Two Classes Of Numerical Methods For Solving Fractional Integro-differential Equations

In recent years,when dealing with complex practical problems,compared with integer derivative theory,the global correlation of fractional derivative theory makes the model established by fractional equations more accurate to simulate the objective world,which are used as research models in many questions in physics,chemistry and other disciplines.So the fractional derivative theory is widely concerned by scholars.However,in the research of that,it is generally difficult to find the exact solution of fractional equations.So it becomes particularly important to solve them numerically.Therefore,there are many numerical methods for solving fractional integro-differential equations,such as collocation method,operation matrix method,reproducing kernel method,etc.In this paper,two numerical methods are used to solve fractional integro-differential equations,fractional weighted reproducing kernel method and minimum residual method,respectively.In Chapter 1,the background of fractional integro-differential equations and the research status of numerical methods are introduced,and the preliminary knowledge is briefly described.In Chapter 2,a fractional weighted reproducing kernel space is constructed,and a class of fractional integro-differential equations with weak singular kernels are solved by using the fractional weighted reproducing kernels in this space.Firstly,the series form of the exact solution of the equation is given.Secondly,by finite truncation of the series form of the exact solution of the equation,a numerical method for approximating the original equation is constructed,and the convergence of the numerical method is analyzed.Finally,numerical examples are given to verify the correctness of the theory and the effectiveness of the numerical method.In Chapter 3,firstly,a set of orthonormal Legendre multiwavelet basis in 2L[0,1] is transformed into a set of orthonormal basis in 12W[0,1] by integral,and then the minimum residual method is established.Secondly,a class of multi-order fractional integro-differential equations is approximated by using the modified multiwavelet basis combined with the minimum residual method,and the convergence and stability of the numerical method are analyzed in detail.Finally,numerical examples are given to verify the effectiveness and stability of the minimum residual method.