Numerical Methods For Two Classes Of Fractional Differential Equations

As is well known, the study of fractional order differential equation pervades many fields, such as physics, biology and engineering etc. Recently, many researchers find that applying the fractional calculus to describe anomalous diffusion is very suitable, and many kinds of fractional anomalous diffusion equations are developed. Besides, people realize that it is usually not easy to get the exact solutions of most fractional differential equations due to the existence of fractional derivative. Therefore, more and more scholars focus on the investigation of numerical solution of such equations.This paper aims to consider the numerical methods for two kinds of fractional differential equations which are nonlinear fractional Fokker-Planck equation and two-dimensional variable time fractional anomalous subdiffusion equation. To our knowledge, published works on numerical solutions of these two kinds of equations are limited. Consequently, establishing efficient numerical methods to solve them are worth further investigating. Main content of this work are as follows.Firstly, based on reproducing kernel theory, the approximate solution of nonlinear fractional Fokker-Planck equation is given in the form of series, which is named by ε-approximate solution. And then, the existence of any ε-approximate solution is proved. At the same time, a framework approach to get the ε-approximate solution is developed. The numerical results verify the theoretical outcome is effective enough.Secondly, differing from the previous articles, in this thesis, by combining reproducing kernel theory and the idea of spline, a new basis in the reproducing kernel space is constructed. Furthermore, a new method to solve two-dimensional variable time fractional anomalous subdiffusion problems is proposed. What is more, by using the idea of collocation technology, an optimal solution for obtaining the coefficients of approximation solution is given. The numerical examples confirm that the new approach is accurate, easily implemented and well suited to get numerical solutions with high precision.