The Research Of Wavelet Numerical Method For Supersingular Integrals And Fractional Defferential Equations

Wavelet analysis is a new subject developed in recent decades, it is a developing ofFourier analysis. It has been played important roles in all kinds of numerical calculation.Approximate values of supersingular integrals and numerical solutions of fractionaldifferential equations are always important topics in recent years. In this paper, we mainlystudy the applications of wavelet base functions in the approximate values ofsupersingular integrals and the numerical solutions of fractional order differentialequations. Our purpose is to reduced the singularity of supersingular integrals, todecreased the computation of supersingular integrals, and to improved the computation byusing the properties of wavelet functions. We can transform the problem of solve thenumerical solution of fractional differential equations into solving the algebra equations,and the algebra equations can be solved easily by Matlab. Meanwhile we make the furtheranalysis to the error estimate of the numerical solutions of fractional differential equationsobtained by wavelet method.Firstly, in view of solving the approximate values of supersingular integrals ofarbitrary positive integer order on the interval. In order to simplify the problem and getmore precise approximate values, we take advantage of Legendre wavelet and itsproperties to convert the supersingular integrals on the interval into the supersingularintegrals at the endpoint of interval. The convergence of this approach is also discussed inconvergence analysis.Secondly, making full use of the orthogonality of Chebyshev wavelet, thecomputability of the wavelet functions and the important formula in generalized function,we have solved effectively the supersingular integrals on circle and the singular integralswith Hilbert kernel.Thirdly, we adapt Haar wavelet to further study a class of fractional differentialequations with variable coefficients, the corresponding error analysis is given and theconvergence of the method is also obtained at the same time. Numerical examples provethat the theory is correct and the method is effective.Finally, a new Haar wavelet operational matrix of fractional order integration isderived by combining the definition of fractional integral with the idea of operational matrix. Also, the Haar wavelet operational matrix of the fractional order differentiation isobtained. We have discussed the numerical method of a class of fractional partialdifferential equations by using the obtained operational matrix, the error estimation of thismethod is also given in error analysis.