that five to sixdigit accuracy can be obtained in a competitive computational time without resorting to powerful workstations.
 Terranova, N., & Ganapol, B. D. (2012). An adaptive natural multigroup for neutron slowing down. Transactions of the American Nuclear Society, 106, 388391.
 Ganapol, B. D. (2011). An analytical multigroup benchmark for (n, γ) and (n, n′, γ) verification of diffusion theory algorithms. Annals of Nuclear Energy, 38(9), 20172023.More infoAbstract: The angular flux for the "rod model" describing coupled neutron/gamma (n, γ) diffusion has a particularly straightforward analytical representation when viewed from the perspective of a onegroup homogeneous medium. Cast in the form of matrix functions of a diagonalizable matrix, the solution to the multigroup equations in heterogeneous media is greatly simplified. We shall show exactly how the onegroup homogeneous medium solution leads to the multigroup solution. © 2011 Elsevier Ltd. All rights reserved.
 Ganapol, B. D. (2011). Radiative transfer with internal reflection via the converged discrete ordinates method. Journal of Quantitative Spectroscopy and Radiative Transfer, 112(4), 693713.More infoAbstract: In response to the challenge of establishing highly accurate solutions to the plane layer radiative transfer equation with the simplest of methods, the converged discrete ordinates method is presented. With this algorithm of only finite difference, quadrature and acceleration, we show how to obtain highly accurate intensities for radiative transfer in a finite layer with internal surface reflection. The method features angular smoothing and angular interpolation through "faux" quadrature. In addition, a manufactured solution demonstrates the high accuracy of the method for forward peaked scattering. We consider scattering in a heterogeneous medium as a final demonstration. © 2010 Elsevier Ltd.
 Ganapol, B. D., & Picca, P. (2011). A highly accurate benchmark for reactor point kinetics with feedback, invited. Transactions of the American Nuclear Society, 104, 885887.
 Ganapol, B. D., Tye, B., Battistini, E., & Mostacci, D. (2011). Implementation of an analytical multigroup diffusion solution in 1D heterogeneous media. Transactions of the American Nuclear Society, 104, 379381.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2011). A Hybrid Transport PointKinetic method for simulating source transients in subcritical systems. ANNALS OF NUCLEAR ENERGY, 38(12), 26802688.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2011). A hybrid transport pointkinetic method for simulating source transients in subcritical systems. Annals of Nuclear Energy, 38(12), 26802688.More infoAbstract: This paper describes a novel hybrid technique for the solution of the timedependent neutron transport equation in source switchon transients. The Hybrid Transport Point Kinetics (HTPK) method is derived from a multicollision description of the transport equation and is based on the observation that the external source mainly influences the first few neutron generations and, through collisions and multiplications, an ensemble behavior statistically emerges at higher order. The HTPK is designed to separately describe the propagation phenomenon from the ensemble response of the system, using both the rigorous transport equation and simplified kinetic models. Numerical simulations performed on a typical acceleration driven system configuration confirm the basic physical intuition of HTPK and show a significant gain in accuracy when compared to classical point kinetics. The proposed HTPK technique is also shown to be attractive because of its flexibility and competitive computational effort. © 2011 Elsevier Ltd. All rights reserved.
 Previti, A., Furfaro, R., Picca, P., Ganapol, B. D., & Mostacci, D. (2011). Solving radiative transfer problems in highly heterogeneous media via domain decomposition and convergence acceleration techniques. APPLIED RADIATION AND ISOTOPES, 69(8), 11461150.
 Previti, A., Furfaro, R., Picca, P., Ganapol, B. D., & Mostacci, D. (2011). Solving radiative transfer problems in highly heterogeneous media via domain decomposition and convergence acceleration techniques. Applied Radiation and Isotopes, 69(8), 11461150.More infoPMID: 21131208;Abstract: This paper deals with finding accurate solutions for photon transport problems in highly heterogeneous media fastly, efficiently and with modest memory resources. We propose an extended version of the analytical discrete ordinates method, coupled with domain decompositionderived algorithms and nonlinear convergence acceleration techniques. Numerical performances are evaluated using a challenging case study available in the literature. A study of accuracy versus computational time and memory requirements is reported for transport calculations that are relevant for remote sensing applications. © 2010 Elsevier Ltd.
 Battistini, E., Ganapol, B. D., Furfaro, R., & Mostacci, D. (2010). Convergence acceleration for multipanel fine group finite difference numerical benchmarks. Transactions of the American Nuclear Society, 103, 362364.
 Battistini, E., Ganapol, B. D., Furfaro, R., & Mostacci, D. (2010). Paneling strategy for ultrafine group slowing down numerical benchmarks. Transactions of the American Nuclear Society, 103, 365367.
 Ganapol, B. (2010). Onegroup steady state diffusion in a heterogeneous slab. Mathematics and Computers in Simulation, 80(11), 21422158.More infoAbstract: We derive a customized solution to the diffusion equation in a heterogeneous slab with an arbitrary contiguous source. While aspects of solution have previously been reported in the literature, its full explicit form as presented here by solving a 3term recurrence, has not. We conclude with a series of exercises demonstrating the utility of the solution. © 2010 IMACS.
 Ganapol, B. D., & Nigg, D. W. (2010). High order finite difference approximations to the onegroup neutron diffusion equation in 1D heterogeneous media part I: Theory in plane media. International Conference on the Physics of Reactors 2010, PHYSOR 2010, 4, 32443266.More infoAbstract: Verification that a numerical method performs as intended is an integral part of code development. Semianalytical benchmarks enable one such verification modality. Unfortunately, a semianalytical benchmark requires some degree of analytical forethought and treats only relatively idealized cases making it of limited diagnostic value. In the first part of our investigation (Part I), we establish the theory of a straightforward finite difference scheme for the 1D, monoenergetic neutron diffusion equation in plane media. We also demonstrate an analytically enhanced version that leads directly to the analytical solution. The second part of our presentation (Part II, in these proceedings) is concerned with numerical implementation and application of the finite difference solutions. There, we demonstrate how the numerical schemes themselves provide the semianalytical benchmark. With the analytical solution known, we therefore have a test for accuracy of the proposed finite difference algorithms designed for high order.
 Ganapol, B. D., & Nigg, D. W. (2010). High order finite difference approximations to the onegroup neutron diffusion equation in 1D heterogeneous media part II: Implementation and application. International Conference on the Physics of Reactors 2010, PHYSOR 2010, 4, 32673286.More infoAbstract: Verification that a numerical method performs as intended is an integral part of code development. Semianalytical benchmarks enable one such verification modality. Unfortunately, a semianalytical benchmark requires some degree of analytical forethought and treats only relatively idealized cases making it of limited diagnostic value. In the first part of our investigation (Part I, in these proceedings), we established the theory of a straightforward finite difference scheme for the 1D, monoenergetic neutron diffusion equation in plane media. We also demonstrated an analytically enhanced version that leads to the analytical solution. The second part of our presentation (Part II) concerns the numerical implementation and application of the finite difference solutions of Part I. Here, we demonstrate how the numerical schemes themselves provide the semianalytical benchmark. With the analytical solution known, we therefore have a test for accuracy of the proposed finite difference algorithms designed for high order.
 Ganapol, B., DeHart, M., & Park, H. (2010). Accelerating the K_{eff} calculation in the NEWT lattice physics code. Transactions of the American Nuclear Society, 103, 335337.
 Hadad, K., Hajizadeh, A., Jafarpour, K., & Ganapol, B. D. (2010). Neutronic study of nanofluids application to VVER1000. Annals of Nuclear Energy, 37(11), 14471455.More infoAbstract: The change in neutronic parameters of the VVER1000 nuclear reactor core attributable to the use of nanoparticle/water (nanofluid) as coolant is presented in this paper. Optimization of type and volume fraction of nanoparticles in water that affect the safety enhancement of core primary parameters is intended in this study. Reactivity change, radial and axial local peaking factors (LPPF), and the consequence of nanoparticle deposition on fuel clad are investigated. We considered five nanoparticles which have been studied extensively for their heat transfer properties including Alumina, Aluminum, Copper oxide, Copper and Zirconia. The results of our study show that at low concentration (0.001 volume fraction) Alumina is optimum nanoparticle for normal operation. The maximum radial and axial LPPF were found to be invariant to the type of nanofluid at low volume fractions. With an increase in nanoparticle deposition thickness on fuel clad, a flux and Keff depression occurs and Al2O3 has the lowest rate of drop off. © 2010 Elsevier Ltd. All rights reserved.
 Hadad, K., Yousefnia, M., & Ganapol, B. D. (2010). Evaluation of power peaking factor in Iranian VVER1000 using MCNP. Transactions of the American Nuclear Society, 102, 499500.
 Picca, P., Furfaro, R., Ganapol, B. D., Dulla, S., & Ravetto, P. (2010). Application of artificial neural networks to infer subcriticality level through kinetic models. International Conference on the Physics of Reactors 2010, PHYSOR 2010, 1, 397407.More infoAbstract: The paper presents some recent advances in the study of the inverse kinetics for subcritical systems. A neuralbased approach is adopted to predict the reactivity of the multiplying medium through the analysis of the reactor response to a source pulse. An artificial neural network is designed to infer the subcriticality level through the analysis of power evolution. The training set is computed using an approximate model and its performances are then tested directly on experimental measures, here simulated through a detailed spaceenergy kinetic model. In order to improve the accuracy of the reactivity estimation, various strategies are proposed and compared, including a multitransient inversion and the use of different kinetic models for the training. The issue of robustness of the inversion scheme to experimental noise is also addressed.
 Picca, P., Ganapol, B. D., & Furfaro, R. (2010). A hybrid transportpoint kinetic method for the analysis of source transients in subcritical systems. International Conference on the Physics of Reactors 2010, PHYSOR 2010, 1, 387396.More infoAbstract: The paper describes a novel technique for approximating the timedependent neutron transport equation in the simulation of sourcedriven transients. The methodology, based on a physical intuition, adopts a multigenerational approach to improve the accuracy of lumped parameter approximations and better account for the propagation effects typical of a source switchon. More precisely, the first few neutron generations are simulated collision by collision through the Boltzmann equation while a simplified lumped parameter model is adopted for the multiply collided neutrons. Numerical simulations prove that a significant gain in accuracy is obtained compared to classical point kinetics, especially for detector signal analysis. In terms of computational costs, the technique is more time consuming than point kinetics but, at low orders, remains sensibly faster than fulll time dependent transport solution or quasistatic approximations.
 Ganapol, B. D. (2009). A REFINED WAY OF SOLVING REACTOR POINT KINETICS EQUATIONS FOR IMPOSED REACTIVITY INSERTIONS. NUCLEAR TECHNOLOGY & RADIATION PROTECTION, 24(3), 157166.
 Ganapol, B. D., & Hadad, K. (2009). The converged Sn algorithm for nuclear criticality. American Nuclear Society  International Conference on Mathematics, Computational Methods and Reactor Physics 2009, M and C 2009, 3, 17131729.More infoAbstract: A new discrete ordinates algorithm to determine the multiplication factor of a 1D nuclear reactor, based on Bengt Carlson's S n method, is presented. The algorithm applies the Romberg and Wynnepsilon accelerators to accelerate a 1D, onegroup S n solution to its asymptotic limit. We demonstrate the feasibility of the Converged Sn (CSn) solution on several onegroup criticality benchmark compilations. The new formulation is especially convenient since it enables highly accurate critical fluxes and eigenvalues using the most fundamental transport algorithm.
 Ganapol, B. D., & Kornreich, D. E. (2009). Threedimensional transport theory: An analytical solution for the internal beam searchlight problem, II. ANNALS OF NUCLEAR ENERGY, 36(8), 12421255.
 Ganapol, B. D., & Kornreich, D. E. (2009). Threedimensional transport theory: An analytical solution for the internal beam searchlight problem, II. Annals of Nuclear Energy, 36(8), 12421255.More infoAbstract: Multidimensional semianalytical particle transport benchmarks to provide highly accurate standards of assessment are few and far between. Because of a wellestablished 1D theory for the analytical solution of the transport equation, it is sometimes possible however, to "bootstrap" 1D solutions to give more comprehensive solution representations. Here, we propose the internal searchlight problem in a half space, designated ISLP/HS, as a multidimensional benchmark to be constructed from 1D solutions. This is a variation of the usual SLP/HS where a source emits within the half space rather than striking its surface. Our primary interest is in the exiting intensity at the free surface established through a new Fn formulation. The benchmark features true 2/3D particle transport through integration of a point kernel to simulate 2/3D source emission. In this way, we accommodate a solid or hollow cylindrical source and a general line source in addition to the standard point, ring and disk sources featured in previous investigations. © 2009 Elsevier Ltd.
 Ganapol, B. D., & Nigg, D. W. (2009). A high order finite difference algorithm for the onegroup diffusion equation. Transactions of the American Nuclear Society, 101, 383386.
