小波伽辽金方法应用于变系数波动方程

我们考虑问题K（x）uxx=ua.0＜X〈1，t≥0，其中K（x）≥a≥0，u（0，t）=g，ix（0，t）=0．这是一个不适当的方程，因为当解存在时在边界g上一个小的扰动将对它的解造成很大的改变．我们考虑存在解u（x，·）∈L^2（R）用小波伽辽金方法和Meyer多分辨分析去滤掉高频部分，从而在尺度空间Vj上得到适定的近似解．我们也可以得到问题的准确解与它在Vj上的正交投影之间的误差估计．

A Wavelet Galerkin Method Applied to Wave Equations with Variable Coefficients

We consider the problem K(x)u tt=u tt,0＜x＜1,t≥0, where K(x) is bounded below by a positive constant.The solution on the boundary x=0 is a known function g and ux(0,t)=0. This is an ill-posed problem in the sense that a small disturbance on the boundary specification g can produce a big alteration on its solution,if it exists.We consider the existence of a solution u(x,·) ∈L2(R) and we use a wavelet Galerkin method with the Meyer multi-resolution analysis,to filter away the high-frequencies and to obtain well-posed approximating problems in the scaling spaces Vj.We also derive an estimate for the difference between the exact solution of the problem and the orthogonal projection onto Vj.