Wavelet Numerical Method For Solving Differential And Integral Equations

Wavelet analysis is a new field of current mathematics, which has been rapidly developed and has a wide range of applications in numerical analysis, signal processing, graphics, quantum field and many other areas. Differential and integral equations and their numerical solutions are very important subjects of mathematical research, and come from their physical problems, and are used as an important mathematical tool in physical problem, electrostatics, elasticity, electromagnetic theory, radiation science and other subjects, so that mathematical problems can be reduced to solving the corresponding integral equations. Wavelets with regularity, vanishing moments, short sets and other features can make the simplified equation to form matrix, and then to be solved with a quick and efficient numerical method.In the first part of this paper, we introduce the background and development of wavelet analysis, and early researches on the current direction of wavelet applications, also introduce the organizational structure of this paper. Starting from the Fourier analysis, we introduce the basic theory of wavelet analysis, including the properties of wavelets, continuous wavelet transform, discrete wavelet transform, multiresolution analysis, scaling function and wavelet function and so on, which serve as a theoretical foundation of the paper.And then, we introduce the history of the differential and integral equations, clas-sification, numerical algorithms and applications of wavelets in solving differential and integral equations.Finally, the Haar wavelet collocation method are used to solve linear and nonlin-ear Fredholm integral equations, and Shannon wavelet are used to solve differential equations of initial boundary value problems. Experimental examples show the high-precision of numerical method of Haar wavelet and the effectivity of Shannon wavelet function approximation method.