 Ganapol, B. D., Maldonado, G. I., & Williams, M. L. (2009). An ultrafine group slowing down benchmark. American Nuclear Society  International Conference on Mathematics, Computational Methods and Reactor Physics 2009, M and C 2009, 2, 870881.More infoAbstract: We suggest a new solution to the neutron slowing down equation in terms of multienergy panels. Our motivation is to establish a computational benchmark featuring an ultrafine group calculation, where the number of groups could be on the order of 100,000. While the CENTRM code of the SCALE code package has been shown to adequately treat this many groups, there is always a need for additional verification. The multipanel solution principle is simply to consider the slowing down region as sub regions of panels, with each panel a manageable number of groups, say 100. In this way, we reduce the enormity of dealing with the entire spectrum all at once by considering many smaller problems. We demonstrate the solution in the unresolved U 238 resonance region.
 Hadad, K., Ganapol, B. D., Hamilton, R. J., Watchman, C. J., & Yang, X. u. (2009). Dose assessment to heart and lungs in HDR breast brachytherapy. Transactions of the American Nuclear Society, 101, 10971099.
 Hadad, K., Ganapol, B. D., Hamilton, R., & Watchman, C. (2009). Dose verification for accelerated partial breast irradiation. American Nuclear Society  International Conference on Mathematics, Computational Methods and Reactor Physics 2009, M and C 2009, 2, 13711381.More infoAbstract: Noticeable advances have been made in reducing the reoccurrence of breast cancer after a lumpectomy through invasive irradiation of the surrounding participating tissue. One effective postoperative procedure introduces a radionuclide applicator surrounded by an inflatable balloon into the evacuated breast cavity. Through, a series of polyethylene guide tubes, radioactive sources are introduced in a timecontrolled program to irradiate tissue surrounding the lumpectomy site to a desired integrated dose of about 30 Gy at a depth of 1cm over a 5day period in 10 treatment fractions. To be most effective and to minimize collateral radiation damage, reliable and accurate dose estimates must be performed in the patient treatment planning stage. The current treatment uses the TG43 protocol [1] where radiation transport is estimated in a standard tissue phantom to provide best estimates of the delivered dose. Given today's computational power and comprehensive radiation transport algorithms, it is generally thought by those in the radiation oncology community that we should be able to more precisely predict doses, which is the subject of this presentation.
 Hadad, K., Pirouzmand, A., & Ganapol, B. D. (2009). Verification of cellular neural network (CNN) applied to TN equation in slab geometry. Transactions of the American Nuclear Society, 101, 407410.
 Luciani, A., Panfili, P., Furfaro, R., Ganapol, B. D., & Mostacci, D. (2009). Estimating water and ice content on planetary soils using neutron measurements: A neural network approach. Radiation Effects and Defects in Solids, 164(56), 345349.More infoAbstract: A modelbased neural network methodology to estimate water and ice content in planetary soils using neutron fluxes detected by in situ and/or airborne deployment of neutron detectors is proposed and shown to be effective. Focusing of epithermal and thermal energy regimes, the neutron fluxes are computed [Panfili, P.; Luciani, A.; Furfaro, R.; Ganapol, B.D.; Mostacci, D. Radiat. Eff. Defects Solids 2009, 164 (56), 340344.] as functions of the medium physical properties and used to train neural networks in the inverse mode. For homogeneous soil, the modelbased neural network shows satisfactory performances in retrieving the percentage of water. For soil modelled as layered, neural networks designed to retrieve both the depth and thickness of an ice layer beneath the soil surface provide good results only in a limited range of configurations. However, it has been found that training the two networks to independently retrieve the two parameter results more accurately. It has also been found that multiple measurements help improve the accuracy of the inversion for this configuration.
 Panfili, P., Luciani, A., Furfaro, R., Ganapol, B. D., & Mostacci, D. (2009). Modelling neutron transport in planetary media via analytical multigroup diffusion theory. Radiation Effects and Defects in Solids, 164(56), 340344.More infoAbstract: A novel analytical solution to the 1D, steadystate, multislab, multigroup diffusion equation is proposed as a mean to compute the energydependent galactic cosmic rayinduced neutron fluxes established in planetary media. More specifically, the proposed algorithm is implemented to allow fast and highly accurate determination of lowenergy cosmic ray neutrons inside the Earth's surface and atmosphere. Two sets of experimental measurements have been considered to validate our model. In both cases, a good agreement between the calculated and observed neutron fluxes is achieved. Subsequently, neutron diffusion calculations have been performed for various Earthbased scenarios comprising (a) twoslab (airsoil) configuration and (b) threeslab (airsoilice) configuration to investigate the functional relationship between soil composition and neutron spatial distribution.
 Park, H., Knoll, D. A., & Ganapol, B. D. (2009). Application of a nonintrusive convergence acceleration to criticality calculations. Transactions of the American Nuclear Society, 101, 444446.
 Picca, P., Furfaro, R., Ganapol, B. D., Dulla, S., & Ravetto, P. (2009). Subcriticality determination by neuralbased inversion of spaceenergy neutron kinetic equations. American Nuclear Society  International Conference on Mathematics, Computational Methods and Reactor Physics 2009, M and C 2009, 5, 30493058.More infoAbstract: The determination of the reactivity level in ADS represents nowadays an important open issue for the development of subcritical systems. Several approaches were developed in the past based on simplified models, such as point kinetics, for which an analytical inversion is feasible. On the other hand, numerical simulations shows that the evolution of the power and detector signals are not pointlike. In the present work, we propose a neuralbased inversion that can invert more realistic physical models. As a result, the spatial and spectral effects connected to the position and the energy selectivity of detectors are directly accounted for. An outline of the theory is presented in the following and some preliminary results show the performance of the technique proposed in the interpretation of pulsed experiments.
 Picca, P., Ganapol, B. D., & Furfaro, R. (2009). Timedependent neutral particle transport in spherical geometry. Transactions of the American Nuclear Society, 101, 396398.
 Ganapol, B. D. (2008). A neutron transport benchmark in onedimensional cylindrical geometry: Revisited. Nuclear Science and Engineering, 159(2), 169181.More infoAbstract: A new benchmark for monoenergetic neutron transport in onedimensional cylindrical geometry is presented. In the past, several accurate benchmarks (i.e., numerical solutions) in cylindrical geometry, based on the singular eigenfunction expansion of the solution to the corresponding pseudoproblem, have appeared in the literature. In the new formulation, called the direct F N method in cylindricalgeometry, we base the FN solution directly on the integrodijferential equation satisfied by the pseudo problem. Through appropriate projections, a straightforward FN formulation results in singular integral equations for both the flux and current. Enhanced by convergence acceleration, the FN approximation accurately reproduces published benchmark solutions for both fixed sources and criticality. Thus, we have developed an entirely pedagogical selfcontained and highly accurate benchmark based on an alternative application of FN theory.
 Ganapol, B. D. (2008). Multigroup caseology in 1D via the Fourier transform. Progress in Nuclear Energy, 50(8), 886907.More infoAbstract: An alternative view of the Fourier transform inversion provides a new solution to the ageold problem of multigroup neutral particle transport equation in onedimensional plane geometry. Through analytical continuation, the inversion contour shifts from the real line to accommodate pole contributions as discrete matrix eigenfunctions with continuum contributions appear from the branch cut. We recast the solution in terms of continuum singular matrix eigenfunctions to derive a singular eigenfunction expansion. Closure is an immediate consequence of the Fourier transform inversion. Finally, based on knowledge of the matrix weight factors and closure, we construct a singular eigenfunction expansion, incorporating orthogonality. The significance of this work is that a concise and consistent multigroup eigenfunction expansion emerges for anisotropic scattering that is as easily applied as in the onegroup case. © 2008.
 Ganapol, B. D., & Furfaro, R. (2008). The art of analytical benchmarking. Lecture Notes in Computational Science and Engineering, 62 LNCSE, 105134.More infoAbstract: The most primitive numerical method of solving the radiative transfer equation forms the basis of an algorithm to generate high quality numerical solutions. We apply convergence acceleration to the discrete ordinates balance equation derived from a diamond difference approximation. Wynnepsilon and Romberg accelerations then rearrange low order solutions to produce a high order solution. Published semianalytical benchmarks from the C.E. Siewert portfolio establish the credibility of the new converged discrete ordinates method (CDOM). © 2008 SpringerVerlag Berlin Heidelberg.
 Ganapol, B. D., & Kornreich, D. E. (2008). Interior beam searchlight semianalytical benchmark. International Conference on the Physics of Reactors 2008, PHYSOR 08, 1, 595603.More infoAbstract: Multidimensional semianalytical benchmarks to provide highly accurate standards to assess routine numerical particle transport algorithms are few and far between. Because of the wellestablished 1D theory for the analytical solution of the transport equation, it is sometimes possible to "bootstrap" a 1D solution to generate a more comprehensive solution representation. Here, we consider the searchlight problem (SLP) as a multidimensional benchmark. A variation of the usual SLP is the interior beam SLP (IBSLP) where a beam source lies beneath the surface of a half space and emits directly towards the free surface. We consider the establishment of a new semianalytical benchmark based on a new FN formulation. This problem is important in radiative transfer experimental analysis to determine cloud absorption and scattering properties.
 Ganapol, B. D., & Mund, E. H. (2008). Acceleration of the numerical solution of the reactor kinetics equations in plane geometry. International Conference on the Physics of Reactors 2008, PHYSOR 08, 3, 18651870.More infoAbstract: We apply the mathematical procedure of convergence acceleration to the reactor kinetics equation in plane geometry. The featured concept is to take the most fundamental (consistent) finite difference numerical algorithm and show how, by extrapolating a sequence of solutions, a considerably more accurate solution emerges. We demonstrate this new algorithm on the time evolution of a reactor from an initial critical system to another through perturbation of the cross sections. In the demonstration, we consider convergence of the centreline flux at a specific time and convergence of keff.
 Ganapol, B. D., & Nigg, D. W. (2008). Extension of the 1D fourgroup analytic nodal method to full multigroup. International Conference on the Physics of Reactors 2008, PHYSOR 08, 1, 582588.More infoAbstract: In the mid 80's, a fourgroup/tworegion, entirely analytical 1D nodal benchmark appeared. It was readily acknowledged that this special case was as far as one could go in terms of group number and still achieve an analytical solution. In this work, we show that by decomposing the solution to the multigroup diffusion equation into homogeneous and particular solutions, extension to any number of groups is a relatively straightforward exercise using the mathematics of linear algebra.
 Ganapol, B. D., & Townsend, L. W. (2008). A benchmark solution for heavy galactic cosmic ion cascade. International Conference on the Physics of Reactors 2008, PHYSOR 08, 4, 29052913.More infoAbstract: With our everincreasing interest in establishing humankind's presence in space, the protection of personnel from highly energetic radiation in the space environment has become acutely relevant. A significant factor limiting human travel in space is the absorbed dose in tissue resulting from galactic cosmic rays. NASA's new initiatives to colonize the moon and send an expedition to Mars, has greatly increased interest in reliably simulating the cumulative whole body dose received during space missions. Dose simulations require proper verification, which is the primary theme of our presentation. In the following analysis of ion cascades from Galatic Cosmic Rays, we present a mathematically tractable, yet representative analytical benchmark.
 Ganapol, B. D., Gougar, H. D., & Ougouag, A. M. (2008). A 2D benchmark for the verification of the PEBBED code. International Conference on the Physics of Reactors 2008, PHYSOR 08, 2, 10861091.More infoAbstract: A new benchmarking concept is presented for verifying the PEBBED 3D multigroup finite difference/nodal diffusion code with application to pebble bed modular reactors (PBMRs). The key idea is to perform convergence acceleration, also called extrapolation to zero discretization, of a basic finite difference numerical algorithm to give extremely high accuracy. The method is first demonstrated on a 1D cylindrical shell and then on an r,θ wedge where the order of the second order finite difference scheme is confirmed to four places.
 Ganapol, B. D., Yoon, W. Y., & Nigg, D. W. (2008). Verification of the INL/COMBINE7 neutron energy spectrum code. International Conference on the Physics of Reactors 2008, PHYSOR 08, 1, 626634.More infoAbstract: We construct semianalytic benchmarks for the neutron slowing down equations in the thermal, resonance and fast energy regimes through mathematical embedding. The method features a fictitious timedependent slowing down equations solved via Taylor series expansion over discrete "time" intervals. Two classes of benchmarks are considered the first treats methods of solution and the second the multigroup approximation itself. We present several meaningful benchmark methods comparisons with the COMBINE7 energy spectrum code and a simple demonstration of convergence of the multigroup approximation.
 Morris, R. D., Kottas, A., Taddy, M., Furfaro, R., & Ganapol, B. D. (2008). A Statistical Framework for the Sensitivity Analysis of Radiative Transfer Models. IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, 46(12), 40624074.
