Method For Solving PDE Numerically By Using Delta-sequence

The traditional methods for numerically solving PDE include local methods (such as finite difference or finite element method) and global methods (such as spectrum methods). Recently, Wei and Hoffmann et.al. have proposed a new method called Discrete Singular Convolution method, which both has the high accuracy of global methods and flexibility of local methods. Different DSC kernels and relative methods have been successfully used in many realms such as hydrodynamics and beam analysis. DSC kernel can be regard as a kind of derivative operator but it also has strong tie to wavelet theory.Wavelet theory is the perfect combination of functional analysis,harmonic analysis,time-frequency analysis,numerical analysis,theory of approximations and generalized functional theory. It has the character of widely applicable and strongly theoretics. Because of including the conception such as measurement division,space of approximations and interpolation cardinal function, which equal with the conception of numerical solution. Therefore, this article tries to combine with the context of the theory of wavelet and the algorithm of PDE to obtain delta-sequence, which has the property of fast attenuation. This can be the supplement and advancement of the algorithm of DSC.This thesis will be expanded from two aspects: first, delta-sequence obtained by cubic spline interpolation cardinal function showed the properties of symmetry, Riesz basis and interpolation. Non linear convection diffusion equation (Burgers′equation) was used as an example. Then we modified this delta-sequence to improve the attenuation, the same example was used in numerical application. Second, a new delta-sequence with the properties of interpolation, symmetry and compact support is constructed based on the Daubechies wavelet function. The computational accuracy is tested by convetion-dominated equation. Using delta-sequence to solve Two-Point Boundary Problem is simplicity and accuracy.