Interpolating Wavelets Collocation Method For Hyperbolic Partial Differential Equation

Wavelet analysis is a new academic subject in recent years, which has been applied in many different fields. In 1990, J.Liandrat, P.Tchamitchian began to solve Bugers equation using a spatial wavelet approximation. Later many specialist and scholars made scientific researches on this new field, and got a few effective algorithms. The reason why wavelet can be used in solving partial differential equation is that wavelet can concentrate on local characteristics of a function in both spatial and frequency domain, and it can adaptively take different sample intervals according to different frequency parts, therefore it can decompose signals into different frequency parts, and can focus any details, which plays an important role in solving equations with sharp transitions and steep changes solution.This paper first presents the development of the wavelet analysis and some classic subjects such as Multiresolution analysis . Daubechies wavelets are also introducedSecondly, this paper presents the theory of wavelet interpolation. It mainly presents the conformation of interpolating wavelet function, interpolating wavelet multiresolution analysis, the interpolating wavelet function which bases on interpolating polynomial in dyadic points. Each wavelet is uniquely associated with one dyadic point, and points out that Daubechies autocorrlation function is one kind of interpolating wavelet function.Finally, following the theory of interpolation wavelet, this paper presents an adaptive algorithm for solving partial differential equation based on Daubechies wavelets. By using polynomial interpolation on dyadic points, an approximation of initial function is given. Then wavelet coefficients is thresholded and all the wavelets whose coefficients are larger than the given threshold are reserved, thus a good approximation of initial function is given with only fewer wavelet coefficients. Then we use collocation point method, and calculation of the space derivatives can be done using centered finite differences. The resulting system of ODEs can then be solved by the standard forth order Runge-Kutta method. In this way we get an approximation at next time step. Periodically we can get all the approximate analytic solution at all the multiple time steps.