The Numerical Solution Of Fractional Integral Equation By The Rational Haar Wavelet

As the development of calculus theory, people have already put forward the concept of fractional calculus. Because of fractional calculus are used more and more widely in various fields and received a lot of scholars’attentions and research, fractional calculus applications in various fields, mainly through calculus equation. With respect to the integer-order calculus model, using fractional calculus of differential and integral equations not only very simple but also the results closer to the actual. Therefore, the solution of fractional calculus equations become a very important job. Wavelet meth-ods is one of the main methods for solving integral equations of fractional numerical solution. The Haar wavelet function is the earliest and most simple of orthogonal wavelet function be used in the analysis. On the basis of the Haar wavelet solution, by removing irrational numbers and introducing the dual integral base to get a new numerical method-numerical solution of rational Haar wavelet. It preserves all the properties of the Haar transform, but it can be effectively and more quickly to deal with the quasi-binary numbers problem. The transform is much faster than the Fourier, and it is even faster than the Walsh transform. So this paper considers solving several types of fractional integral equations by using rational Haar wavelet.This paper consists of five parts.In Chapter1, brief statement of article research purpose, significance and research status are given. In Chapter2, the different forms definition and operational properties of fractional calculus, the definition and nature of rational Haar wavelet are briefly introduced.In Chapter3, by use of the definition and nature of rational Haar wavelet, an operational matrix of fractional integration and product operator matrix are given, and then combination the colloca-tion method to solve the fractional Volterra integral equation of the second kind and weakly singular Volterra integral equation, a error estimate analysis is also obtained, finally, the numerical exam-ples demonstrate the effectiveness of this method, and the results are compared with other wavelet methods.In Chapter4, two types of fractional nonlinear integral equation are solved by the rational Haar wavelet, the existence and local attractivity of solutions of a quadratic Volterra-Hammerstein integral equation of fractional order on unbounded intervals. The results of numerical examples show that rational Haar wavelet method is better convergence than Haar wavelet and Legendre wavelet.In Chapter5, the main work done in the paper is summarized and the future outlook of the work is proposed.