Stable numerical results to a class of time-space fractional partial differential equations via spectral method

In this paper, we are concerned with finding numerical solutions to the class of time-space fractional partial differential equations: under the initial conditions. and the mixed boundary conditions. where $$ is the arbitrary derivative in Caputo sense of order p corresponding to the variable time t. Further, $$ is the arbitrary derivative in Caputo sense with order p corresponding to the variable space x. Using shifted Jacobin polynomial basis and via some operational matrices of fractional order integration and differentiation, the considered problem is reduced to solve a system of linear equations. The used method doesn't need discretization. A test problem is presented in order to validate the method. Moreover, it is shown by some numerical tests that the suggested method is stable with respect to a small perturbation of the source data $$ . Further the exact and numerical solutions are compared via 3D graphs which shows that both the solutions coincides very well.