Numerical Approximation Of R-L Fractional Integral Equation And Fractional Integro-Differential Equation

In real life, most of the materials is neither an ideal solid nor perfect fluid, it is the situation that between the ideal solid and perfect fluid. Using fractional calculus characterize these materials’ con-stitutive more precisely, to better meet the actual needs. As the important supplement of the Calculus, fractional calculus’ unique properties make it apply in the population growth model, the PID control theory, a viscoelastic material, chaotic phenomena and so on. these problems are transformed into large most of fractional differential equations, integral equations and fractional fractional integral differential equations by modeling. Now, for the study of the numerical solution of such equations is still in its infancy. Therefore, this paper will give several kind of numerical solutions of fractional integral equations and fractional differential integral equations.This paper is composed of the six chapters.In Chapter1, Statement of research background, status and significance of fractional integral equations and fractional integro-differential equations is given briefly.In Chapter2, Definition, important properties and the basic functional is introduced.In Chapter3, Taylor series and reversibility of fractional integral operators is used for solving a class of R-L fractional integral equations.In Chapter4, Series approximation, Picard and Adomian decomposition for solving linear and nonlinear fractional second kind Volterra integral equations. Since the analytic solution of such equations is not easy to obtain, so the uniqueness and convergence of solution were analyzed, and make integer order equation for example to verify the effectiveness of the algorithm.In Chapter5, Selecting Taylor series, Bessel functions Respectively is to solving a class of frac-tional differential-integro equations, numerical examples show the effectiveness of approximation methods.In Chapter6, The main work done of this paper is summarized and the future outlook of the work is proposed.