Numerical algorithm to Caputo type time–space fractional partial differential equations with variable coefficients

Abstract

This article assigns to a new numerical algorithm as an appropriate tool that deals with the time–space fractional partial differential equations in Caputo sense with variable coefficients. In the current algorithm, firstly, we presume that the approximate solution of the main problem is expandable along space variable via shifted Chebyshev polynomials with time-dependent coefficients. In the second step, we employ operational matrices of space-fractional derivatives to transform (reduce) the expanded problem to a system of time-fractional ordinary differential equations (FODEs) with initial value conditions. Indeed, the solutions of this system are required to obtain the time-dependent coefficients of the mentioned expansion. To solve this system, we define some independent secondary initial value problems and solve them analytically. At the final step, we find an optimal linear combination of this particular solutions to obtain an approximate solution of the main problem such that the residual error function forced to vanish in an average sense over the desired region, and the approximate solution satisfies in initial/boundary conditions of the main problem. The convergence property of the presented algorithm is demonstrated by the residual error analysis. The reliability and accuracy of the new algorithm are confirmed by solving some illustrative test problems. In order to perform a premier analysis with more details for the convergence property of the new algorithm, we compute the observed convergence order indicators in each test problem. Moreover, we evaluate our computed results with other numerical schemes in the literature to emphasize the promising performance of the proposed algorithm.