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

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.

arXiv comment23 pages, 3 figures