Reproducing Kernel Galerkin Method For Some Fractional Differential Equations

The numerical solution of fractional differential equations is a topic concerned by many researchers all over the world.One important reason is that fractional differential equations may provide better mathematical explanations for many natural physical phenomena.Due to the non-locality of multi-term fractional differential equations,it has important theoretical research and wide application in many fields.On the basis of exploring the existence of solutions of fractional differential equations,this paper continues to deeply study the numerical simulation of three kinds of multi-term fractional order problems.In this paper,the existence and uniqueness of solutions of fractional differential equations are proved theoretically by using the fixed point theorem first.In addition,the reproducing kernel spline Galerkin method is used to study multi-term fractional differential equations.The reproducing kernel spline function is constructed as the basis function of the solution space.The approximate solution of the equation is established through the specific linear combination of the basis functions,and the exact solution is approximated by Galerkin method.Lastly,the error estimation and convergence analysis of the algorithm are carried out through the effective derivation process.Numerical examples show the accuracy and effectiveness of the method.The full text consists of the following five chapters.The first chapter briefly introduces the background and development of the problems to be studied,and puts forward the contents to be studied.The second chapter gives the preliminary knowledge needed to study the problem,and introduces the contents and properties of reproducing kernel,spline function,fractional differential operator and fixed point theorem so as to prepare for the next calculation and proof.In Chapter 3,the existence of fractional differential solutions of different orders is proved by Banach fixed point theorem and Krasnoselskii fixed point theorem,which paves the way for exploring its numerical solutions.In Chapter 4,the reproducing kernel spline Galerkin method is used to construct the numerical solutions of multi-term fractional differential equations with constant coe cients and quasilinear terms and fractional differential equations with variable coe cients and quasilinear terms.On this basis,the error estimation and convergence analysis of the algorithm are given.The stability and effectiveness of the algorithm are proved by several numerical examples,and the convergence rate is calculated.In Chapter 5,the variable order fractional differential equations are analyzed by using the reproducing kernel spline Galerkin method.The error estimation and convergence analysis are also demonstrated,and the corresponding numerical examples are calculated.