Alternating Direction Implicit Method For A Class Of Time Fractional Diffudion Equations

Now, more and more people begin to pay their attention to the development and research of fractional differential equations, because they can describe many phenomenon, physical and chemical process more accurately than classical integer-order differential equations. However, researchers find that most of the analytical solutions are indicated by some special functions. Meanwhile, most of problems can not be solved analytically. To develop numerical methods for solving fractional differential equations seems to be more important.The alternating direction implicit method is a kind of finite difference methods, which is useful for heat conduction equations and diffusion equations with two dimensions or mitiple demensions. It is well known that alternating direction implicit method works well for multiple demensional problem, since it deals with multiple demensional problem by solving a series of smaller, independent one-dimensional problems. For large problems, the alternating direction implicit method reduces the storage requirements and computational complexities greatly.We consider a class of two-dimensional fractional sub-diffusion equations with non-homogeneous source term. The main content is as follows:1. A class of two-dimensional sub-diffusion equations are studied in a bounded domain. We first construct an alternating direction implicit schemes based on the 1L approximation. Then we analyze the truncation error, the solvability, unconditional stability and convergence are rigorously proved.2. We construct a new alternating direction implicit scheme based on backward Euler method. The truncation error, solvability, unconditional stability and convergence are presented as before.3. Numerical experiments are carried out for the schemes we have constructed. The corresponding convergence orders are calculated. Compared the numerical solutions with the exact solutions, we obtain the reasonableness of two schemes.