Volume 65, pp. 281-307, 2026.

Numerical approximation of Caputo-type advection-diffusion equations via shifted Chebyshev polynomials

Francisco de la Hoz and Peru Muniain

Abstract

In this paper, using a pseudospectral approach, we develop operational matrices based on the shifted Chebyshev polynomials to numerically approximate Caputo fractional derivatives and Riemann–Liouville fractional integrals. To make the generation of these matrices stable, we use variable precision arithmetic. Then, we apply the Caputo differentiation matrices to numerically solve Caputo-type advection-diffusion equations in one and in multiple spatial dimensions, which involves transforming the discretization of the equation under consideration 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.

Full Text (PDF) [824 KB], BibTeX , DOI: 10.1553/etna_vol65s281

Key words

Caputo fractional derivative, Riemann–Liouville fractional integral, Caputo-type advection-diffusion equations, pseudospectral methods, shifted Chebyshev polynomials, Sylvester equations, Sylvester tensor equations

AMS subject classifications

15A24, 15A69, 26A33, 35R11, 65M70