An Explicit Forward-Euler Type Method for Time-Space Fractional Advection Equations

Abstract

This paper introduces an explicit finite-difference scheme (FDS) for time–space fractional-order (FO) advection equations whose temporal and spatial derivatives have different FOs. The discretization is constructed from a FO Taylor expansion and produces a forward-Euler–type approximation of Caputo derivatives. We provide a rigorous treatment of the scheme’s consistency, establish stability via Fourier analysis, and derive convergence by combining stability with local truncation error estimates. Numerical experiments corroborate the theoretical findings, showing the method efficiently captures the nonlocal memory effects