 Morris, R. D., Kottas, A., Taddy, M., Furfaro, R., & Ganapol, B. D. (2008). A statistical framework for the sensitivity analysis of radiative transfer models. IEEE Transactions on Geoscience and Remote Sensing, 46(12), 40624074.More infoAbstract: Process models are widely used tools, both for studying fundamental processes themselves and as elements of larger system studies. A radiative transfer model (RTM) simulates the interaction of light with a medium. We are interested in RTMs that model light reflected from a vegetated region. Such an RTM takes as input various biospheric and illumination parameters and computes the upwelling radiation at the top of the canopy. The question we address is as follows: Which of the inputs to the RTM has the greatest impact on the computed observation? We study the leaf canopy model (LCM) RTM, which was designed to study the feasibility of observing leaf chemistry remotely. Its inputs are leaf chemistry variables (chlorophyll, water, lignin, and cellulose) and canopy structural parameters (leaf area index, leaf angle distribution, soil reflectance, and sun angle). We present a statistical approach to the sensitivity analysis of RTMs to answer the question previously posed. The focus is on global sensitivity analysis, studying how the RTM output changes as the inputs vary continuously according to a probability distribution over the input space. The influence of each input variable is captured through the "main effects" and "sensitivity indices." Direct computation requires extensive computationally expensive runs of the RTM. We develop a Gaussian process approximation to the RTM output to enable efficient computation. We illustrate how the approach can effectively determine the inputs that are vital for accurate prediction. The methods are applied to the LCM with seven inputs and output obtained at eight wavelengths associated with Moderateresolution Imaging Spectroradiometer bands that are sensitive to vegetation. © 2008 IEEE.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2008). Optimization of the extrapolatediterative method for the multislab transport problem. International Conference on the Physics of Reactors 2008, PHYSOR 08, 1, 651659.More infoAbstract: The transport equation in 1D slab geometry and in the twoangle formulation is analysed in the general case of both anisotropic scattering and a heterogeneous medium. The approach is based on the analytical discrete ordinates technique, which is capable of dealing with highly forwardpeaked scattering. The algorithm is extended to deal with multilayered structures by applying a method derived from the domain decomposition technique, where each single slab is separately solved and the consistency of interface conditions is iteratively sought. To enhance performance, a convergence acceleration of the solution of the linear system is proposed and a study of the dependence of the performance on the adjustable parameters is presented. The optimization of the calculation in terms of computational time and memory requirements is discussed through case studies. The accuracy of the results show great potential as this approach may generate benchmark quality solutions, even for media whose properties change continuously.
 Picca, P., Furfaro, R., Ganapol, B. D., Dulla, S., & Ravetto, P. (2008). Inverse point kinetics with neural networks. Transactions of the American Nuclear Society, 99, 348349.
 Picca, P., Furfaro, R., Kargel, J., & Ganapol, B. D. (2008). Forward and inverse models for photon transport in soilice mixtures and their application to the problem of retrieving optical properties of planetary surfaces. Proceedings of SPIE  The International Society for Optical Engineering, 6960.More infoAbstract: Remote sensing studies are often based on simplified approaches describing the photon transport in absorbing and scattering media. The main purpose of the present paper is to show the potentiality of modeling directly the transport phenomena by mean of linear Boltzmann equation. Some details about the solution method of the integrodifferential equation are reported with a collection of results of relevance in planetary study domain. An inverse approach based on artificial neural network is also proposed to retrieve the optical properties of planetary surfaces and its performances are tested in various cases.
 Picca, P., Furfaro, R., Kargel, J., & Ganapol, B. D. (2008). Forward and inverse models for photon transport in soilice mixtures and their application to the problem of retrieving optical properties of planetary surfaces. SPACE EXPLORATION TECHNOLOGIES, 6960.
 Picca, P., Ganapol, B. D., & Furfaro, R. (2008). Accelerated quasistatic method for neutron kinetics. Transactions of the American Nuclear Society, 99, 335337.
 Picca, P., Ganapol, B. D., & Furfaro, R. (2008). Applications of convergence acceleration to particle transport in spherical geometry. International Conference on the Physics of Reactors 2008, PHYSOR 08, 1, 660668.More infoAbstract: The Boltzmann equation in 1D spherical geometry presents features that are generally encountered in the solution of particle transport in curvilinear geometry. The present paper describes a possible method to obtain highly accurate results by progressively building the solution from lowerorder intermediate discretization schemes. Results prove the efficiency of this approach in terms of reduction of inner iterations and computational time. The concept of convergence acceleration can be suitably introduced into the algorithm to improve its performance and to speed up the inner iteration cycle. Finally, the application of extrapolation to the sequence of partial solutions is applied to seek the meshindependent limiting solution. Several results, for both source and critical problems, are reported and compared to established benchmarks.
 Furfaro, R., & Ganapol, B. D. (2007). Spectral theory for photon transport in dense vegetation media: Caseology for the Canopy equation. Transport Theory and Statistical Physics, 36(13), 107135.More infoAbstract: In this paper, the spectral theory for the transport of photons in dense vegetation structure is developed. The onedimensional, oneangle, timeindependent canopy equation, which models the passive response of green canopies to incident sunlight, is analyzed with spectral methods borrowed from conventional transport theory. The nonrotational invariance characteristic of canopy architectures precludes the use of Legendre polynomials to expand the scattering kernel. Nevertheless, it is shown that Case's method can be applied with proper modification to area canopy equations with a finite rank kernel. "Caseology" is developed to characterize the complete spectrum (discrete and continuous) as well as to determine the eigenfunctions. Orthogonality and fullrange completeness are shown to be inherent properties of the full set of modes. The spectral theory was used to derive the two integral equations that form the backbone of the Fn method. Caseology provides a natural bridge between numerical computation and remote sensing applications that finally justifies the need for such theoretical analysis.
 Furfaro, R., Ganapol, B. D., Johnson, L. F., & Herwitz, S. R. (2007). Neural network algorithm for coffee ripeness evaluation using airborne images. Applied Engineering in Agriculture, 23(3), 379387.More infoAbstract: A NASA unmanned aerial vehicle (UAV) was deployed over a commercial coffee plantation during the 2002 harvest season. An onboard digital camera system collected a set of highresolution surface reflectance images in three spectral bands (580, 660, and 790 nm). An intelligent and robust algorithm operated on the multispectral images to estimate absolute percentages of underripe (green), ripe (yellow), and overripe (brown) coffee cherries displayed on the canopy surface. The procedure was based on a coupled leaf/canopy radiative transfer model (LCM2), modified to include fruiting bodies as photon scattering and absorbing elements. A neural network (NN) set was trained on simulated data, and then used to invert LCM2 for retrieval of fruit and leaf percentages from empirical canopy reflectance data. A projection technique was implemented to systematically mitigate situations where the observed reflectance data fell outside the NN training set domain and the inversion thus initially rendered nonphysical solutions (fruit percentages outside of range 0 to 100%). The algorithm was applied to three study fields representing a broad gradient of mature (ripe plus overripe) fruit ranging from 28% to 61%. Correlation between predictions and yield data across all ripeness levels was 0.78, with a mean absolute error of 11% (range 1% to 26%). By comparison, a standard groundbased harvest readiness assessment produced a correlation 0.64 with yield, mean absolute error of 13% (range 5% to 23%). The procedure was designed to operate on a reasonably modest set of a priori specifications and, by coupling with remote sensing, potentially represents an efficient method for monitoring ripeness progression or other agricultural phenomena that alter visible and nearinfrared crop canopy reflectance. © 2007 American Society of Agricultural and Biological Engineers.
 Ganapol, B. (2007). A further comment on "a resolution of the stiffness problem of reactor kinetics" or "it's all about nothing?". Nuclear Science and Engineering, 157(2), 245247.
 Ganapol, B. D. (2007). A further comment on "A resolution of the stiffness problem of reactor kinetics" or "it's all about nothing?". NUCLEAR SCIENCE AND ENGINEERING, 157(2), 245247.
 Ganapol, B. D. (2007). The analytical solution to the multigroup diffusion equation in onedimensional plane, cylindrical and spherical geometries. Joint International Topical Meeting on Mathematics and Computations and Supercomputing in Nuclear Applications, M and C + SNA 2007.More infoAbstract: By viewing the multigroup diffusion equation in plane, cylindrical and spherical geometries as matrix equations, standard solution techniques for second order ordinary differential equations can be applied to find analytical solutions. By adjusting the boundary conditions appropriately, a solution with the simplicity of the onegroup case in the three geometries is found. Diffusion in cylindrical geometry is used as a demonstration of critical and fixed source problems. No special considerations are required when fission is present or not in any region. The solution is new, and, because of its generality, completely eliminates the need for numerical multigroup solutions of the diffusion equation in heterogeneous plane, spherical and cylindrical geometries.
 Ganapol, B. D., Clarno, K. T., Johnson, S., Oliveira, C. D., Hamilton, S., Ganapol, B. D., & Oliveira, C. D. (2007). An embedded semianalytical benchmark via iterative interpolation for neutron transport methods verification. Joint International Topical Meeting on Mathematics and Computations and Supercomputing in Nuclear Applications, M and C + SNA 2007.More infoAbstract: A new multidimensional semianalytical benchmark capability is developed. The key feature in the solution is the point kernel formulation. The 3D nature of the source is inherited in the flux making this a true multidimensional test. In addition, an efficient numerical scheme, called iterative interpolation, is used to evaluate the required point kernel solution and maintain benchmark accuracy. The EVENT finite element transport algorithm is compared to the point source solution as the first step of embedding the benchmark directly with the EVENT code. Additional code comparisons will be presented.
 Picca, P., Furfaro, R., & Ganapol, B. D. (2007). Extrapolated iterative solution of the transport equation in inhomogeneous media. Transactions of the American Nuclear Society, 97, 627629.
 Picca, P., Ganapol, B. D., & Furfaro, R. (2007). Convergence acceleration for keff using power iteration. Transactions of the American Nuclear Society, 97, 391393.
 Dulla, S., Ganapol, B. D., & Ravetto, P. (2006). Space asymptotic method for the study of neutron propagation. Annals of Nuclear Energy, 33(10), 932940.More infoAbstract: Space asymptotic theory is shown to be a suitable model for the study of pulsed experiments in neutron multiplying systems. After a short revisitation of the basic aspects of space asymptotic theory applied on the Laplace transformed onegroup transport equation, the full solution is derived. It is shown how results are exact in representing localized pulse propagation in the first portion of the transient, until the boundary is reached by the neutron signal, since it propagates with a finite velocity. Approximate models are then derived starting from the exact formulation and the BN method is used to account for anisotropy effects. Numerical results are presented for onedimensional systems, discussing the physical phenomena and noting the distortions introduced by approximate models, which may then turn out to be inadequate for the simulation of realistic pulsed experiments situations. © 2006 Elsevier Ltd. All rights reserved.
 Ganapol, B. D. (2006). A one grouponedimensional transport benchmark in cylindrical geometry. Transactions of the American Nuclear Society, 94, 525527.
 Ganapol, B. D. (2006). Mining the discrete velocity method for high quality solutions for onedimensional Poiseuille flow. Zeitschrift fur Angewandte Mathematik und Physik, 57(6), 10111024.More infoAbstract: A fully discretized solution for Poiseuille flow in a onedimensional channel is presented. Unlike previous semianalytical methods, such as the Analytical DiscreteOrdinates (ADO) or the FN methods, which have been specifically designed to avoid spatial discretization error, no analytical advantage is assumed. Instead, the solution is "mined" in a process where each discrete approximation is an element in a sequence of solutions whose convergence to the solution is accelerated. This process leads most straightforwardly to high quality benchmark results for use in algorithm verification with a minimum of theoretical and numerical complexity. © Birkhäuser Verlag, Basel 2006.
 Ganapol, B. D. (2006). Thermal neutron diffusion in an array of plates. Nuclear Science and Engineering, 152(3), 284291.More infoAbstract: The exact solution to the diffusion equation for thermal neutrons emitted into an infinite medium containing absorbing and scattering plates is obtained. A fully explicit analytical representation is found through the solution of the resulting recurrence relation for the interfacial fluxes. Departure from Galanin's theory, which assumes infinitesimally thin absorbing plates, is demonstrated with increasing absorber thickness and decreasing diffusion coefficient.
 Ganapol, B. D. (2006). Vegetation canopy reflectance modeling with turbid medium radiative transfer. Lecture Notes in Computational Science and Engineering, 48, 173210.More infoAbstract: Biophysical considerations for vegetation canopy reflectance modeling are presented. Included is a brief overview outlining strengths and weaknesses of four possible canopy reflectance models. The overview is followed by the description of the LCM2 coupled leaf/canopy turbid medium reflectance model based on natural averaging. The model follows conventional radiative transfer theory with modification for canopy architecture as characterized by leaf orientation. The presentation concludes with a demonstration of LCM2 in a multiple pixel mode to estimate the amount of ripe coffee cherries at harvest in the fields of the Kauai Coffee Company and to detect targets hidden beneath canopies.
 Ganapol, B. D., & Ougouag, A. M. (2006). The boundary element formulation of the one group 1D nodal equations. Transactions of the American Nuclear Society, 94, 521524.
 Furfaro, R., & Ganapol, B. (2005). Probabilistic and generalized regression neural networks for nonmultiplying material identification. Transactions of the American Nuclear Society, 93, 514516.
