Caputo derivatives of fractional variable order: numerical approximations

TL;DR

This paper introduces a new numerical method for approximating Caputo derivatives of fractional variable order, enabling the solution of related partial differential equations with error estimates and practical examples.

Contribution

It provides novel approximation formulas for three types of Caputo fractional derivatives of variable order using standard derivatives, along with error analysis and application to PDEs.

Findings

Approximation formulas for three Caputo fractional operators

Error estimates for the numerical approximations

Successful application to solve partial fractional differential equations

Abstract

We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable order.