The numerical solution of fractional integral equations via orthogonal polynomials in fractional powers

TL;DR

This paper introduces a spectral method using Jacobi fractional polynomials for solving linear fractional integral equations, achieving exponential convergence even with complex fractional orders and variable coefficients.

Contribution

The paper develops a new spectral method with Jacobi fractional polynomials and algorithms for fractional integration matrices, improving stability and efficiency over existing methods.

Findings

Achieves exponential convergence for fractional integral equations.

Outperforms sparse spectral methods in stability and efficiency.

Provides algorithms for fractional integration matrix construction.

Abstract

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.