Numerical Solutions For The Fractional Partial Differential Equations

In recent years, The numerical methods of the fractional partial differential equation has been one of the most popular and challengeable areas of differential equations, the value of it for each domain also attract many researchers attention. In this paper, we discuss the implicit difference approximation for the time-fractional order reaction-diffusion equation and the space-time fractional partial differential equation respectively, and the corresponding theoretical analysis about convergence and stability is presented.The main ideas of this article comes from the study of Sun Zhizhong?Yu Qiang?Liu Fawang and the reference [50]. In this paper, we make a corresponding adjustment about the literature [22]'s equation and then obtain the equation is studied in the chapter two and three. The following we get the corresponding difference scheme by means of the study of Sun Zhizhong about the fractional order derivative, and these difference format order of convergence is higher than the result of the study of Yu Qiang and Liu Fawang which is also an advantage of this article.The fourth chapter, we discuss the finite difference scheme of the space-time fractional partial differential equation. The main train of thought of the fourth chapter is derived from the literature [26] about the difference scheme of fractional order derivative and the literature [50] about the discrete fractional Laplace operator. And on the basis of this we have obtained the finite difference scheme and also presents a corresponding analysis of the convergence and stability.