Numerical approximation of Caputo-type advection-diffusion equations in one and multiple spatial dimensions via shifted Chebyshev polynomials

TL;DR

This paper introduces a pseudospectral method using shifted Chebyshev polynomials to numerically solve Caputo-type advection-diffusion equations in multiple dimensions, demonstrating high accuracy and stability.

Contribution

It develops stable operational matrices for fractional derivatives using shifted Chebyshev polynomials and applies them to solve multi-dimensional fractional PDEs with detailed MATLAB implementation.

Findings

Effective numerical approximation of Caputo derivatives

Stable computation via variable precision arithmetic

Successful application to multi-dimensional advection-diffusion equations

Abstract

In this paper, using a pseudospectral approach, we develop operational matrices based on the shifted Chebyshev polynomials to approximate numerically Caputo fractional derivatives and Riemann-Liouville fractional integrals. In order to make the generation of these matrices stable, we use variable precision arithmetic. Then, we apply the Caputo differentiation matrices to solve numerically Caputo-type advection-diffusion equations in one and multiple spatial dimensions, which involves transforming the discretization of the concerning equation into a Sylvester (tensor) equation. We provide complete Matlab codes, whose implementation is carefully explained. The numerical experiments involving highly oscillatory functions in time confirm the effectiveness of this approach.