Numerical Solutions for Stochastic Differential Equations Driven by Fractional Caputo-Hadamard Operator with Simulation on Physics

Abstract

This paper presents a novel contribution focused on constructing numerical solutions for a stochastic fractional differential equation governed by two distinct operators: the Caputo-Hadamard and the Caputo-Katugampola fractional derivatives. These operators enable the evaluation and validation of the most accurate approximations for the given problem. Building on these results, we extend our approach to solve fractional neutron point kinetic

equations, which model anomalous slow diffusion in nuclear reactor dynamics. Numerical techniques, such as the explicit finite difference method, are employed to efficiently and effectively derive solutions. These solutions are analyzed and graphically represented to demonstrate their accuracy and applicability. The fractional neutron point kinetic model introduces a fractional-order exponent to the relaxation time associated with neutron flux variation, providing a more refined understanding of neutron dynamics. This model is used to analyze the dynamic behavior of neutron diffusion under subcritical, supercritical, and critical reactivities. The results are compared with those obtained from classical neutron point kinetic equations, highlighting the advantages of the fractional approach in capturing complex neutron behavior. This work contributes to the broader field of nuclear reactor dynamics by offering a robust framework for modeling and simulating anomalous diffusion processes.