 Furfaro, R., Ganapol, B. D., Johnson, L. F., & Herwitz, S. (2005). Modelbased neural network algorithm for coffee ripeness prediction using helios UAV aerial images. Proceedings of SPIE  The International Society for Optical Engineering, 5976.More infoAbstract: Over the past few years, NASA has had a great interest in exploring the feasibility of using Unmanned Aerial Vehicles (UAVs), equipped with multispectral imaging systems, as longduration platform for crop monitoring. To address the problem of predicting the ripeness level of the Kauai coffee plantation field using UAV aerial images, we proposed a neural network algorithm based on a nested LeafCanopy radiative transport Model (LCM2). A modelbased, multilayer neural network using backpropagation has been designed and trained to learn the functional relationship between the airborne reflectance and the percentage of ripe, overripe and underripe cherries present in the field. LCM2 was used to generate samples of the desired map. Postprocessing analysis and tests on synthetic coffee field data showed that the network has accurately leam the map. A new Domain Projection Technique (DPT) was developed to deal with situations where the measured reflectance fell outside the training set. DPT projected the reflectance into the domain forcing the network to provide a physical solution. Tests were conducted to estimate the error bound. The synergistic combination of neural network algorithms and DPT lays at the core of a more complex algorithm designed to process UAV images. The application of the algorithm to real airborne images shows predictions consistent with postharvesting data and highlights the potential of the overall methodology.
 Ganapol, B. D. (2005). A new formulation of criticality for multigroup diffusion theory in heterogeneous plane geometry. Transactions of the American Nuclear Society, 92, 645647.
 Ganapol, B. D. (2005). Onegroup reactor kinetics in cylindrical geometry?. Transactions of the American Nuclear Society, 92, 564567.
 Ganapol, B. D. (2005). Progress report on the development of time dependent neutral particle transport benchmarks in two and three dimensions. Transactions of the American Nuclear Society, 93, 458460.
 Wei, Z., & Ganapol, B. D. (2005). Reliability quantification of electronics in the space environment. American Nuclear Society Embedded Topical Meeting  2005 Space Nuclear Conference, 626634.More infoAbstract: Space environment naturally can damage electronics through two failure mechanisms, totalionizingdose and singleevent effects (SEE). In this presentation, a reliability quantification method is proposed by considering both failure mechanisms, The reliability of an electronic device due to the first failure mechanism is defined as the probability of the cumulative effect of the particles greater than an allowable threshold. The reliability due to the second failure mechanisms is defined as the probability of a highenergy particle impacting the device and causing the hard failure of the electronics. The reliability of the entire device is obtained by using Baye's theory. A cascade model for the cumulative deposited energy of Galactic ions is presented. Finally, a comprehensive example is given to illustrate the usage of the proposed method.
 Ganapol, B. D. (2004). FN approximation of the solution to a singular integral equation of classical reactor physics. Annals of Nuclear Energy, 31(17), 20172024.More infoAbstract: The iterated FN method is applied to a singular integral equation arising from a classical problem of reactor physics to determine the distribution of fissile material giving a spatially uniform flux. The FN iterations are accelerated toward convergence through the Wynnε algorithm  but first  Happy Birthday "Fast Eddie" Larsen Why do I refer to the well known, most proper and exquisitely accomplished Edward W. Larsen as "Fast Eddie". Well our story begins in a small back bar room in the lobby of one of Los Alamos' finest and most luxurious hotels. Two young men were having a transport theoretic discussion while they were engaged in a serious game of pool with monetary benefits going to the winner. In addition, the two were sipping their most favorite lavation in rather large quantities  one, a short stocky man with thinning hair, was sipping to forget the cost of his recent divorce, and the other, a shorter stockier man also with thinning hair, was drinking, well because he liked to drink and it just made him silly. As they continued their transport discussion, one stocky man turned to the other and said, "I wonder what "Fast Eddie" Larsen would say to our transport question". The other stocky man immediately thought the "Fast Eddie" reference was to Paul Newman who played "Fast Eddie", an expert at applied particle transport theory (a pool player) in the movie the Hustler and asked if indeed this was the case. The first stocky man said "No. I call everyone with the name Ed 'Fast Eddie' "  and that's the story of how "Fast Eddie" Larsen got his name. Happy 60th Ed and thanks for all the great transport theory  from one of your biggest fans. © 2004 Elsevier Ltd. All rights reserved.
 Ganapol, B. D. (2004). Yet another variation of the 1D discrete ordinates algorithm of neutron transport. Transactions of the American Nuclear Society, 91, 154157.
 Ganapol, B. D. (2003). A semianalytical benchmark for 2d time dependent neutron transport. Transactions of the American Nuclear Society, 88, 324.
 Ganapol, B. D. (2003). Fourier Transform Transport Solutions in Spherical Geometry. Transport Theory and Statistical Physics, 32(57), 587605.More infoAbstract: New transport solutions in infinite spherical geometry related to solutions in plane geometry are obtained through a Fourier transform analysis. The analysis leads to the true Green's function for spherical symmetry which is used in combination with Placzek's lemma to treat a finite spherical shell.
 Ganapol, B. D. (2003). Neutron Transport in an Infinite Medium with Forward/Backward/Anisotropic Scattering. Transactions of the American Nuclear Society, 89, 500.
 Parsons, D. K., & Ganapol, B. D. (2003). Adaptive Binning for Extreme Scattering in Monte Carlo Calculations. Transactions of the American Nuclear Society, 89, 367.
 Radtke, G., Ortega, A., & Ganapol, B. (2003). Measurement of the unsteady thermal Green's function in a boundary layer flow: A preliminary theory. American Society of Mechanical Engineers, Heat Transfer Division, (Publication) HTD, 374(4), 389396.More infoAbstract: Compared to the heat transfer coefficient, the thermal Green's function concept is a more fundamental method of describing the relationship between local wall heat transfer and wall temperature. It is far more amenable to situations involving strong spatial, temporal, and boundary condition variability. The utility of this methodology has been established, in particular for the analysis of the conjugate heat transfer problem. A necessary element in this technique is an inverse theoretical model to infer the Green's function from laboratory thermal response data. This paper presents preliminary results from a first attempt to develop such a measurement theory to extract local approximations to the unsteady thermal Green's function (UTGF) in 3D boundary layer flows. The flow model used is a linear shear flow, which is a solution valid in the near wall region of laminar flows, as well as the viscous sublayer region of a turbulent boundary layer. This model is governed by the shear velocity, which is a measure of the local wall shear stress. The solution methodology employs the mathematical theory of Green's function solutions to the energy equation for a general 3D boundary layer flow, where the UTGF of interest is the thermal response to an impulsive heat load. Analytic methods are used to condense the equation from a 3D to a 2D transient PDE, and the reduced equation is solved using a PetrovGalerkin Finite Element Method. These data are used to construct a numerical UTGF uniquely determined by the shear velocity, flow angle, and the thermodynamic properties of the fluid. An error minimization scheme is proposed to find the appropriate value of the shear velocity, thus providing local UTGF, and shear velocity measurements.
 Ganapol, B. D., Picard, R. H., Winick, J. R., Wintersteiner, P. P., & Woolf, S. (2002). Incorporation of azimuthal dependence into the LCM2 coupled leaf/canopy reflectance model. Proceedings of SPIE  The International Society for Optical Engineering, 4542, 214222.More infoAbstract: Recently, a coupled leaf/canopy radiative transfer model, called LCM2, has been developed with NASA and AFRL (Air Force Research Laboratory) funding to investigate the feasibility of detecting spectral signatures from within and under a vegetation canopy. The model is unique in that it features a direct radiative transfer coupling between the leaf scattering elements and the canopy reflectance. The coupling is through the withinleaf radiative transfer model LEAFMOD. The leaf radiative transfer model characterizes photon scattering within a participating homogeneous leaf taking into account both biochemical composition and leaf thickness. The leaf model is incorporated as the scattering element in a canopy reflectance model (CANMOD) assuming a biLambertian leaf scattering phase function. The primary distinction between conventional and canopy radiative transfer is canopy architecture which is introduced through leaf angle distributions. A Lambertian partially reflecting background is assumed to lie beneath the canopy. The influences of biochemical composition, average leaf thickness, reflecting background and canopy architecture on canopy reflectance can therefore be investigated. The model is extended here to include azimuthal dependence by considering the uncollided and first collided radiances separately. The LCM2 model with the new azimuthal correction will be coupled to an atmospheric radiative transfer code MODTRAN4 to simulate a satellite imager response at orbital altitude.
 Ganapol, B. D. (2001). From Fourier transforms to singular eigenfunctions for multigroup transport. Nuclear Science and Engineering, 137(3), 400410.More infoAbstract: A new Fourier transform approach to the solution of the multigroup transport equation with anisotropic scattering and isotropic source is presented. Through routine analytical continuation, the inversion contour is shifted from the real line to produce contributions from the poles and cuts in the complex plane. The integrand along the branch cut is then recast in terms of matrix continuum singular eigenfunctions, demonstrating equivalence of Fourier transform inversion and the singular eigenfunction expansion. The significance of this paper is that it represents the initial step in revealing the intimate connection between the Fourier transform and singular eigenfunction approaches as well as serves as a basis for a numerical algorithm.
 Ganapol, B. D. (2000). A consistent theory of neutral particle transport in an infinite medium. Transport Theory and Statistical Physics, 29(12), 4368.More infoAbstract: The solution to the ageold infinite medium Green's function is revisited using the hindsight of the past coupled with the wisdom of the present. A new solution is found based on a blend of the theories developed by F. Vanmassenhove and C. Grosjean, E. Inonu, K. Case and J. Mika. The solution is obtained from wellknown recurrence relations and is shown to reduce to the singular eigenfunction representation of Case. Closure is shown to be a consequence of the infinite medium Green's function.
 Yoshioka, H., Miura, T., Huete, A. R., & Ganapol, B. D. (2000). Analysis of vegetation isolines in redNIR reflectance space. Remote Sensing of Environment, 74(2), 313326.More infoAbstract: The characteristic behavior of rednearinfrared (NIR) reflectancebased vegetation isolines were analyzed by focusing on its three features: the slope, NIRintercept, and the intersection between the vegetation isoline and the soil line. These properties are the key factors in understanding variations of vegetation index values with changes of canopy background brightness, known as background noise. The analysis was conducted based on a vegetation isoline equation derived by using the representation of canopy reflectance by the adding method. The isoline parameters, slopes, and NIRintercepts of vegetation isolines were numerically obtained by the SAIL canopy model. Some of the known behaviors of the vegetation isoline were simulated and analyzed in detail. (C) Elsevier Science Inc., 2000.
 Ganapol, B. D., Johnson, L. F., Hlavka, C. A., Peterson, D. L., & Bond, B. (1999). LCM2: A coupled leaf/canopy radiative transfer model. Remote Sensing of Environment, 70(2), 153166.More infoAbstract: Two radiative transfer models have been coupled to generate vegetation canopy reflectance as a function of leaf chemistry, leaf morphology (as represented by leaf scattering properties), leaf thickness, soil reflectance, and canopy architecture. A model of radiative transfer within a leaf, called LEAFMOD, treats the radiative transfer equation for a slab of optically uniform leaf material, providing an estimate of leaf hemispherical reflectance and transmittance as well as the radiance exiting the leaf surfaces. The canopy model then simulates radiative transfer within a mixture of leaves, with each having uniform optical properties as determined by LEAFMOD, assuming a biLambertian leaf scattering phase function. The utility of the model, called LCM2 (Leaf/Canopy Model version 2), is demonstrated through predictions of radiometric measurements of canopy reflectance and sensitivity to leaf chlorophyll and moisture content.
 Ganapol, B. D., Johnson, L. F., Hammer, P. D., Hlavka, C. A., & Peterson, D. L. (1998). LEAFMOD: A new withinleaf radiative transfer model. Remote Sensing of Environment, 63(2), 182193.More infoAbstract: We describe the construction and verification of a withinleaf radiative transfer model called LEAFMOD (Leaf Experimental Absorptivity Feasibility MODel). In the model, the onedimensional radiative transfer equation in a slab of leaf material with homogeneous optical properties is solved. When run in the forward mode, LEAFMOD generates an estimate of leaf reflectance and transmittance given the leaf thickness and optical characteristics of the leaf material (i.e., the absorption and scattering coefficients). In the inverse mode, LEAFMOD computes the total withinleaf absorption and scattering coefficient profiles from measured reflectance, transmittance, and leaf thickness. Inversions with simulated data demonstrate that the model appropriately decouples scattering and absorption within the leaf, producing fresh leaf absorption profiles with peaks at locations corresponding to the major absorption features for water and chlorophyll. Experiments with empirical input data demonstrate that the amplitude of the fresh leaf absorption coefficient profile in the visible wavebands is correlated with pigment concentrations as determined by wet chemical analyses, and that absorption features in the nearinfrared wavebands related to various other biochemical constituents can be identified in a dryleaf absorption profile.
 Ganapol, B. D., Pomraning, G. C., & Zweifel, P. F. (1998). Fifteenth international conference on transport theory: Göteborg, Sweden, June 17, 1997. Transport Theory and Statistical Physics, 27(2), 177197.
 Pomraning, G. C., & Ganapol, B. D. (1998). Simplified radiative transfer for combined Rayleigh and isotropic scattering. Astrophysical Journal Letters, 498(2 PART I), 671688.More infoAbstract: We consider the vector equation of radiative transfer describing the Stokes parameters of light, with a scattering law corresponding to a combination of Rayleigh and isotropic scattering. We show that this vector description can be reduced to a renormalized scalar equation of transfer for the intensity I in the asymptotic limit of either nearthermodynamic equilibrium or that corresponding to a sourcefree, weakly absorbing system. A simple quadrature result is also obtained for the state of polarization of the light. We apply this analysis to the classic diffuse reflection problem, and numerical results indicate an improvement in accuracy over the usual scalar equations of transfer predictions for this scattering law. As part of our solution methodology for the diffuse reflection problem, we introduce a new analytic technique based upon the source function in the equation of transfer. © 1998. The American Astronomical Society. All rights reserved.
