Development and implementation of stochastic neutron transport equations and development and analysis of finite difference and Galerkin methods for approximate solution to Volterra's population equation with diffusion and noise



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Texas Tech University


Many systems in this world are influenced by stochastic (random) processes either from within the system or from external agents. When modeling these systems, these processes and their derivatives arise naturally in a field of study called stochastic differential equations (SDEs). SDEs find application in diverse areas of engineering, chemistry, physics, economics and finance, population dynamics, pharmacology and medicine, and social sciences, to name a few. This research is divided into two parts, the common thread being SDEs. In the second chapter, a new system of SDEs for modeling the random behavior of neutron travel is derived. Numerical methods are developed to solve this system and shown to be accurate when compared with the Monte Carlo method. In the third chapter, two independent numerical methods are developed to solve Volterra's population equation with diffusion and noise. Error analyses are performed on the two methods which prove convergence of the approximations to the exact solution. Three numerical examples are given which confirm the results of the error analyses.