The Application of Wavelet Analysis in the Solving of Singular Integrals and Fractional Order Differential Equations 

Author  GengWanHai 
Tutor  ChenYiMing 
School  Yanshan University 
Course  Computational Mathematics 
Keywords  Fractional differential equations Singular integral Haar wavelet Numerical solution Legendre operator matrix 
CLC  O241.8 
Type  Master's thesis 
Year  2012 
Downloads  62 
Quotes  0 
The wavelet is a kind of function which meets the specific nature. Its advantagelies in the smooth and local compactly supported sex in essence. Therefore, it is betterthan the traditional Fourier analysis with a more detailed video analysis capability tobetter deal with the problem of local existence of singularities. Nonlinear fractionaldifferential equation and its solution of the nonlinear science research as acuttingedge research topics and hot issues, with great challenge.Firstly, this paper introduces the development history and current situation offractional computation, and some work at present. While the paper also gives thehistory and development status of wavelet analysis. Then giving some priorknowledge about fractional calculation the definition and properties of wavelet.Secondly, in this work, we use the wavelet to cope with the singular point in thesingular integral, making use of the operational matrix of Haar wavelet to gives amethod which solves the approximation of the definite integral and singular integral,this paper also gives the comparison of the numerical and the exact solution in eachinterval though mapping in the integration process. Then we can apply the wavelet tosolve the nonlinear differential equations of fractional order, using the orthogonally,tight support, and rapid attenuation characteristic of Haar wavelet, transforming thesolution of the equation upon the original coordinate system into wavelet system. Theequation was shown a simple sparse form by a few wavelet functions appropriately.Making the matrix multiplication in operator calculation translated into sparse matrixmultiplication, we can also give a computing format which solves the numericalsolution of the timefractional order partial differential equations.Finally, this paper may obtain the numerical solution of fractional differentialequations by using Legendre operator matrix. The original problem which solves thefractional differential equations is translated into the problem which solves algebraicequations and computation became convenient.