A Finite Difference Method With Non-uniform Timesteps For Variable-order Fractional Partial Equations

Four kinds of variable-order fractional partial equations are investigated in this paper. All of them are discretized by the finite difference method with non-uniform timesteps.The first equation is the time-dependent variable-order fractional diffusion equation. The stability is analyzed by Fourier Methods.In the second part, we get the approximation scheme for time variable-order fractional diffusion equation, and analyze the unconditional stability in maximum norm.Thirdly, the time variable-order fractional advection-diffusion equation are studies. Our scheme increases the error to h2. Theoretical Analysis is followed.The last equation is the space variable-order fractional advection-diffusion equation with nonlinear source term. The stability and convergence are ana-lyzed by Maximum norm Methods. When q(x, t) is independent of t, we can analyze the convergence for the two equations above.In the end, a numerical example has demonstrated the effectiveness of the non-uniform meshes.