Research On Adaptive Grid Algorithms For Two Types Of Fractional Differential Equations

Fractional calculus has been widely used in recent years,and it has solved many problems that integer order calculus cannot solve well.In general,it is difficult to obtain the analytical solution of fractional differential equations.Therefore,it is of great significance to study the effective numerical methods of fractional differential equations.At present,for the study of fractional differential equations,almost all numerical methods are constructed on uniform meshes.Only a few literatures give the adaptive moving mesh algorithm of fractional differential equations and the corresponding theoretical analysis.Therefore,this paper will study the adaptive moving mesh method of fractional differential equations.The paper is composed of four chapters.In Chapter ?,the research background of fractional differential equations,research progress,and the main work of this dissertation and the definition of fractional derivatives are introduced.In Chapter ?,some special functions and properties used in fractional differential equations are introduced,and the Riemann–Liouville fractional derivative definition and Caputo fractional derivative definition are given.In Chapter ?,the adaptive moving grid method for the Caputo fractional order derivative initial value problem is investigated.First,the problem is transformed into a Volterra integral equation,the right rectangular formula is used to discretize the Volterra integral equation,and the corresponding prior error analysis is given.Then,an appropriate mesh control function is obtained by using local truncation error estimation.Finally,the numerical results verify the correctness of the theoretical analysis presented in this chapter.In Chapter ?,the adaptive moving grid method of Caputo fractional derivative initial value problem is studied.First,use Taylor expansion to linearize the initial value problem to obtain the prior information for the exact solution of the continuous problem.Then,the discrete format of this kind of problem is constructed,and the corresponding adaptive moving grid algorithm is designed.Finally,the numerical results verify the effectiveness of the algorithm.