Numerical Solution Of Integral Equations Via Waveles

Integral equations have been widely utilized in such many fields as Engineeringand Dynamics. Due to the difficulty in getting accurate solutions, how to get numericalsolutions or approximate solutions of integral equations has become one of the mosthot research pots in this sphere. Wavelet analysis is a new branch of modernMathematics, which inherits the good advantage of Fourier analysis, but avoids itsdisadvantage in doing local analysis of a signal. The virtues of wavelet analysis makeit significant in obtaining the numerical solutions of integral equations.Here, our main work is how to utilize wavelet analysis in integral equations andget the numerical solutions, especially, no any projection method, such as Collation orGalerkin underlying. The whole work will be discussed in the following three aspects.1. Introducing the background and development of wavelet analysis, mainly focu-sing on its application and research status in getting solutions of integral equations,specifically in the second kind of Volterra-Fredholm integral equations and Integro-differential equations.2. Presenting the definition of Block-Pulse functions (BPFs), exploring itsproperties and deducing the operational matrix of integration. Trying to illustrate themethod how to solve the integral equations and obtain their numerical solutions byBPFs. Then, doing error analysis when BPFs is approaching the equations. At last,analyzing the precision degree between the numerical solutions and the analyticalsolutions by means of figures and tables of the examples.3. Appling this method into resolving the Integro-differential equations, exploringits numerical method of this kind of equation.In this paper, we solved the numerical solutions of Volterra-Fredholm integralequations and Integro-differential equations via BPFs with no projection methodunderlying. The results showed that the approximation effect of the numericalsolutions and accurate solutions were good enough, and the accuracy degree got asbetter as the BPFs increased, which had verified that our method was feasible andreasonable. The study has a great sense to obtain the numerical solutions of others integral equations, such as fractional order integral equations, singular integralequations.