 Ganapol, B. D., & McCormick, N. J. (1997). Radiative transfer halfspace problems by the source function method. Transport Theory and Statistical Physics, 26(45), 507532.More infoAbstract: Numerical solutions for the classical, monoenergetic albedo and Milne problems for a halfspace are obtained from a numerical Laplace transform inversion and the nonlinear integrodifferential equation for the source function. Anisotropic scattering and an azimuthallydependent incident illumination are treated.
 Hlavka, C. A., Peterson, D. L., Johnson, L. F., & Ganapol, B. (1997). Analysis of forest foliage spectra using a multivariate mixture model. Journal of Near Infrared Spectroscopy, 5(3), 167173.More infoAbstract: Wet chemical measurements and near infrared spectra of dry ground leaf samples were analysed to test a multivariate regression technique for estimating component spectra. The technique is based on a linear mixture model for log(1/R) pseudoabsorbance derived from diffuse reflectance measurements. The resulting unmixed spectra for carbohydrates, lignin and protein resemble the spectra of extracted plant carbohydrates, lignin and protein. The unmixed protein spectrum has prominent absorption peaks at wavelengths that have been associated with nitrogen bonds. It therefore appears feasible to incorporate the linear mixture model in whole leaf models of photon absorption and scattering so that effects of varying nitrogen and carbon concentration on leaf reflectance may be simulated.
 Kornreich, D. E., & Ganapol, B. D. (1997). Numerical evaluation of the threedimensional searchlight problem in a halfspace. Nuclear Science and Engineering, 127(3), 317337.More infoAbstract: The linear Boltzmann equation for the transport of neutral particles is investigated with the objective of generating a benchmarkquality calculation for the threedimensional searchlight problem in a semiinfinite medium. The derivation assumes stationarity, one energy group, and Isotropic scattering. The scalar flux (both surface and interior) and the current at the surface are the quantities of interest. The source considered is a pencilbeam incident at a point on the surface of a semiinfinite medium. The scalar flux will have twodimensional variation only if the beam is normal; otherwise, it is threedimensional. The solutions are obtained by using Fourier and Laplace transform methods. The transformed transport equation is formulated so that it can be related to a onedimensional pseudo problem, thus providing some analytical leverage for the inversions. The numerical inversions use standard numerical techniques such as GaussLegendre quadrature, summation of infinite series, Hfunction iteration and evaluation, and EulerKnopp acceleration. The numerical evaluations of the scalar flux and current at the surface are relatively simple, and the interior scalar flux is relatively difficult to calculate because of the embedded twodimensional Fourier transform inversion, Laplace transform inversion, and Hfunction evaluation. Comparisons of these numerical solutions to results from the MCNP probabilistic code and the THREEDANT discrete ordinates code are provided and help confirm proper operation of the analytical code.
 Kornreich, D. E., & Ganapol, B. D. (1997). The Green's function method for nuclear engineering applications. Nuclear Science and Engineering, 126(3), 293313.More infoAbstract: The Green's function method (GFM) is employed to obtain scalar and angular flux distributions in heterogeneous slab geometry with isotropic scattering. All solutions utilize the infinitemedium Green's function to obtain results infinite media. Past Green's function analyses that do not resort to expansions of the angular flux in basis functions have been performed for nonmultiplying media only; in this paper, results are provided, for the first time, for both multiplying and nonmultiplying media using the GFM. Several source configurations are considered, including a beam source on the leftmost face, isotropic incidence on any face, and constant inhomogeneous volume sources in internal materials. Scalar and angular flux distributions compare favorably with those obtained using the FN method as well as the ONEDANT discrete ordinates code. In addition, the single and heterogeneous critical slab problems are investigated and solved using the GFM.
 Ganapol, B. D. (1996). Solution to the linear BGK equation via the principles of invariance. Journal of Physics A: Mathematical and General, 29(6), 12271244.More infoAbstract: The solution to the linearized BGK equation of kinetic theory is obtained via the principles of invariance as formulated by Chandrasekhar. The solution represents an alternative to the singular eigenfunction method featuring regular integral equations which are more easily evaluated. Laplace transforms are obtained for the net flow velocity and velocity distribution for both the albedo and Milne problems with wall reflection. The transforms are then numerically inverted to a high degree of accuracy. Results are compared to those of Kainz and Titulaer. © 1996 IOP Publishing Ltd.
 Ganapol, B. D., & Kornreich, D. E. (1996). The Green's function method for the monoenergetic transport equation with forward/backward/isotropic scattering. Annals of Nuclear Energy, 23(45), 301320.More infoAbstract: The monoenergetic transport equation for slab geometry with isotropic/forward/backward scattering is solved using the Green's function method. The general formulation is found in terms of the infinite medium Green's function for isotropic scattering only. Results are presented for single and multiple slab configurations and include results from a model of the photon transport in a heterogeneous leaf.
 Ganapol, B. D., & Pomraning, G. C. (1996). The tworegion Milne problem. Nuclear Science and Engineering, 123(1), 110120.More infoAbstract: We consider the tworegion Milne problem, defined as the steadystate monoenergetic linear transport problem for two adjoining homogeneous sourcefree halfspaces, with a particle source coming from infinity in one of the halfspaces. We demonstrate that the asymptotic (Case discrete mode) component of the solution for the scalar flux is easily and explicitly written in terms of Chandrasekhar's Hfunction for each medium. This asymptotic solution is shown to exhibit a discontinuity in both the scalar flux and current at the interface between the two halfspaces. Numerical benchmark results for the linear extrapolation distance and the discontinuities are given for various combinations of the mean number of secondaries (c) characterizing the two media. Contact is also made with a variational treatment. In particular, the variational formalism is shown to predict the linear extrapolation distance and these asymptotic discontinuities correct to first order in the difference between the values of c characterizing the two halfspaces.
 Kornreich, D. E., & Ganapol, B. D. (1996). The suite of analytical benchmarks for neutral particle transport in infinite isotropically scattering media. Nuclear Science and Engineering, 125(1), 2450.More infoAbstract: The linear Boltzmann equation for the transport of neutral particles is investigated with the objective of generating benchmarkquality evaluations of solutions for homogeneous infinite media. In all cases, the problems are stationary, of one energy group, and the scattering is isotropic. The solutions are generally obtained through the use of Fourier transform methods with the numerical inversions constructed from standard numerical techniques such as GaussLegendre quadrature, summation of infinite series, and convergence acceleration. Consideration of the suite of benchmarks in infinite homogeneous media begins with the standard onedimensional problems: an isotropic point source, an isotropic planar source, and an isotropic infinite line source. The physical and mathematical relationships between these source configurations are investigated. The progression of complexity then leads to multidimensional problems with source configurations that also emit particles isotropically: the finite line source, the disk source, and the rectangular source. The scalar flux from the finite isotropic line and disk sources will have a twodimensional spatial variation, whereas a finite rectangular source will have a threedimensional variation in the scalar flux. Next, sources emitting particles anisotropically are considered. The most basic such source is the point beam giving rise to the Green's function, which is physically the most fundamental transport problem, yet may be constructed from the isotropic point source solution. Finally, the anisotropic plane and anisotropically emitting infinite line sources are considered. Thus, a firm theoretical and numerical base is established for the most fundamental neutral particle benchmarks in infinite homogeneous media.
 Caraffini, G. L., Ganapol, B. D., & Spiga, G. (1995). A multigroup approach to the nonlinear extended boltzmann equation. Il Nuovo Cimento D, 17(2), 129142.More infoAbstract: Following a procedure which is typical of linear (neutron) transport theory, a multigroup approach is proposed for the nonlinear extended Boltzmann equation in the presence of removal, a background medium, an external source and an external force field. The relevant multigroup equations, corresponding to a discretization of the speed variable only, are derived and discussed, especially in connection with the socalled semidiscrete models recently introduced in kinetic theory. © 1995 Società Italiana di Fisica.
 Ganapol, B. D. (1995). Radiative transfer in a semiinfinite medium with a specularly reflecting boundary. Journal of Quantitative Spectroscopy and Radiative Transfer, 53(3), 257267.More infoAbstract: A consistent solution of the radiative transfer equation characterizing photon transport in a semiinfinite medium of refractive index greater than or equal to one is obtained following the method of Sobolev. Fresnel specular reflection, Snell's law and isotropic scattering are assumed. An algorithm is developed and its accuracy is demonstrated. A numerical Laplace transform inversion leads to an efficient evaluation for the interior flux and source function distributions. © 1995.
 Ganapol, B. D. (1995). The Milne problem for a specularly reflecting boundary. Journal of Quantitative Spectroscopy and Radiative Transfer, 54(3), 495508.More infoAbstract: The standard Milne problem, serving as an analytical benchmark for almost 50 years, is reconsidered in light of the numerical Laplace transform inversion. The application of the numerical inversion greatly reduces the effort in generating highly accurate numerical evaluations of solutions to particular transport problems. Le Caine's results for the integrated intensity are shown to Contain only 3 or 4 correct digits. The Milne problem has also been generalized to include a specularly reflecting boundary. © 1995.
 Ganapol, B., Johnson, L., Hammer, P., Hlavka, C., Peterson, D., & Baret, F. (1995). New withinleaf radiative transfer model: preliminary results. International Geoscience and Remote Sensing Symposium (IGARSS), 2, 14571459.More infoAbstract: As part of the NASA Ecological Processes and Modeling Program, researchers at NASA/Ames and INRA in the LEAFMOD group are developing a new withinleaf radiative transfer model The model is based on a radiative transfer formulation and on the assumption of the separation of the origins of the scattering and absorption transport effects. Preliminary results are presented.
 Pomraning, G. C., & Ganapol, B. D. (1995). Asymptotically consistent reflection boundary conditions for diffusion theory. Annals of Nuclear Energy, 22(12), 787817.More infoAbstract: It is well known that the classic diffusion equation is an asymptotic limit of the onespeed linear transport equation. In this paper, we carry out the analysis required to obtain an asymptotically consistent boundary condition for this diffusion equation in the case of partial surface reflection. The specification of this boundary condition requires the transport theory solution to a purely scattering halfspace problem. A variational treatment is used to obtain explicit results for this halfspace problem in the general case. In the special case of purely diffuse reflection, simple considerations give exact halfspace results, and a more involved exact halfspace analysis is presented in the case of specular reflection. These exact results are used to assess the accuracy of the variational treatment. © 1995.
 Ganapol, B. D. (1994). Reflection for dense plant canopies from the oneangle radiative transfer equation. International Geoscience and Remote Sensing Symposium (IGARSS), 3, 16291630.More infoAbstract: The solution to the radiative transfer equation is complicated by the complexity of the photon scattering interactions. The paper presents the solution to the one angle radiative transfer equation for a dense canopy, with the leaves assumed to scatter as Lambertian surfaces, is solved using a technique applied to the conventional radiative transfer equations.
 Ganapol, B. D., Kornreich, D. E., Dahl, J. A., Nigg, D. W., Jahshan, S. N., & Wemple, C. A. (1994). Searchlight problem for neutrons in a semiinfinite medium. Nuclear Science and Engineering, 118(1), 3853.More infoAbstract: The solution to the searchlight problem for monoenergetic neutrons in a semiinfinite medium with isotropic scattering illuminated at the free surface is obtained through the numerical evaluation of an analytical expression for the scalar flux at various positions within the medium. The sources considered are normally incident pencil beam and isotropic point sources as well as a longitudinal uniformly distributed source. The analytic solution is effected by a recently developed numerical inversion technique applied to the FourierBessel transform. The transform inversion results from the solution method of Rybicki, where the twodimensional problem is solved by casting it as a variant of a onedimensional problem. The numerical inversion results in a highly accurate solution. Comparisons of the analytic solution with results from Monte Carlo (MCNP) and discrete ordinates transport codes (DORT, TWODANT, and SMARTEPANTS) show excellent agreement. These comparisons, which are free from any associated data or crosssection set dependencies, provide significant evidence of the proper operation of the transport codes tested.
 Ganapol, B. D. (1993). Equivalence of the single and doubleintegral formulations of the searchlight problem. Journal of Quantitative Spectroscopy and Radiative Transfer, 50(5), 551554.More infoAbstract: The equivalence of two integral equations associated with the solution of the searchlight problem in a halfspace is shown by application of contour integration. © 1993.
 Ganapol, B. D., Kelley, C. T., & Pomraning, G. C. (1993). Asymptotically exact boundary conditions for the P_{N} equations. Nuclear Science and Engineering, 114(1), 1219.More infoAbstract: It was recently demonstrated that in planar geometry, the classic PN equations are an asymptotic limit of the transport equation. A corresponding boundary layer analysis established the asymptotically consistent boundary conditions. These boundary conditions were evaluated variationally, and it was conjectured that these variational approximations are quite accurate for all values of N. Here, we evaluate these boundary conditions exactly (numerically) and show that the previous variational results are indeed accurate to a few percent. The exact results were computed using numerical methods previously developed for solving Chandrasekhar's H equations.
