Finite Difference Method For Fractional Partial Differential Equations

In this paper, we mainly apply finite difference method to solving numerical solutions of fractional differential equations. As a new tool, fractional calculus has been paid much attention by many math researchers for its better capacity of describing material and process which have chariceristics of memory and heredity. The applications of fractional calculus in various areas lead to appearance of large amount of fractional differential equations. And because of it is difficult to obtain the analytical solution of these problems, numerical methods for them become a hot spot in computational mathematics.First of all, we retrospects the development history and actuality of fractional differential equations briefly, and some basic knowledge of fractional derivatives, in addition, the research background and significance of this task are also analized.Secondly, we discuss the numerical calculus of the typical fractional differential equation—time-fractional Fokker-Planck equation by finite difference method, and test the solvability and uniqueness of the difference scheme by the method of analizing matrix. Meanwhile, we discuss stability and convergence of the method in the sense of ?∞and give some stability and convergence conditions which are more broad, and we also do an experiment to test the feasibility of the method.Next, based on the discussion of numerical solution for linear fractional equation, we further briefly discuss the numerical method of nonlinear fractional equation—time-fractional KdV equation, and show that finite difference method is applicable to nonlinear fractional equation as well by numerical simulation, and the numerical results have great agreement with realistic problem.Lastly, we simply do some conclusion for the numerical method discussed above, and point that the method has traits of convenient and efficient. The theory issues to be perfected have been proposed, and we anticipate better foreground.