| Fractional differential equation is a class of differential equations which are abstracted from practical problems. Compared with integer order differential equation, the most important advantage of fractional differential equation is that it can simulate some natural physical phenomena and dynamic system process better, so it has been widely applied into physics, engineering, finance, groundwater and environmental problems. However, compared with integer order differential equation, fractional differential equation is less perfect. It is short for scientific solution formula. At present the study of it is still in its infancy.Like integer order differential equation, only a few types of fractional differential equation can find out its analytic solution. In most cases, it can only use the numerical method to compute its solution. Therefore, the research of numerical evaluation has very important significance.In this paper, we mainly study the numerical methods for the one dimensional two-sided space fractional advection-diffusion equations. All the fractional derivatives refer to Riemann-Liouville definition in terms of the fractional derivative in this article. The main work is as follows:In Chapter one, an introduction to the history of the fractional calculus, the research significance of fractional differential equation and the current research status at home and abroad of the numerical method for fractional differential equations are given.In Chapter two, some preliminary knowledge is given, including fractional derivative, Toeplitz and circulant matrices and the related theorem.In Chapter three, the finite difference method for the one dimensional two-sided space fractional advection-diffusion equations is studied. According to the thought of the finite difference method proposed by some scholars, we construct the center weighted C-N scheme for the equation which can achieve two order accuracy both in time and space. The stability and convergence of the method are theoretically established. Finally, the numerical example verifies the effectiveness, accuracy and reliability of the method. But the method can not guarantee that the coefficient matrix of the discrete system strictly diagonally dominant, which can lead to some difficulties to computing.In Chapter four, on the base of the center weighted C-N scheme, we develop a new weighted C-N scheme which can also achieve two order accuracy both in time and s-pace. The new method can guarantee that the coefficient matrix of the discrete system strictly diagonally dominant. And then we analyze the existence and uniqueness of the solution, stability and convergence of the format. Also the numerical example verifies the effectiveness, accuracy and reliability of the method.In Chapter five, we study the fast finite difference method for the one dimensional two-sided space fractional advection-diffusion equations. By the fast Fourier transform, the fast method only requires storage of O(K) and computational work of O{K log2 K) per time step, while retaining the same accuracy as the center or new weighted C-N scheme. Numerical experiment is presented to verify the efficiency and accuracy of the fast method. |
