Two Implicit Meshless Finite Point Schemes for the Two-Dimensional Distributed-Order Fractional Equation

Abstract In this paper, the distributed-order time fractional sub-diffusion equation on the bounded domains is studied by using the finite-point-type meshless method. The finite point method is a point collocation based method which is truly meshless and computationally efficient. To construct the shape functions of the finite point method, the moving least square reproducing kernel approximation is employed. Two implicit discretisation of order O ⁢ ( τ ) {O(\tau)} and O ⁢ ( τ 1 + 1 2 ⁢ σ ) {O(\tau^{1+\frac{1}{2}\sigma})} are derived, respectively. Stability and L 2 {L^{2}} norm convergence of the obtained difference schemes are proved. Numerical examples are provided to confirm the theoretical results.

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