Difference Methods For Two Kinds Of Time Distributed-order Diffusion Equation

In this article, we consider the numerical solution of one-dimensional time distributed-order diffusion equation combined with suitable initial condition and boundary condi-tions. Firstly, by homogeneous initial conditions, the Caputo fractional derivatives and Riemann-Liouville fractional derivatives could be inter-transformed freely. Then, Ap-plying the composite rectangular quadrature formula and the composite cotes quadra-ture formula to approximate distributed-order integral item, respectively. For time fractional derivative, we use the weighted and shifted Grunwald-Letnikov scheme to approximate. The spatial derivative is discretized by second-order central difference scheme, and obtain the two kinds of numerical methods. The stability and the conver-gence of the obtained schemes are analyzed by the energy method, and the schemes is convergent of orders O(?2+h2+??2) and O(?2+h4+??6), respectively, where ?, h and Aa are the step sizes in time, space and distributed-order variables. Finally, the results of numerical examples support the theoretical analysis.