波动方程的小波解法

The wavelet solution to the wave equations

探讨应用 Wavelet-Galerkin方法求解一维波动方程的初边值问题 ,通过修改边界上的小波函数 ,得到满足齐次边界条件的有限区域的小波基 ,用 Wavelet－ Galerkin方法离散微分方程后 ,得到一个确定小波系数的线性方程组 ,此方程组的系数矩阵在一维情况下是一个带状矩阵 ,且其中还有许多小的元素 ,其逆矩阵有类似的性质。数值实验表明，小波为求解微分方程提供了一个新的强有力的工具 ,用它来求解方程得到的小波近似解能很好地满足各种边界条件，且解的精度可以通过增加小波函数或增加尺度而得到提高。

This article mainly discusses the Daubechies wavelet solution to the problems of the initial boundary value of the wave equations in one dimensional space by using the Wavelet Galerkin technique. The numerical experiment shows that wavelet provides a powerful new tool for solving the differential equations; and the approximate wavelet solutions can satisfactorily meet all kinds of boundary conditions. The accuracy of the solution can be improved by increasing the wavelet function or the scale. Numerical results are also presented to illustrate the effectiveness of the new method.