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:Dtpu(t,x)+κDxpu(t,x)+τu(t,x)=g(t,x),1<p<2,(t,x)∈[0,1]×[0,1],under the initial conditions.u(0,x)=θ(x),ut(0,x)=ϕ(x),and the mixed boundary conditions.u(t,0)=ux(t,0)=0,where Dtp is the arbitrary derivative in Caputo sense of order p corresponding to the variable time t. Further, Dxp 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

• Fractional partial differential equations
• Caputo fractional derivative
• Shifted Jacobin polynomials
• Operational matrices
• Numerical solution
• Stability