Finite Difference Method For Time-and-space Fractional Diffusion Equations

The time fractional diffusion equation is obtained by replacing the first-order time derivative term of the classical diffusion equation with the time fractional derivative term(0<??1),and the spatial fractional diffusion equation,the s-patial second derivative term of the equation is replaced by the space fractional derivative term(1<??2),which is widely used in natural science and engineering applications.In the first part,this paper derives a new spatial fourth-order finite difference scheme based on L1-2 interpolation approximation.The numerical results show that in this paper are more accurate.When using the Caputo derivative to approximate the ? order time fractional derivative,We approximate the fractional derivative with the L1 operator on the first inter-cell[t0,t1],and use the L1-2 operator to approx-imate the fractional derivative on other small intervals[tj-1,tj](j?2),the second derivative of space is approximated by the five-point central difference scheme,and the stability and convergence of the scheme are proved.Finally,the numerical exam-ples are programmed by Matlab,and the results show that the scheme is effective.In the second part,Based on the idea of weighted average,this paper deduces the discrete scheme of the variable coefficient space fractional diffusion equation.The spatial fractional derivative for the ? order is approximated using the standardG—L formula and the shifted G-L formula weighted average.In(?)-1/2???2,the spatial direction converges to the second order.The unconditional stability of the difference scheme is proved.Finally,the numerical example also shows the result,the method has a wide stability domain and a fast convergence rate.