Numerical Algorithms For A Class Of Fractional Order Singularly Perturbed Problems

Fractional differential equations have importantly used in physics,hydrology,signal processing,quantum mechanics,cybemetics and other fields,the study of acquiring solution has been a hot focus in fractional differential equations.Many achievements have been made in the study of fractional differential equations,but there are still some problems to be solved,like the fractional differential equation which with singularly perturbed parameters.In this paper,we consider a class of singularly perturbed fractional differential equations,and construct a finite difference scheme to acquire solutions,then study the convergence of the numerical method.The theoretical results are illustrated by numerical examples at last.In Chapter 1,we introduce the backgrounds and overviews of fractional differential equations,and describe the current research situation of this kind of problem,at the same time,the content structure of the article is explained.In Chapter 2,we introduce some basic knowledge needed in the research process,including the definition of several kinds of fractional derivatives and some kinds of meshes used in constructing differential equation numerical schemes.In Chapter 3,the discrete schemes for a class of fractional order singularly perturbed differential problems are studied.First,the time derivative is semi-discretized by using the L1 approximation algorithm based on the uniform grid in time,then we select the Vulanovicbakhvalov grid in space,and derive the fully discrete expression of the problem(1.1)by upwind finite difference method.In this paper,we give the concrete calculation process and discuss the validity of the numerical algorithm,and prove the stability and convergence of the method.Finally,the theoretical results are verified by numerical examples.The Chapter 4 is an extension of the Chapter 3.The problems we consider in this chapter are similar to those in the third chapter.The main innovation is that the grid algorithms proposed in this discretization process are different.First,the fractional derivative of Caputo type is discretized in the time direction by using the L 1 approximation method,and the approximation effect of the numerical solution in the process is discussed.Then,a kind of adaptive algorithm is designed based on the principle of equal distribution in the spatial direction,combining with the semi-discrete expression derived before,using the classical difference quotient formula to discretize the spatial derivative on the adaptive grid,then the stability and local truncation error of the numerical method are analyzed,and the convergence of the method is proved in this paper.At last,we summarize the results and give the future work of the fractional singular perturbation differential equation.