Stable numerical results to a class of time-space fractional partial differential equations via spectral method
Kamal Shah, Fahd Jarad, Thabet Abdeljawad
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.