Numerical solution of fractional evolution equations of Caputo type

• Main Author: Khor, Sze Chong
• Other Authors: Dusit Niyato
• Format: Final Year Project
• Language: English
• Published: Nanyang Technological University 2024
• Subjects: Mathematical Sciences; Partial differential equations; Monte Carlo simulation; Fractional calculus
• Online Access: https://hdl.handle.net/10356/175651
• Institution: Nanyang Technological University

In this project, we investigate Caputo-type fractional derivatives, time-fractional evolution equations similar to the heat equation, and their numerical solutions using a Feynman-Kac type representation. We discuss several applications of PDEs involving the Caputo derivative. We review results by Meerschaert et al. and Bonaccorsi et al., and implement them in R to attain Monte Carlo estimates to solutions of the PDEs involving the Caputo derivative. We compare the results of our simulation to those by Meerschaert, and apply the Monte Carlo method on a point initial distribution and uniform initial distribution. We perform some analysis on the errors and runtime of the Monte Carlo scheme. Finally, we discuss future work that can be done in the same spirit of this project, performing some simple exploratory work on a simple variation of the heat-like equation.