Wavelet-Galerkin方法在微分方程中的应用

Wavelet-Galerkin Method for Differential Equation

运用小波理论,针对某一类变系数微分方程,首次将Littlewood-paley小波引入到变系数微分方程求解中,得到了Littlewood-paky小波ψ的尺度函数(?),构造了L2[0,1]中的正交小波基ψj,kfold,证明了该正交小波基满足方程的初始条件.运用Galerkin方法求出了方程在子空间中的逼近解,得出了变系数微分方程的准确解.拓宽了小波理论的适用范围,并为微分方程的求解问题提供了新的理论空间.

Using the theory of the wavelet, it solves the differential equation originally with the Littlewood - Palery wavelet bases. First it gets the scaling function (?) of the Littlewood - Paley wavelet ψ. At the same time, it proves the constructed orthonormal wavelet bases ψj,kfold for L2[0, 1] satisfying the boundary conditions of the equation. Using wavelets in conjunction with the Galerkin method, we look for the approximation uj to the actaul solution u on the subspace Vj. So we can get the actual solution in the end. It not only widens the space of the wavelet, but also provides a new view for solving the differential equation.