 Ganapol, B. (1992). Radiative transfer in a onedimensional single angle plant canopy. Nuclear Science and Engineering, 112(3), 270282.More infoAbstract: The radiative transfer equation for photons interacting with the phytoelements (primarily leaves) of a plant canopy of finite height is solved by application of Siewert's FN numerical algorithm. A onedimensional, oneangle transport model is assumed with the Lambertian scattering leaves all oriented in the same direction. In addition, a Lambertian reflecting soil is assumed at the lower canopy boundary. Since the focus of this work is on the development of the FN algorithm, emphasis is given to the derivation of the algorithm and the algorithmic accuracy; however, a comparison with a field experiment is also presented to indicate the potential usefulness of the FN solution.
 Ganapol, B. D. (1992). Distributed neutron sources in a semiinfinite medium. Nuclear Science and Engineering, 110(3), 275281.More infoAbstract: A novel application of the numerical Laplace transform inversion to the solution of the onegroup neutron transport equation in a halfspace with a distributed source is presented. Embedded isotropic sources are considered as well as distributed beam sources. The simplicity of the method in generating numerical results is demonstrated.
 Ganapol, B. D., & Myneni, R. B. (1992). The application of the principles of invariance to the radiative transfer equation in plant canopies. Journal of Quantitative Spectroscopy and Radiative Transfer, 48(3), 321339.More infoAbstract: Solutions of the radiative transfer equation describing photon interactions with vegetation canopies are important in remote sensing since they provide the canopy reflectance distribution required in the interpretation of satellite acquired information. The general onedimensional twoangle transport problem for a finite copy of arbitrary leaf angle distribution is considered. Analytical solutions are obtained in terms of generalized Chandrasekhar's X and Yfunctions by invoking the principles of invariance. A critical step in the formulation involves the decomposition of the integral of the scattering phase function into a product of known functions of the incident and scattered photon directions. Several simplified cases previously considered in the literature are derived from the generalized solution. Various symmetries obeyed by the scattering operator and reciprocity relations are formally proved. © 1992.
 Ganapol, B., & Myneni, R. (1992). The F_{n} method for the oneangle radiative transfer equation applied to plant canopies. Remote Sensing of Environment, 39(3), 213231.More infoAbstract: Solutions of the radiative transfer equation describing photon interactions with vegetation canopies are important in remote sensing since they provide the canopy reflectance distribution required in the interpretation of satelliteacquired information. Although the most widely used transport models such as the moments formulations collapse the photon directional information into a limited number of directions (usually two or four), the more sophisticated methods such as discrete ordinates can, in principle, employ an unlimited number of discrete directions. These discrete methods, however, also contain spatial discretization error and lack testing against more accurate Numerical formulations for specific vegetation canopy scattering kernels. In this article, we consider a semianalytical approach to the solution of the oneangle radiative transfer equation in slab geometry called the Fn method. This method has a its basis two integral equations specifying the intensities exiting the vegetation canopy boundaries. The solution is then obtained through an expansion in a set of basis functions with expansion coefficients to be determined. These coefficients are obtained from a collocation procedure resulting in a set of algebraic equations solved by matrix inversion. The advantage of this method is that only discretization in the angular variable is required, thus avoiding spatial truncation error entirely. The paper begins by considering a canopy where all the leaves are oriented at the same angle. Lambertian scattering will be assumed with unequal leaf reflectance and transmittance. This simple model contains all the difficulties of the more complex model incorporating a general leaf angle distribution which is to be considered in the latter part of the presentation. A sensitivity analysis is performed including variation of the numerical and model parameters. In addition, discrete ordinates calculations as well as field measurements are compared to the Fn results. © 1992.
 Myneni, R., Ganapol, B., & Asrar, G. (1992). Remote sensing of vegetation canopy photosynthetic and stomatal conductance efficiencies. Remote Sensing of Environment, 42(3), 217238.More infoAbstract: The problem of remote sensing the canopy photosynthetic and stomatal conductance efficiencies is investigated with the aid of one and threedimensional radiative transfer methods coupled to a semiempirical mechanistic model of leaf photosynthesis and stomatal conductance. Desertlike vegetation is modeled as clumps of leaves randomly distributed on a bright dry soil with partial ground cover. Normalized difference vegetation index (NDVI), canopy photosynthetic (Ep), and stomatal efficiencies (Es) are calculated for various geometrical, optical, and illumination conditions. A base case is defined to investigate the dynamics of Ep and Es with respect to ground cover, clump leaf area index, soil reflectance, and atmospheric conditions. The contribution of various radiative fluxes to estimates of Ep is evaluated and the magnitude of errors in bulk canopy formulation of problem parameters are quantifieid. The nature and sensitivity of the relationship between Ep and Es to NDVI is investigated and an algorithim is proposed for use in operational remote sensing. © 1992.
 Myneni, R. B., & Ganapol, B. D. (1991). A simplified formulation of photon transport in leaf canopies with scatterers of finite dimensions. Journal of Quantitative Spectroscopy and Radiative Transfer, 46(3), 135140.More infoAbstract: A formalism for photon transport in leaf canopies with finitedimensional scattering centers that mutually crossshade was recently proposed resulting in an equation set which is rather complicated. In this paper, we propose a simplification of the correction factors that describe the correlation in photon fates. These simplified correction factors permit the derivation of a modified integrodifferential transfer equation with two additional terms that arise from considerations of finite size scatterers. A numerical procedure for the solution of this transfer equation in the framework of a successive collisions approach is also presented. © 1991.
 Vanderbilt, V. C., Ustin, S. L., Berger, K. M., Caldwell, W. F., Clark, J. A., Ganapol, B. D., Kasischke, E. S., Martens, S. N., Pettigrew, R. E., & Rousseau, R. A. (1991). Biophysical information in asymmetric and symmetric diurnal bidirectional canopy reflectance. IEEE Transactions on Geoscience and Remote Sensing, 29(6), 875889.More infoAbstract: The authors present a theory for partitioning the information content in diurnal bidirectional reflectance measurements in order to detect differences potentially related to biophysical variables. The theory, which divides the canopy reflectance into asymmetric and symmetric functions of solar azimuth angle, attributes asymmetric variation to diurnal changes in the canopy biophysical properties. The symmetric function is attributed to the effects of sunlight interacting with a hypothetical average canopy which would display the average diurnal properties of the actual canopy. The authors analyzed radiometer data collected diurnally in the Thematic Mapper wavelength bands from two walnut canopies that received differing irrigation treatments. The reflectance of the canopies varied with sun and view angles and across seven bands in the visible, nearinfrared, and middle infrared wavelength regions. Although one of the canopies was permanently water stressed and the other was stressed in midafternoon each day, no water stress signature was unambiguously evident in the reflectance data.
 Ganapol, B. (1990). Numerical treatment of the halfspace problem of radiative transfer. Journal of Quantitative Spectroscopy and Radiative Transfer, 44(2), 289297.More infoAbstract: The fundamental halfspace problem of radiative transfer is reconsidered in the light of several recent articles appearing in this journal. A realistic computational method making use of an offtheshelf numerical Laplacetransform inversion is presented and demonstrated. © 1990.
 Ganapol, B. D. (1990). The determination of the reflected intensity for a single leaf angle plant canopy via Chandrasekhar's method. Digest  International Geoscience and Remote Sensing Symposium (IGARSS), 2, 13771380.More infoAbstract: A solution based on Chandrasekhar's method is applied to the oneangle photon transport equation in a 1D finite plant canopy where all leaves are oriented at the same inclination. By superposition, the turbid medium transport equation with a beam source and a reflecting soil condition is decomposed into two equationsone for the beam source with a vacuum condition at the soil boundary and the other with no source and the original reflecting soil condition. The solution to the second equation is obtained using the solution to the first, yielding an analytical expression for the reflected intensity in terms of generalized Chandrasekhar X and Yfunctions. The FN method is then applied to obtain a numerical evaluation of the reflected intensity. Results are presented in order to explain the canopy response in the context its architectural properties.
 Ganapol, B. D., Ravetto, P., & Sumini, M. (1990). Application of a numerical Laplace transform inversion technique to a problem in reactor dynamics. Annals of Nuclear Energy, 17(1), 4957.More infoAbstract: A newly developed numerical technique for the Laplace transform inversion is applied to a classical timedependent problem of reactor physics. The dynamic behaviour of a multiplying system has been analyzed through a continuous slowing down model, taking into account a finite slowing down time, the presence of several groups of neutron precursors and simplifying the spatial analysis using the space asymptotic approximation. The results presented, show complete agreement with analytical ones previously obtained and allow a deeper understanding of the model features. © 1990.
 Wilson, J. W., Townsend, L. W., Lamkin, S. L., & Ganapol, B. D. (1990). A closedform solution to HZE propagation. Radiation Research, 122(3), 223228.More infoPMID: 2356274;Abstract: An analytic solution for highenergy heavy ion transport assuming straightahead and velocityconserving interactions with constant nuclear crossreactions is given in terms of a Green's function. The series solution for the Green's function is rapidly convergent for most practical applications. The Green's function technique can be applied with equal success to laboratory beams as well as to galactic cosmic rays allowing laboratory validation of the resultant space shielding code.
 Ganapol, B. D., Spiga, G., & Oggioni, S. (1989). Evaluation of the electric field generated by longitudinal plasma waves. Physics of Fluids B, 1(11), 21492152.More infoAbstract: The stabilizing effects of the addition of shortrange collisions to a partially ionized plasma are investigated through a recently developed numerical Laplace transform inversion algorithm. The technique is demonstrated using a twostream equilibrium distribution and double Maxwellian equilibrium distributions. The regions of plasma stability obtained in previous studies are verified. © 1989 American Institute of Physics.
 Ganapol, B. D., Townsend, L. W., & Wilson, J. W. (1989). Benchmark solutions for the galactic ion transport equations. Energy and spatially dependent problems. NASA Technical Paper.More infoAbstract: Nontrivial benchmark solutions are developed for the galactic ion transport (GIT) equations in the straightahead approximation. These equations are used to predict potential radiation hazards in the upper atmosphere and in space. Two levels of difficulty are considered: (1) energy independent and (2) spatially independent. The analysis emphasizes analytical methods never before applied to the GIT equations. Most of the representations derived have been numerically implemented and compared with more approximate calculations. Accurate ion fluxes (to 3 to 5 digits) are obtained for nontrivial sources. For monoenergetic beams, both accurate doses and fluxes are found. The benchmarks presented herein are useful in assessing the accuracy of transport algorithms designed to accommodate more complex radiation protection problems. In addition, these solutions can provide fast and accurate assessments of relatively simple shield configurations.
 Wilson, J. W., Townsend, L. W., Nealy, J. E., Chun, S. Y., Hong, B. S., Buck, W. W., Lamkin, S. L., Ganapol, B. D., Khan, F., & Cucinotta, F. A. (1989). Bryntrn: A Baryon transport model. NASA Technical Paper.More infoAbstract: This report describes the development of an interaction data base and a numerical solution to the transport of baryons through an arbitrary shield material based on a straight ahead approximation of the Boltzmann equation. The code is most accurate for continuousenergy boundary values but gives reasonable results for discrete spectra at the boundary using even a relatively coarse energy grid (30 points) and large spatial increments (1 cm in H2O). The resulting computer code is selfcontained, efficient, and easy to use. The code requires only a very small fraction of the computer resources required for Monte Carlo codes.
 Wilson, J. W., Townsend, L. W., Barry, G., Chun, S. Y., & Buck, W. W. (1988). CHARGEDPARTICLE TRANSPORT IN ONE DIMENSION.. Nuclear Science and Engineering, 99(3), 285287.More infoPMID: 11539743;Abstract: A numerical solution to highenergy chargedparticle transport is founded by evaluation of the integral equations obtained by inverting the Boltzmann differential operator. An algorithm has been written for either continuous or discrete spectra at the boundary, which allows efficient and accurate evaluation of this rather complicated problem. Present results are compared with analytic solutions based on the perturbation theory.
 Ganapol, B. D., & Pomraning, G. C. (1987). On the use of the Rosseland and Planck mean absorption coefficients in the nonequilibrium radiative transfer equation. Journal of Quantitative Spectroscopy and Radiative Transfer, 37(3), 297309.More infoAbstract: The use of the Rosseland and Planck mean absorption coefficients in the nonequilibrium (timedependent) radiative transfer equation is investigated in the gray approximation for a prescribed temperature distribution. The corresponding gray radiative flux at the surface of a halfspace is compared to an exact (multifrequency) solution obtained from multiple collision theory. An improved mean absorption coefficient is then obtained using the exact angle integrated intensity at the surface, and the corresponding gray approximation is shown to be markedly accurate. © 1987.
 Ganapol, B. D. (1986). SOLUTION OF THE ONEGROUP TIMEDEPENDENT NEUTRON TRANSPORT EQUATION IN AN INFINITE MEDIUM BY POLYNOMIAL RECONSTRUCTION.. Nuclear Science and Engineering, 92(2), 272279.More infoAbstract: The numerical solution to the onegroup timedependent neutron transport equation in infinite plane, spherical, and cylindrical geometries is obtained via an expansion in Legendre polynomials. The computation features general anisotropic scattering, isotropic and beamsources, and a power law timedependent crosssection variation. Results for test problems are compared with previously obtained numerical solutions and with the diffusion approximation.
