Numerical Algorithms Of Fractional Order Equations With Nonlocal Boundary Conditions

The non-locality of fractional differential equation can well describe some complex phenomena,so mathematical models based on fractional order theory can be established in many fields such as biology and physics.The research on fractional differential equations in recent years shows that it is of great practical significance to study the numerical algorithm for solving fractional partial differential equations.The research of this paper is to establish a numerical algorithm to obtain the approximate solutions of time fractional partial differential equations with nonlocal boundary conditions.In this paper,the existence of solutions of linear fractional-order equations is analyzed,a set of substrates for the reproducing kernel space is constructed by analyzing two nonlocal boundary conditions.Then,in order to solve the complexity of fractional integral involved in calculation,this paper uses piecewise parabolic interpolation to discrete the integral time variable,thus the equation to be solved is transformed into a system of linear ordinary differential equations with only spatial variable,and the ? approximate solution of the equations is further constructed.Finally,the improved minimum residual method is used to solve the system of linear equations,and the stability of the method is obtained by analyzing the system of normal equations.Numerical examples show the effectiveness and feasibility of the method.Nonlinear model is always the focus and difficult point of research.Therefore,based on the analysis of the linear model,this paper further studies the nonlinear model with nonlocal boundary.First,by understanding the relevant reproducing kernel space theory,the paper selects suitable two dimensional reproducing kernel space and constructs orthonormal basis of the space.Then,in order to deal with the nonlinear terms of the equation,the Newton iteration method is introduced.Due to the local convergence of Newtonian iteration method,the selection of initial iteration value is particularly important.This paper constructs the initial iteration value through two nonlocal boundary conditions to ensure the convergence of algorithm iteration process.By the above process,the nonlinear equations are transformed into a series of linear equations.Finally,the ? approximate solution of the equations is constructed,and the stability of the method is proved.Several numerical examples are selected to verify the validity and feasibility of the method.