小波伽辽金方法应用于变系数波动方程(英文)

A Wavelet Galerkin Method Applied to Wave Equations with Variable Coefficients

我们考虑问题K(x)uxx=utt,0<x <1,t≥0,其中K(x)≥α≥0,u(0,t) =g,ux(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,·) ∈L2(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.

We consider the problem K（x）ua = ua, 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,·） ∈ L^2 （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 V~ . We also derive an estimate for the difference between the exact solution of the problem and the orthogonal projection onto Vj .