 Ganapol, B. D., & Matsumoto, M. (1986). Numerical evaluation of timedependent reflected intensity from an anisotropically scattering semiinfinite atmosphere. Journal of Quantitative Spectroscopy and Radiative Transfer, 35(1), 7178.More infoAbstract: A numerical scheme, designed to provide four or five digit accuracy for the reflected photon intensity from an anisotropically scattering semiinfinite medium, is presented. The atmosphere is characterized by both a mean free time between collisions and a mean time of temporal capture. Tables of reflected intensities for a testphase function are included for use as benchmarks to which other numerical solutions can be compared. © 1986.
 Ganapol, B. (1985). Reconstruction of the timedependent monoenergetic neutron flux from moments. Journal of Computational Physics, 59(3), 468483.More infoAbstract: The solution to the monoenergetic timedependent neutron transport equation in infinite spherical geometry with isotropic scattering is reconstructed from moments. By application of the transformation between spherical and cylindrical geometry, the corresponding solutions is obtained directly. In addition, the form of the solution allows generalization to the anisotropic scattering case. © 1985.
 Ganapol, B. D. (1985). EXACT TIMEDEPENDENT MOMENTS FOR THE MONOENERGETIC NEUTRON TRANSPORT EQUATION WITH ISOTROPIC SCATTERING.. Nuclear Science and Engineering, 89(3), 256260.More infoAbstract: Exact expressions for the moments of the timedependent angular flux describing the transport of monoenergetic neutrons are derived. The moments are then used to reconstruct the angular flux which is compared to diffusion theory.
 Ganapol, B. D. (1985). Explicit expressions for the moments of the monoenergetic neutrontransport equation for isotropic scattering in an infinite plane medium. Annals of Nuclear Energy, 12(1), 3538.More infoAbstract: The analytical determination of the spatial and angular moments of the monoenergetic neutrontransport equation in an infinite medium initiated by Case, de Hoffmann and Placzek 30 years ago is completed. Analytical expressions for the moments that are convenient for numerical evaluation are obtained, and several highorder moments are evaluated as a demonstration. © 1985.
 Ganapol, B. D. (1985). SOLUTION OF THE ONEGROUP TIMEDEPENDENT NEUTRON TRANSPORT EQUATION IN AN INFINITE MEDIUM BY POLYNOMIAL RECONSTRUCTION.. Array, 696707.More infoAbstract: The numerical solution to the onegroup timedependent neutron transport equation in infinite plane, spherical and cylindrical geometries is obtained via an expansion in Legendre polynomials. The computation features general anisotropic scattering, isotropic and beam sources and a power law timedependent cross section variation. Results for test problems are compared with previously obtained numerical solutions and with the diffusion approximation.
 Ganapol, B. D. (1985). SPENCERLEWIS EQUATION OF ELECTRON TRANSPORT THEORY: A BENCHMARK.. Transport Theory and Statistical Physics, 15(67), 871895.More infoAbstract: The SpencerLewis equation of electron transport theory with an energy independent screenedRutherford scattering kernel in an infinite medium is solved using a moment reconstruction technique. The solution is evaluated to a high degree of accuracy for use as a benchmark. The evaluation features general anisotropic scattering for a localized source or for a uniformly distributed source in a halfspace. These results are relevant to shielding integrated circuits against electrons.
 Royl, P., Ganapol, B. D., & Bell, C. R. (1985). STUDY OF FUEL REMOVAL VIA INNER BLANKET INTERSUBASSEMBLY GAPS DURING THE DISRUPTION PHASE OF HYPOTHETICAL LOSS OF FLOW ACCIDENTS IN HETEROGENEOUS LMFBR CORES.. Nuclear Technology, 71(1), 145161.More infoAbstract: The potential for fuel removal through the inner blanket intersubassembly gaps before the formation of interconnected pools of fuel and steel has been investigated for the Clinch River Breeder Reactor heterogeneous core under conditions of an unprotected loss of flow accident. The relevant physical phenomena were simulated with the SIMMERII accident analysis code with the necessary model adaptations idntified. A base case has only been set up for the four inner subassembly rings. Based on the understanding gained from this case and an evaluation of a sensitivity study, combined variations for an enhancement of fuel removal are examined. For low flow resistance in the gap, sustained pressurization in the driver assemblies, and delayed blanket gap wall failure, a substantial amount of fuel can be removed from the active core through the inner blanket gaps. The analyzed problem is closely related to the thermal propagation issue.
 Spiga, G., & Ganapol, B. D. (1985). INFLUENCE OF SHORTRANGE COLLISIONS ON THE VLASOV EQUATION FOR PLASMAS.. Transport Theory and Statistical Physics, 14(5), 595617.More infoAbstract: The linearized VlasovMaxwell equations, describing longitudinal plasma waves, are studied with the addition of a simple shortrange collision term and for a given equilibrium distribution. The problem of stability, and of existence and location of isolated eigenvalues determining the asymptotic time behavior of the plasma following a small plane wave initial perturbation with given arbitrary wave number are investigated.
 Woolf, S., Filippone, W. L., Ganapol, B. D., & Garth, J. C. (1985). ELECTRON TRANSPORT CALCULATIONAL METHODS FOR A PLANE SOURCE EMBEDDED IN AN INFINITE MEDIUM.. Array, 788799.More infoAbstract: Two discrete ordinates methods, the Sn method and the method of streaming rays (SR) are applied to electron transport problems. Calculational results in the form of energy deposition profiles are compared with those obtained by the method of moments for the case of a 200 keV plane perpendicular source embedded in infinite aluminum. In a second set of calculations, Sn and SR flux data are compared with results of analytical benchmark technique applied to the solution of the energyindependent form of the SpencerLewis equation for electron transport.
 Ganapol, B. (1984). NUMERICAL EVALUATION OF THE VELOCITY PERTURBATION AT THE WALL FOR THE RAYLEIGH PROBLEM WITH SPECULAR AND DIFFUSE REFLECTION.. Array, 1, 107116.More infoAbstract: An analytical expression for the perturbation at the wall for the classical Rayleigh problem of gas kinetics is obtained via the multiple collision method. The wall can reflect both diffusely and specularly and its acceleration can be explicitly taken into account. The analytical expression is numerically evaluated for uniform wall motion and compared to an asymptotic result valid at large times.
 Ganapol, B. D. (1984). MULTICOMPONENT PHASE TRANSITION MODELING FOR THE SIMMERII LMFBR SAFETY ANALYSIS CODE.. Transactions of the American Nuclear Society, 46, 513514.More infoAbstract: The objective of this paper was to develop a vaporization/condensation model that reliably describes the multicomponent phase transition process for core materials in sufficient physical detail for use in LMFBR accident analysis. Advantage was taken of the large body of chemical engineering literature that exists for the design of multicomponent tube coolercondensers with noncondensible components currently in industrial use.
 Ganapol, B. D., & Larsen, E. W. (1984). ASYMPTOTIC EQUIVALENCE OF FOKKERPLANCK AND DIFFUSION SOLUTIONS FOR LARGE TIME.. Transport Theory and Statistical Physics, 13(5), 635641.More infoAbstract: Beals and Protopopescu proved the asymptotic equivalence of the solution of a FokkerPlanck problem (describing the Brownian motion of a gas) and the solution of the corresponding diffusion problem. Some of their results are obtained for a different class of problems using a simpler method of proof.
 Ganapol, B. D., & Parlette, E. B. (1984). The generalized function solution of the FokkerPlanck equation for monoenergetic charged particle transport. Journal of Mathematical Physics, 25(4), 10761079.More infoAbstract: The solution of the FokkerPlanck equation describing the motion of an isotropic burst of ions originating at the center of an infinite medium and undergoing small angle deflections is obtained in terms of generalized functions using the multiple collision approach. The resulting solution can be shown to define a functional on the test function space Sαβ for α≥1/2 and β≤1/2 and the space of polynomials specifying the dynamic moments. © 1984 American Institute of Physics.
 Ganapol, B. D., Filippone, W. L., & Garth, J. (1984). RECONSTRUCTION OF THE TIMEDEPENDENT MONOENERGETIC NEUTRON FLUX IN AN INFINITE MEDIUM FROM ITS MOMENTS.. Transactions of the American Nuclear Society, 46, 430431.More infoAbstract: In this paper, two conventional orthogonal polynomial expansions will be used to determine the timedependent monoenergetic scalar flux in an infinite medium. The difficulties associated with this reconstruction technique were then identified since a highly accurate benchmark already exists for this problem.
 Ganapol, B. D., Filippone, W. L., & Garth, J. C. (1984). TRANSPORT DESCRIPTION OF XRAY PHOTOELECTRON EMISSION IN THICK TARGETS.. Transactions of the American Nuclear Society, 46, 438439.More infoAbstract: The proper description of energy deposition and transfer of photoelectrons generated in microelectronic structures is described as essential in assessing the vulnerability of very large scale integrated circuits (VLSI) to penetrating radiation. Diffusion models coupled with the continuous energy deposition approximation have found widespread use primarily because of simplicity and general applicability. This paper attempts to address model reliability by first establishing a realistic benchmark solution based on electron transport theory and then comparing the transport result to several well know diffusion models.
 Filippone, W. L., & Ganapol, B. D. (1983). TIMEDEPENDENT ONEDIMENSIONAL TRANSPORT CALCULATIONS USING THE STREAMING RAY METHOD.. Nuclear Science and Engineering, 83(3), 366373.More infoAbstract: The streaming ray technique was developed as a method for solving twodimensional neutron stationary transport problems. A formulation for applying a similar approach to the solution of onedimensional timedependent problems is given. As in the stationary case, use is made of an S//Ntype grid and a second independent grid comprised of streaming rays. For timedependent problems, the streaming ray grid enables accurate tracking of wave fronts, while the S//N grid provides a mechanism for the calculation of the scattering source. Sample calculations are given, and the results are in excellent agreement with analytical benchmark solutions.
 Ganapol, B. D. (1983). CONVERGENCE OF THE GENERALIZED FUNCTION SOLUTION OF KRAMERS' EQUATION FOR BROWNIAN MOTION.. Transport Theory and Statistical Physics, 13(12), 6983.More infoAbstract: The generalized function solution of Kramers' equation describing Brownian motion is shown to be valid in type S function spaces of entire functions for at least a finite time. The method also has been applied to the FokkerPlanck equation describing the motion of charged particles undergoing scattering without energy loss. The theoretical results obtained in this paper are expected to guide that development.
 Ganapol, B. D., & Pomraning, G. C. (1983). The nonequilibrium Marshak wave problem: A transport theory solution. Journal of Quantitative Spectroscopy and Radiative Transfer, 29(4), 311320.More infoAbstract: An analytic solution to a particular Marshak problem is given. The radiative transfer model used is the onegroup grey transport description coupled with the material energy balance. This solution provides a benchmark for validating timedependent radiative transfer algorithms and constitutes a transport solution to the same problem previously solved using diffusion and loworder PN approximations. Typical numerical results are given for surface and integral quantities and comparisons are made with the previously reported diffusion solution, as well as with a Monte Carlo result. © 1983.
 Ganapol, B., & Spiga, G. (1983). The determination of the timedependent intensity at a specularly and diffusely reflecting surface. ZAMP Zeitschrift für angewandte Mathematik und Physik, 34(1), 101111.More infoAbstract: An analytical expression for the timedependent angular intensity at a specularly and diffusely reflecting surface of a semiinfinite medium is determined by the multiple collision formulation. The ratio of the net currents exiting the medium with and without reflection is then evaluated. © 1983 Birkhäuser Verlag.
 Sanchez, R., & Ganapol, B. D. (1983). BENCHMARK VALUES FOR MONOENERGETIC NEUTRON TRANSPORT IN ONEDIMENSIONAL CYLINDRICAL GEOMETRY WITH LINEARLY ANISOTROPIC SCATTERING.. Nuclear Science and Engineering, 84(1), 6167.More infoAbstract: The integral transform method (IT//N) has been extended to the treatment of onedimensional homogeneous media with linearly anisotropic scattering. A previously obtained formula linking the isotropic and the anisotropic onedimensional kernels allows for calculation of the anisotropic matrix elements in the form of linear combinations of a few isotropic matrix elements. The method is applied to the calculation of critical parameters for bare cylinders. Highly accurate values, to be used as benchmarks, are obtained and illustrate the precision and fast convergence rate of the method.
 Filippone, W. L., & Ganapol, B. D. (1982). S//NMONTE CARLO HYBRID TRANSPORT METHOD.. Transactions of the American Nuclear Society, 41, 487489.
 Ganapol, B. D. (1982). MULTIPLE COLLISION SOLUTION OF KRAMER'S EQUATION FOR BROWNIAN MOTION.. Transport Theory and Statistical Physics, 11(3), 217231.More infoAbstract: The solution to Kramer's equation describing Brownian motion is obtained using the multiple collision formulation. The solution is represented in terms of generalized functions; and using examples with known solutions, interpretations are given.
 Ganapol, B. D. (1982). TIMEDEPENDENT SURFACE ANGULAR FLUX FOR A SEMIINFINITE MEDIUM WITH SPECULAR REFLECTION.. Nuclear Science and Engineering, 80(3), 412415.More infoAbstract: The angular flux for a semiinfinite medium with a specularly reflecting surface illuminated by a pulsed source is obtained via the multiple collision method of solution. Numerical results for the scalar flux are presented.
