Galerkin operational approach for multi-dimensions fractional differential equations

Abstract

The current manuscript introduces a novel numerical treatment for multi-term fractional differential equations with variable coefficients. The spectral Galerkin approach is developed with the operational matrix of fractional derivatives to get a system of algebraic equations that can be solved using a suitable technique. The operational matrix is introduced based on a combination of the shifted Jacobi polynomials. The proposed approach is applied also for 1+1 and 2+1 multi-term time-fractional diffusion and diffusion-wave equations. In addition, the convergence analysis for the presented approaches is investigated. Some specific numerical examples are given to ascertain the wide applicability and the good efficiency of the suggested algorithms, and to confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.