Research On Numerical Algorithm For Fractional Integro-differential Equations

In recent decades,the theory of fractional calculus has attracted attention and extensive attention from the mathematics community under the influence of a strong application background.It has gradually become a new and active research field.At present,fractional integro-differential equations have been widely used in many fields such as fractional physics,chaos and turbulence,polymer chain melting and stochastic processes,and some promising research results have been obtained.The study of fractional integro-differential equations has very important theoretical significance and practical application value.Especially the problem of fractional integro-differential equations abstracted from practical problems has become a hot topic for many mathematicians.However,it is difficult to obtain analytical solutions for most fractional integro-differential equations.For a long time,scholars have been devoted to the study of their numerical solutions.Although scholars have proposed some numerical methods for solving fractional integro-differential equations,the theoretical system still needs further improvement.Therefore,the numerical solution of nonlinear fractional integro-differential equations and multidimensional Fredholm fractional integral equations is discussed in this paper.This paper is mainly based on the generalized hat functions and the modification of hat functions of fractional integro-differential equations and multi-dimensional fractional fredhlom integral equations of high-precision algorithm.First,for the nonlinear fractional integral differential equation,the integral operator matrix of the fractional integral differential equation is derived by using the generalized hat functions.The numerical solution is transformed into an algebraic equation group,and the numerical solution is obtained.Then,the error analysis is carried out by numerical examples.And compare it with the numerical results of CAS wavelet method,the numerical results show that the proposed method is effective and feasible.Second,multi-dimensional fractional fredhlom integral equations.The existence and uniqueness of the solution of the equation is proved,and then briefly explain the basic theory of collocation methods.The collocation methods based on hat functions are used to solve multi-dimensional fractional fredhlom integral equations.The integral equations can be reduced to a system of algebraic equations by using modification of hat functions and collocation methods.According to the theories of projection operator,the convergence theories and error analysis can be established.The numerical examples show that the algorithm is high accuracy.