The solution to the neutron Boltzmann equation is separated into a straightahead component dominating at high energies and an isotropic component dominating at low energies. The high-energy solution is calculated using HZETRN-05, and the low-energy isotropic component is modeled by two non-coupled integro-differential equations describing both forward and backward neutron propagation. Three different solution methods are then used to solve the equations. The collocation method employs linear I3-splines to transform each equation into a system of ODES; the resulting system is then solved exactly and evaluated using numerical integration techniques. Wilson's method uses a perturbational approach in which a fundamental solution is obtained by solving a simple ODE, a new source term is generated by the fundamental solution, and the collocation method is then used to solve the remaining equation. The fixed-point series method extends Wilson's method by continuing the perturbational procedure until desired convergence criteria are met. In all three cases, the total neutron flux is found by adding the forward and backward components. Comparisons are made between the three methods in one, two and three layer configurations in various space environments and compared to Monte Carlo data where available.
","abstract_html":"<p>The solution to the neutron Boltzmann equation is separated into a straightahead component dominating at high energies and an isotropic component dominating at low energies. The high-energy solution is calculated using HZETRN-05, and the low-energy isotropic component is modeled by two non-coupled integro-differential equations describing both forward and backward neutron propagation. Three different solution methods are then used to solve the equations. The collocation method employs linear I3-splines to transform each equation into a system of ODES; the resulting system is then solved exactly and evaluated using numerical integration techniques. Wilson's method uses a perturbational approach in which a fundamental solution is obtained by solving a simple ODE, a new source term is generated by the fundamental solution, and the collocation method is then used to solve the remaining equation. The fixed-point series method extends Wilson's method by continuing the perturbational procedure until desired convergence criteria are met. In all three cases, the total neutron flux is found by adding the forward and backward components. Comparisons are made between the three methods in one, two and three layer configurations in various space environments and compared to Monte Carlo data where available.</p>","abstract_has_math":false,"creators":["Slaba, Tony Charles"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["John Tweed","John H. Heinbockel","Gordon Melrose","Francis Badavi","Steve Blattnig"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2007,"date_issued":"2007-04-01T07:00:00Z","date_published":"2007-04-01T07:00:00Z","updated_at":"2026-07-24T03:35:00Z","subjects":["Boltzmann equations","Neutron Boltzmann equation","Ordinary differential equations","Ordinary Differential Equations and Applied Dynamics"],"languages":[],"rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["9780549255642","https://digitalcommons.odu.edu/mathstat_etds/57"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The solution to the neutron Boltzmann equation is separated into a straightahead component dominating at high energies and an isotropic component dominating at low energies. The high-energy solution is calculated using HZETRN-05, and the low-energy isotropic component is modeled by two non-coupled integro-differential equations describing both forward and backward neutron propagation. Three different solution methods are then used to solve the equations. The collocation method employs linear I3-splines to transform each equation into a system of ODES; the resulting system is then solved exactly and evaluated using numerical integration techniques. Wilson's method uses a perturbational approach in which a fundamental solution is obtained by solving a simple ODE, a new source term is generated by the fundamental solution, and the collocation method is then used to solve the remaining equation. The fixed-point series method extends Wilson's method by continuing the perturbational procedure until desired convergence criteria are met. In all three cases, the total neutron flux is found by adding the forward and backward components. Comparisons are made between the three methods in one, two and three layer configurations in various space environments and compared to Monte Carlo data where available.
"]},{"key":"dc:title","label":"Title","values":["Three Methods for Solving the Low Energy Neutron Boltzmann Equation"]}]}],"canonical_facts":{"dc:contributor":["John Tweed","John H. Heinbockel","Gordon Melrose","Francis Badavi","Steve Blattnig"],"dc:creator":["Slaba, Tony Charles"],"dc:date.available":["2019-06-13T07:00:00Z"],"dc:description.abstract":["The solution to the neutron Boltzmann equation is separated into a straightahead component dominating at high energies and an isotropic component dominating at low energies. The high-energy solution is calculated using HZETRN-05, and the low-energy isotropic component is modeled by two non-coupled integro-differential equations describing both forward and backward neutron propagation. Three different solution methods are then used to solve the equations. The collocation method employs linear I3-splines to transform each equation into a system of ODES; the resulting system is then solved exactly and evaluated using numerical integration techniques. Wilson's method uses a perturbational approach in which a fundamental solution is obtained by solving a simple ODE, a new source term is generated by the fundamental solution, and the collocation method is then used to solve the remaining equation. The fixed-point series method extends Wilson's method by continuing the perturbational procedure until desired convergence criteria are met. In all three cases, the total neutron flux is found by adding the forward and backward components. Comparisons are made between the three methods in one, two and three layer configurations in various space environments and compared to Monte Carlo data where available.
"],"dc:identifier":["9780549255642","https://digitalcommons.odu.edu/mathstat_etds/57"],"dc:rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"],"dc:subject":["Boltzmann equations","Neutron Boltzmann equation","Ordinary differential equations","Ordinary Differential Equations and Applied Dynamics"],"dc:title":["Three Methods for Solving the Low Energy Neutron Boltzmann Equation"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:35:00Z"}