 Ganapol, B. D., & Boffi, V. C. (1982). The constant collision frequency model for electrical conductivity. Meccanica, 17(1), 1115.More infoAbstract: The constant collision frequency charged particle transport model for general separable scattering and ionization kernels is generalized for the determination of the spatially uniform electrical current density and the electrical conductivity to include a timedependent electric field, and a constant magnetic field. It is shown that the conductivity is independent of the assumed velocity distribution of both scattering and ionization kernels and the source. © 1982 Pitagora Editrice Bologna.
 Ganapol, B. D., & Filippone, W. L. (1982). ASYMPTOTIC BEHAVIOR OF THE FLUX EXITING A SEMIINFINITE MEDIUM.. Transactions of the American Nuclear Society, 43, 373374.
 Ganapol, B. D., & Filippone, W. L. (1982). Timedependent emergent intensity from an anisotropicallyscattering semiinfinite atmosphere. Journal of Quantitative Spectroscopy and Radiative Transfer, 27(1), 1521.More infoAbstract: The timedependent emergent intensity for an arbitrarily anisotropicallyscattering, semiinfinite medium is obtained in terms of the stationary collided intensities. A numerical example is presented for a Rayleigh scattering atmosphere. © 1982.
 Ganapol, B. O. (1982). Determination of the density perturbation at the wall for the Rayleigh problem. Physics of Fluids, 25(12), 22112217.More infoAbstract: The Rayleigh problem, a fundamental timedependent problem of gas kinetics, is studied in the context of the constant collision frequency BGK model. An analytical result, expressed in terms of the stationary solution, is obtained and numerically evaluated. Asymptotic solutions in both the small and large time limit are also presented and compared to results of other theories. © 1982 American Institute of Physics.
 Ganapol, B. D. (1981). Multiple collision solution of the linearized vlasov equation. Progress in Nuclear Energy, 8(23), 151161.More infoAbstract: The multiple collision or Neumann series solution technique is applied to the linearized Vlasov equation describing collisionless electron motion. A solution in direct rather than in transformed variables is obtained. © 1981.
 Ganapol, B. D. (1981). The numerical evaluation of the timedependent emergent intensity from a semiinfinite atmosphere. Journal of Quantitative Spectroscopy and Radiative Transfer, 25(2), 159166.More infoAbstract: Exact expressions for the timedependent, emergent intensity from a simiinfinite atmosphere are obtained from the steadystate emergent intensity for several source conditions at the free surface. For an isotropic entering intensity, the exact solution is compared to several approximate solutions found in the literature. © 1981.
 Ganapol, B. D., & Boffi, V. C. (1981). Evaluation of the electrical conductivity via the timedependent integral boltzmann equation. International Journal of Engineering Science, 19(5), 689696.More infoAbstract: Following the integral method developed in neutron transport theory, the exact timedependent density distribution and current density describing electron motion under the influence of a conservative force are obtained. From the expression for the current density, the validity of Ohm's law for various scattering interaction models can be verified. © 1981.
 Ganapol, B. D., & Spiga, G. (1981). Timedependent radiative transfer in a semiinfinite medium with a reflecting boundary. ZAMP Zeitschrift für angewandte Mathematik und Physik, 32(5), 555569.More infoAbstract: An exact solution for the intensity due to a pulse radiation on the free surface of a semiinfinite medium with a known surface reflectivity is obtained via the multiple collision method. The timedependent scalar intensity at the free surface is evaluated for several reflectivities and albedos using the corresponding steady state solution. © 1981 Birkhäuser Verlag.
 Ganapol, B. (1980). Introduction. Annals of Nuclear Energy, 7(45), 199.
 Ganapol, B. (1980). On the Rayleigh problem of gas kinetics. Annals of Nuclear Energy, 7(45), 243247.
 Ganapol, B. D. (1980). Pl APPROXIMATION TO THE TIMEDEPENDENT MONOENERGETIC NEUTRON TRANSPORT EQUATION IN INFINITE MEDIA.. Transport Theory and Statistical Physics, 9(4), 145159.More infoAbstract: The multiple collision method of solution is applied to the timedependent P//1 approximation for infinite and semiinfinite plane geometry. For the semiinfinite medium, with an isotropic flux impinging on the surface, it is shown that the P//1 approximation for the exiting neutron flux is identical to the exact solution.
 Ganapol, B. D., & Boffi, V. C. (1980). The multiple collision method in solving the Boltzmann equation for timedependent test particle transport. Meccanica, 15(2), 7581.More infoAbstract: The multiple collision method is applied to solve a timedependent problem of test particle transport as governed by the Boltzmann equation in the frame of the scattering kernel formulation for the collision term. Several features of the solution found are discussed. Curves representing the approach to the steady state are presented. © 1980 Pitagora Editrice Bologna.
 Ganapol, B. D. (1979). The exact solution of the timedependent equation of radiative transfer in the interior of a semiinfinite atmosphere. Journal of Quantitative Spectroscopy and Radiative Transfer, 22(2), 135143.More infoAbstract: An exact solution to the timedependent radiative transfer equation for the interior intensity in a semiinfinite atmosphere exhibiting temporal capture is obtained. The intensity is constructed from a solution found in neutron transport theory and can be expressed in terms of quadratures of elementary functions. The origin of the form of the emergent intensity found by Matsumoto is given, as well as a similarity relation and the relationship of the intensities in absorbing and nonabsorbing media. © 1979.
 Ganapol, B., Smith, M., Clark, B., & Sowers, G. (1979). ANALYSIS OF THE SANDIA UO//2 EQUATION OF STATE EXPERIMENTS.. Proceedings  IEEE Region 6 Conference, 14651474.More infoAbstract: The initial effort in the analysis of the electron beam UO//2 vapor pressure equation of state experiments performed at Sandia is presented. Three computer codes containing different models are used to examine the hydrodynamic behavior of the heated urania sample in an attempt to reproduce the experimental results. Pending further investigations of the differences between experiment and analysis, firm conclusions cannot be stated at this time.
 Murphy Jr., D. J., Farr, W. M., & Ganapol, B. D. (1979). POWER PRODUCTION AND ACTINIDE ELIMINATION BY FAST REACTOR RECYCLE.. Nuclear Technology, 45(3), 299307.More infoAbstract: Elimination of longlived transplutonium actinides by fissioning in a generic actinide burner reactor (a reactor fueled solely with waste actinides) was investigated. The results showed that actinide elimination by fissioning is enhanced by increasing the average energy of the neutron flux spectrum. In addition, the reactivity worths and the fissiontocapture rate ratios of the individual actinide nuclides increased with increasing flux spectrum energy. The fuel value of waste actinides was studied, and the replacement of at least some conventional mixedoxide fast reactor fuel by waste actinides (to conserve a fuel resource) was proposed. It is calculated that the time required to reach equilibrium actinide concentrations in the reactor core, after many refueling periods, is shorter for reactors having higher neutron flux energies. Also, increasing the specific power density within the reactor core both decreases the equilibrium actinide concentrations in the core and increases the time required for equilibrium conditions.
 Ganapol, B. D. (1978). SOLUTION OF THE TIMEDEPENDENT MONOENERGETIC NEUTRON TRANSPORT EQUATION IN A SEMIINFINITE MEDIUM.. Transp Theory Stat Phys, 7(3), 103122.More infoAbstract: The multiple collision technique, also known as the method of successive approximation, is used in conjunction with generalized functions and the principle of invariance for reflection to solve the monoenergetic timedependent neutron transport equation ina semiinfinite medium. General plane geometry with azimuthal dependence and anistropic scattering expressed as a truncated series of spherical harmonics is assumed.
 Ganapol, B. D., Henninger, R. J., & Peddicord, K. L. (1977). TIMELETHARGY EQUIVALENCE RELATIONS IN NEUTRON TRANSPORT THEORY.. Atomkernenergie, 29(4), 247251.More infoAbstract: The equivalent roles of the time and lethargy variables for the following infinite plane geometry transport problems are shown: (1) monoenergetic timedependent neutron transport in an isotropically scattering homogeneous medium; (2) neutrons slowing down assuming a particular scattering kernel in a medium with constant cross section. This relationship is used to generate an analytic benchmark solution to which the S//N multigroup transport code ANISN can be compared. These results are relevant to nuclear reactors.
 Ganapol, B. D., McKenty, P. W., & Peddicord, K. L. (1977). GENERATION OF TIMEDEPENDENT NEUTRON TRANSPORT SOLUTIONS IN INFINITE MEDIA.. Nuclear Science and Engineering, 64(2), 317331.More infoAbstract: The multiple collision technique as applied to the monoenergetic timedependent neutron transport equation for pulsed plan source emission in an infinite medium is used to obtain the flux due to a pulsed point source in the same medium. This result is then integrated to determine the flux due to the corresponding pulsed line source problem. The semiinfinite albedo problem is also shown to be solvable using the multiple collision approach. A generalization to include delayed neutrons follows directly from the multiple collision treatment, as does an equivalence between a monoenergetic timedependent problem and a particular stationary slowing down problem in infinite geometry. Results are tabulated and comparisons are made to provide benchmark solutions to the fundamental timedependent transport problems considered and thus bridge the gap between theory and practice.
 Ganapol, B. D. (1974). ON THE TIMEDEPENDENT GREEN'S FUNCTION FOR MONOENERGETIC NEUTRON TRANSPORT.. Atomkernenergie, 24(1), 7072.More infoAbstract: In an earlier article C. Syros presented a derivation of the timedependent Green's function for monoenergetic neutron transport in plane geometry by applying the Laplace and Fourier transforms to multiple collision equations. It seems, however, that his solution does not correctly take into account the behavior of the transformed flux in the complex plane. The purpose of this communication is to present a corrected solution and to use this solution to derive a closed form for the total flux distribution.
 Ganapol, B. D., & Grossman, L. M. (1973). COLLIDED FLUX EXPANSION METHOD FOR TIMEDEPENDENT NEUTRON TRANSPORT.. Nuclear Science and Engineering, 52(4), 454460.More infoAbstract: A method for evaluating Kholin's solution, specialized to the case of isotropic scattering, is presented. A series of integrals are evaluated numerically by either a recursion relation or a ChebyshevGauss quadrature approximation. The neutron density found by this method serves as an analytic ″benchmark″ to which other solutions to the timedependent transport equation can be compared. A new closed form of the solution is also derived.
Proceedings Publications
 Ganapol, B. D. (2020, November). Solution of the Monoenergetic Neutron Transport Equation in a Half Space via Singular Eigenfunction Expansion. In ANS November Conference.
 Ganapol, B. D. (2019, Fall). Accurate solutions of the radiative transfer problem via theory of connections. In ICTT26.
 Ganapol, B. D. (2019, Fall). Matrix Riccati equation solution of the radiative transfer equation: A Progress Report. In ICTT26.
 Ganapol, B. D. (2019, Fall). Monoenergetic Time Dependent Diffusion in 1D Heterogeneous Media. In ANS.
 Ganapol, B. D. (2019, Fall). On Xenon Fission Product Poisoning. In ICTT26.
 Ganapol, B. D. (2019, Fall). Oneway Coupled Benchmark for CombinedHyperthermiaRadiotherapy Treatment in Slab Geometry. In ICTT26.
 Ganapol, B. D. (2019, Fall). Towards a Multiphysics Model for Tumor Response to Combined HyperthermiaRadiotherapy Treatment. In Math and Computation.
 Ganapol, B. D. (2018, June). An Extreme Benchmark for Monoenergetic Scattering in Hydrogen. In ANS, 4.
 Ganapol, B. D. (2016, Fall). Poiseuille Channel Flow by Adding and Doubling. In 30TH INTERNATIONAL SYMPOSIUM ON RAREFIED GAS DYNAMICS (RGD 30), 1786.
 Ganapol, B. D. (2014, September 1418). A New Solution for the 1D Neutral Particle Transport Equation in Heterogeneous Media. In Topical Meeting of the ANS Radiation Protection and Shielding Division.
Presentations
 Ganapol, B. D. (2018, August). PROGRESS ON PERIDYNAMIC MODELING. NEUP Review. ORNL: DoE.
 Ganapol, B. D. (2016, June). Operational Readiness of the RattlesnakeMultidimensional Sn Transport Code. PHYSOR 2016. Sun VAlley, ID.
 Ganapol, B. D. (2014, October 8). Nucleosynthesis, Nuclear Power and the Solution to the Neutron Transport Equation. AME Graduate Seminar. Tucson, AZ: AME.
 Ganapol, B. D. (2014, September 14). More Than You Wanted to Know About Convergence Acceleration. Radiation Protection and Shielding Division Workshop. Knoxville, TN: ANS.
Poster Presentations
 Ganapol, B. D. (2019, Fall). Precision Benchmark of the COG Monte Carlo Code. ICTT26. Paris: CEA.
Other Teaching Materials
 Ganapol, B. D. (2016. Analytical/Numerical Methods of Monoenergetic Neutron Transport Theory. Idaho National Laboratory.
Others
 Ganapol, B. D. (2016, Feb). Is there a future for small modular reactors in the West?. Arizona Daily Star.