Finite Difference Method For Time-fractional Nonlinear Diffusion Equation With Second Order Time Accuracy

In recent years,fractional order differential equation and the fractional order derivative have been more and more widely used in areas such as mathe-matics, science, and engineering.Now,people has already used it in Viscoelastic material, the signal processing, hydromechanics, biology and many other field-s.More and more evidences show that system models based on fractional order differential equation are more realistic than models based on integer order d-ifferential equation in real problems.That’s the reason why more and more people has realized the importance of fractional order differential equation.At the same time, the research of stability and convergence of fractional order differential equation,which is an important branch of theoretical studying in fractional order differential equation,gets attentions of researchers and has al-ready had some wonderful achievements.This paper studies the finite difference method which having second-order time accuracy in the time fractional nonlinear diffusion equation.We derive fi-nite difference scheme of the equation and reach the two order precision in the time level by using the superconvergence of fractional derivative of traditional Griinwald-Letnikov equation in special point (xi, tk-a/2).We also get the second order finite difference scheme and fourth-order compact difference scheme in space level.We have established the finite difference and the compact scheme of one-dimensional and two-dimensional time fractional nonlinear diffusion e-quation respectively.Theoretical analysis are given.At last,we give a numerical example to verify the feasibility of this scheme.In chapter 1,this paper introduces the development of fractional order calculus as well as background and significants of the research in fractional order differential equation.The definition and properties of the fractional order derivative are also put forward.Core contents is in chapter 2 to chapter 4.First,we introduce the finite difference scheme and its compact difference scheme of one-dimensional and two-dimensional diffusion equation.These schemes are got by using supercon-vergence of fractional derivative in some special points.It’s Riemann-Liouville time fractional derivative in this paper.Second,we use the energy method to prove the stability and convergence of the finite difference scheme after we get it.We discuss the second order finite difference scheme for the one dimen-sional time fractional nonlinear diffusion equation in chapter 2. At the same time, theoretical analysis and specific example is also given.In chapter 3,we introduce the compact difference schemes of one-dimensional diffusion equa-tion.The finite difference scheme of two-dimensional time fractional nonlinear diffusion equation and theoretical analysis are given in chapter 4.Chapter 5 is the conclusion of this paper.