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In this paper, we consider the nonlinear fractional order ordinary differential equa-Fractional order linear multiple step methods are introduced. The high order (2-6) approximations of the fractional order ordinary differential equation with initial value are proposed. The consistence, convergence and stability of the fractional high order methods are proved. Finally, some numerical examples are provided to show that the fractional high order methods for solving the fractional order nonlinear ordinary differential equation are computationally efficient solution methods.

This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.

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