Numerical Solution Of Integral Differential Equations Based On Wavelet Theory

Wavelet analysis, as one of the most exciting topics to emerge from mathematical research, has a wide range of engineering applications due to its time-frequency localization and multi-scale analysis property.Wavelet analysis is the development and perfection of the Fourier analysis.Since the development of the wavelet analysis is the basis to solve some practical problems, and then, it develops into a radioactive multi-disciplined theory, now it has become a hot field in the research internationally.Wavelet transforms complement the shortcomings of Fourier-based techniques because of their flexible time-frequency windows. Wavelets are widely applied in numerical analysis, signal processing,image processing and so on. Because of the high frequency component gradually refined using time domain or frequency domain sampling step, which can be focused on target any details. In this sense, it praised as mathematical microscope, it can be predicted in the future will become a science and technology work Often use an important mathematical tool.Wavelets are a new mathematical technique, arising in 1980' s.The wavelet analysis theory is one of the important branches of mathematics and is a research field developing rapidly in applied mathematics presently. It has abundant mathematical theories and comprehensive application, and is a powerful method and tool in engineering-it brings new ideas to many fields.It has drawn more and more mathematical researchers' attentions.Because the wavelets have the smooth and local compact property, compared with traditional finite element method and finite difference method,it is a more useful method for solving integral and differential equations .The construction of scaling function and wavelet is very important to the theory and application of wavelet analysis.This paper describes in detail the basis theory of wavelet. Firstly, the paper summarizes the characteristics of multi-resolution analysis(MRA) and elicits the essential characteristics of MRA, and then we give the simplest definition of MRA.As well-known, Integral(differential)equation(s) arise in variety fields of sci- ence and engineering technique and play a very important role in these fields.Methods of solving these equations thus become a key factor in such fields.For such equations, except some special cases, exact solutions are difficult to derived by analytical mehtods.As a result, numerical methods or approximate methods remain of much interest. In this paper, we study the application of wavelet analysis to integral and differential equations.This paper is composed of four chapters:The Chapter 1 presents the basic theory of wavelets analysis which includes the definition and properties of wavelets and wavelet transformation,the definition of the multi-resolution analysis and wavelet basis.Finally we introduce the development of integral and differential equations and research status.An improved numerical method based on the wavelet matrix transform method is introduced and analyzed for the Fredholm integral equations of the First kind in Chapter 2.The method combines the multi-wavelet basis to dis-cretize the integral equation with the Galerkin method,following by an iterative method for solving the resulting dense and nonsymmetric linear system.The computational complexity is found to be reduced without sacrificing much accuracy of the solution. Finally, two illustrative examples are included to demonstrate that our wavelets provide is stable.And it is shown that our algorithm yields very accurate results by less computational cost.Chapter 3 introduces the CAS wavelets and the integro-differential equation is approximated by the CAS wavelets.The CAS wavelet operational matrix P of integration is first presented and a general procedure to generate this matrix P is given. With the orthnormal and the operational matrix of integration P of the CAS wavelets, the integro-differential equation is reduced to a system of linear equations which can be solved with Newton iterative method.Finally, three illustrative examples are included to demonstrate that our wavelets provide is stable.And it is shown that our algorithm yields very accurate results by less computational cost.In Chapter 4, we solve nth-order integro-differential equations by changing the problem to a system of ordinary differential equations and using the varia- tional iteration method which we derive. Some examples are given and the results reveal that the method is very effective and simple compared with the Homotopy perturbation method...