The Numerical Solution Of Fractional Integro-differential Equations By Bernoulli Wavelet

Signal processing?fluid mechanics?control theory and many other phenomenas can be de-scribed by fractional integro-differential equation.But to find the analytical solution of this equation is very difficult,so the research scholars focus on the numerical solution of it.Nowadays,there are a lot of numerical methods to solve the fractional integro-differential equations,such as the finite element method,homotopy perturbation method,Adomain decomposition method.However,the wavelet method is applied to solve fractional integro-differential equations is relatively few.There-fore,this paper consider using Bernoulli wavelet methods to solve the numerical solution of several classes of fractional integro-differential equations.This paper is splited into six chapters.In chapter one,the research status and research significance of fractional calculus and the status of its numerical solution which solved by using different methods are summarized.In chapter two,the basic theory of fractional calculus and Bernoulli wavelet are briefly in-troduced and the multiplical integral operator matrix and the fractional integral operator matrix of Bernoulli wavelet are derived.In chapter three,the nonlinear fractional order Fredholm integro-differential equations?linear and nonlinear system of fractional Fredholm integro-differential equations are solved by using frac-tional integral operator matrix of Bernoulli wavelet and proved the existence and uniqueness of their solutions.Also,the convergence analysis of fractional integro-differential equation is given.In chapter four,the linear and nonlinear Fractional Fredholm-Volterra integro-differential Equa-tions and weakly singular fractional integro-differential equations are solved by using the fractional integral operator matrix of Bernoulli wavelet.Meanwhile,some numerical examples shows the fea-sibility of this method on solving these equations.In chapter five,we use the fractional integral operator matrix of Bernoulli wavelet for solve non-linear fractional order Volterra integro-differential equations and systems of fractional order Volterra integro-differential equations whose order is uncertain and satisfy some initial conditions.Mean-while,its convergence is proved and many numerical examples are given to illustrate the validity and accuracy of the method.In chapter six,the work of this paper is summarized and the further research is put forward.