The sine and cosine diffusive representations for the Caputo fractional derivative

TL;DR

This paper introduces two new diffusive representations for the Caputo fractional derivative that improve computational efficiency by addressing the non-local property, supported by error analysis and numerical validation.

Contribution

The paper proposes novel sine and cosine diffusive representations for the Caputo derivative, enhancing approximation accuracy and computational efficiency.

Findings

New diffusive representations for Caputo derivative

Error bounds established for the methods

Numerical examples demonstrate improved performance

Abstract

As we are aware, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order $0<\alpha<1$. Error analysis of the newly presented methods together with some numerical examples are provided at the end.