An Implicit Finite Difference Approximation For The Riesz Fractional Reaction-dispersion Equation

Recently, fractional order differential equations have been applied to many phenomena in physics, engineering, finance and so on. Fractional differential and integral calculus provide a powerful instrument for the description of memory and hereditary properties of difference substances.Fractional order differential equations are more adequate than integer order differential equations in describing the real world. So it is significant not only in theory but also in application to study the fractional order differential equation. However, it is not easy to get an analytical solution for the fractional order differential equation. As a result, more and more researchers are interested in studying the numerical methods for the fractional order differential equations.Reaction-dispersion equation which involves fractional derivative in spatial variable is derived from the model of molecule dispersion. The basic analytic theory for the space fractional differential equation has been developed in 1952 by Feller. At present the works published for the space fractional reaction-dispersion equations mainly use the equivalence of the definitions between Riemann-Liouville and Grunwald-Letnikov,and then use the shifted Grunwald formula to get a difference scheme for this kind of equation. Space Riesz fractional derivative which has two sides Riemann-Liouville fractional derivative is called Riesz potential operator too.In this paper we consider a space Riesz fractional reaction-dispersion mixed problem in a bounded space domain and time domain. The fractional derivative is discredited directly and an implicit finite difference scheme is constructed, which is unconditionally stable and convergent. The local truncation error is O(h2-β)+O(τ)(1<β≤2),but the order of convergence will become one order, when the initial and boundary problems satisfy some conditions.The content of this paper is as follow:In chapter one, we first introduce the generation, development, and the situation of researching for the fractional differential and integral calculus, and then simply introduce the finite difference method.In chapter two, the definition, feature and the reaction-dispersion equation are simply introduced.In chapter three, we discuss the construction of an implicit finite difference scheme for the space Riesz fractional reaction-dispersion mixed problems.In chapter four, the coefficient matrix of the finite difference scheme is estimated and the stability and convergence of the implicit difference scheme are analyzed, respectively. Finally, the order of convergence will become one order, when the initial and boundary problems satisfy some conditions.In chapter five, we use a numerical example to show that the implicit finite difference scheme of this paper is stable and convergent.In chapter six, we give the tag of this paper, summarize the work that has been done and give some advice of further study.