Validation and Analysis of Numerical Methods for Solving Fractional-Order Differential Equations

Abstract

Fractional-order differential equations have been very effective in modeling the behavior of various phenomena that have been otherwise difficult to accurately simulate. A new class of predictor-corrector schemes for solving nonlinear fractional-order differential equations has been proposed with claims of both increased accuracy and decreased computational cost. The purpose of this research is to test and attempt to validate these claims through both an independent replication of the results reported for the new method, as well as a comparison against the performance of an existing standard. Additionally, an operational software package that implements this new method will result as a byproduct of this research.

Independent simulations are run for this new method, and compared against the findings of this new method’s author. This research is broken down into three main areas of interest: (1) Independently develop software for the newly proposed and existing methods, (2) Conduct an error analysis on the methods’ order of accuracy, and (3) Conduct a computational cost analysis of the two methods in their pure form. This research has successfully created a tool for solving fractional-order differential equations with greater accuracy than the existing method, with replicated findings in regard to order of accuracy. The claims of decreased computational cost were neither confirmed or disproved, and require further study. The results of this research have not only increased the validity and usage-case understanding of the newly proposed method, but also raised new areas for future work with respect to optimization and run speed.