Research On High Precision Numerical Algorithms For Fractional Differential Equations

In recent years, as a new mathematical tools, the fractional calculus is required to describe the mathematical model in more and more areas, such as material, financial,mechanical, biological systems, signal and image processing. However, comparing with the integer order differential equations, the numerical solutions of the fractional differential equations are less and some numerical solutions have lower accuracy. Thus,it’s necessary to research high precision numerical solutions of fractional differential equations.In this paper, two high precision numerical algorithms which can also reduce the amount of computation are given to solve the fractional diffusion wave equation and the fractional order differential equations. The full text is organized as follows:Some basic definitions and properties of the fractional calculus are introduced in the first chapter.In the second chapter, based on the equivalent integral form of time fractional diffusion-wave equation, the fractional trapezoidal method and Crank-Nicolson method are applied to construct a stable finite difference scheme, which has second order accuracy in both time and space direction, for solving a class of initial-boundary value problems of time fractional diffusion-wave equations. Numerical examples are given to verify the accuracy and effectiveness of proposed method.In the third chapter, in dealing with the initial value problems of a system of fractional differential equations with Caputo derivative, in the first place, it should be converted into its equivalent Volterra integral equations. An initial approximate solution is obtained by a low-level approach while the residual equation and the error equation is deduced. The spectral deferred correction method is used to improve the numerical accuracy of the solution while the Richard Askey’s integral equation is introduced to reduce the amount of computation. Finally, the accuracy and effectiveness of the new approach is verified through numerical experiments.