Numerical investigation of the two-dimensional space-time fractional diffusion equation in porous media

Abstract

This paper develops the approximate solution of the two-dimensional space-time fractional diffusion equation. Firstly, the time-fractional derivative is discretized with a scheme of order $${\mathcal {O}}({\delta \tau }^{2-\alpha }),~ 0<\alpha <1$$ . Then, the Chebyshev spectral collocation of the third kind is implemented to approximate spatial variables and to obtain full discretization of the equation. Moreover, the unconditional stability and convergence of the proposed method are shown in the perspective $$H^{2}$$ -norm. Two numerical examples are presented to verify the effectiveness and the accuracy of the proposed method. The comparison between our obtained numerical results and the results of existing schemes in the literature shows that the proposed method is more reliable and precise.