The Numerical Solution Of Nonlinear Fractional Volterra Integro-differential Equation By Wavelet

Fractional calculus theory and models are widely applied in various fields,such as the phenomena of heat conduction, fractional order controller settings, population model, which makes the relevant research on fractional calculus has been booming. The behavior of many systems can be described by using fractional integro-differential equtions . It is difficult to find fractional equations analytical solutions, so there has been significant interest in developing numerical schemes for the solution of this kind equations.In this paper we mainly study numerical solution of several categories fractional integro-differential equtions by wavelet. In Chapter 1, the research significance and current status domestic and foreign are briefly introduced. In Chapter 2, one-dimensional Legendre wavelets operational matrix of fractional integration and two-dimensional matrix of operational matrix of fractional integration for Haar wavelets, Legendre wavelet are deduced. In Chapter 3, existences of a unique solution for nonlinear fractional second kind Volterra integral equations is proved. Application of One-dimensional Legendre wavelets fractional operators matrix method is applied to solve under studying problem. Meanwhile, error analysis is given, when exact solution is unknown. In Chapter 4, existences of a unique solution for two-dimensional nonlinear fractional Volterra integral equations is proved. The original equation is solved by two-dimensional Haar wavelets and two-dimensional Legendre wavelets. The obtained numerical results indicate that two-dimensional Legendre wavelets method is more efficient and accurate than two-dimensional Haar wavelets for this kind equations. In Chapter 5, fractional order Volterra integro-differential equations, nonlinear singular fractional Volterra integro-differential equations and systems of nonlinear singular fractional Volterra integro-differential equations are solved by one-dimensional Legendre wavelets fractional operators matrix. Some examples are demonstrated the efficiency of the proposed method. In Chapter 6, the work done are summarized, and the future outlook of the work is proposed.