Adomian Decomposition Method And Legendre Wavelets Solving Nonlinear Fractional Integro-Differential Equations

Adomian decomposition method which has a good convergence is applied for thesolution in the form of infinite series and the solving process is simple. Therefore, after themethod put forward since the1980s, it is widely used to solve all kinds of linear andnonlinear integro-differential equations. Wavelet analysis is a subject which wereproposed and developed in the20th century, its mainly way to solve the problem is toconstruct wavelet basis functions in a particular function space. This method is widelyused in image processing, digital signal analysis, etc. Thesis applies the Adomiandecomposition method and Legendre wavelets to solve the numerical solution of threekinds of nonlinear fractional order integro-differential equations. The purpose of theresearch is to use the characteristics of these methods to transform the solution of thenonlinear fractional order integro-differential equation into a convenient solution form,and then use Matlab software programming to solve, and give the numerical solution tothe desires of convergence analysis and error estimation.Firstly, application the Adomian polynomial to decompose the nonlinear term in theequation, so we can put the solution of equation into the form of infinite series, which notonly can reduce the amount of calculation and calculation time effectively, but also canimprove the convergence speed and reduce the calculate error. At the sime time, we giveconvergence analysis and error estimates of the numerical solution of Adomiandecomposition method.Secondly, we use the advantage of itself characteristics of the Adomiandecomposition method, combining the Adomian polynomial with definition of fractionalcalculus effectively, to obtain the numerical solution of the equation and the maximumabsolute truncation errors. And through numerical example illustrates, the desires ofnumerical solution is able to converge to exact solution.Finally, for Legendre wavelets, we mainly use the local characteristics change of thewavelet to transform the original equations into algebraic equations. To change thecomplex integro-differential equation into a simple matrix equation, not only reduces thedifficulty in solving the problem, but also improves the usability of the algorithm.