Spectral Approximation to Fractional Integral Operators

TL;DR

This paper introduces a fast, stable spectral method for approximating fractional integral operators using Chebyshev polynomials, enabling efficient solutions to various fractional differential equations.

Contribution

It presents a novel recurrence-based spectral approximation technique that outperforms existing methods for fractional integral operators.

Findings

Method significantly improves computational efficiency.

Applicable to boundary value, initial value, and eigenvalue problems.

Demonstrates broad applicability through numerical examples.

Abstract

We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional integrals of mapped Chebyshev polynomials and significantly outperforms existing methods. Through numerical examples, we highlight the broad applicability of these matrix approximations, including the solution of boundary value problems for fractional integral and differential equations. Additional applications include fractional differential equation initial value problems and fractional eigenvalue problems.