Three Kinds Of Wavelet To Solve The Numerical Solution Of A Class Of Linear Fractional Order Differential Equations And Their Convergence Analysis

In recent years, solving the numerical solution of fractional differential equations is ahot issue, with the development of science and technology, more and more engineeringpractical problems need to be described by the fractional order differential equations.More than ten years, wavelet analysis developed from Fourier analysis is a new discipline,which plays a significant role in the field of numerical calculation. This paper mainlystudies three kinds of wavelets to solve the numerical solution of a class of linearfractional order differential equations. The purpose is to apply the wavelet basis functionand the fractional integral operator matrix to translate the problem of solving numericalsolution of fractional order differential equations into system of linear equations, therebyreducing the amount of calculation, also facilitate to solve the problem by the Matlabprogramming. At the same time, this paper presents the convergence analysis ofChebyshev wavelet and Haar wavelet method when solving the fractional orderdifferential equations.First of all, through the application of the orthogonality of Chebyshev wavelet andthe fractional integral operator matrix, to solve the numerical solution of linear fractionalorder differential equations. It introduces the process of solving the numerical solution offractional order differential equations in detail. It also presents the convergence analysis ofthis method, proved its correctness theoretically. The numerical examples verify the givenmethod is effective and practical.Second, with Haar wavelet to solve a class of linear fractional order differentialequations, through the Block Pulse function to find the integral operator matrix of thismethod, then will the target system of equations into algebraic equations, which is easy tosolve. In the cases of the original equations have exact solution and haven’t, the erroranalysis of the method is both presented. There numerical examples are given to prove thevalidity of the theoretical derivation and the effectiveness of the algorithm.Finally, using another kind of wavelet function--Sine-cosine wavelet to solve theproblem of numerical solution of fractional differential equations. By the use of thedefinition of fractional order derivative to get the fractional order integral operator matrix of Sine-cosine wavelet. Using the operator matrix to solve the numerical solution offractional differential equations. Numerical results show the feasibility of the algorithm.