The numerical solution of fractional integral equations via orthogonal polynomials in fractional powers
Abstract
We present a spectral method for onesided 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, nontrivial variable coefficients, and initialboundary 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 highprecision computations, the spectral method nonetheless yields wellconditioned linear systems and is therefore stable and efficient. For timefractional 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.14280
 Bibcode:
 2022arXiv220614280P
 Keywords:

 Mathematics  Numerical Analysis;
 65R20;
 G.1.m;
 G.1.9
 EPrint:
 35 pages, 3 of which are references. 16 figures, all of which can be reproduced by Julia codes at https://github.com/putianyi889/JFPdemo