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

Abstract

In this paper, we are concerned with finding numerical solutions to the class of time-space fractional partial differential equations: $D_t^{p}u(t,x)+\kappa D_x^{p}u(t,x)+\tau u(t,x)=g(t,x),1<p<2,(t,x)\in [0,1]\times [0,1],$ under the initial conditions. $u(0,x)=\theta \left(x\right),u_t(0,x)=\varphi \left(x\right),$ and the mixed boundary conditions. $u(t,0)=u_x(t,0)=0,$ where $D_t^p$ is the arbitrary derivative in Caputo sense of order p corresponding to the variable time t. Further, $D_x^p$ 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 $g(t,x)$ . Further the exact and numerical solutions are compared via 3D graphs which shows that both the solutions coincides very well.

Keywords: Caputo fractional derivative; Fractional partial differential equations; Numerical solution; Operational matrices; Shifted Jacobin polynomials; Stability.