Wavelet Numerical Method For Solving Two Kinds Of Nonlinear Fractional Differential Equations

In practical engineering and physical sciences, many real phenomena found can be needed by fractional integral or fractional differential to describe, so how to solve these fractional integral and differential equations becomes the key to solve these problems. In recent years, wavelet analysis method is used to solve the fractional order nonlinear differential equations become a new numerical method, which has an important role in the field of numerical computation. Therefore, this paper mainly studies the numerical solution of nonlinear fractional order differential equations with three kinds of wavelets. Using wavelet own characteristics and Block Pulse functions related properties derive the corresponding fractional integral operator matrix, thus turning the original fractional order calculus equation numerical solution into solution of solving algebraic equations, thereby greatly reducing the computation of solving process, and it’s convenient to use the matlab program to solve.Firstly, in order to solve the numerical solution of nonlinear fractional differential equations, the paper puts forward a new method that combining Chebyshev polynomial and the variational iteration method is applied to solve the numerical solution of nonlinear fractional differential equations, by choosing appropriate initial approximation, to achieve better approximation effect of non-homogeneous term and nonlinear term, thus reduce the computational work. The algorithm can reduce the amount of computation, improve accuracy and effectively deal with the difficulties of computing complex integral. The numerical examples verify the given method is effective and practical.Secondly, the paper mainly research methods of solving the Fredholm equation by using Sine-cosine wavelets, combined with the properties of block pulse functions, derived from the fractional integral operator matrix, Sine-cosine wavelet fractional integral operator matrix can be used to simplify the integral differential equations into algebraic equations. Finally examples prove the feasibility and validity of the method.Finally, using another kind of wavelet function-Haar wavelet to solve the problem of numerical solution of nonlinear Fredholm integro-differential equation of fractional order, obtained the fractional integral operator matrix of Haar wavelet to solve the problem of nonlinear fractional order calculus, and it’s verified by numerical examples.