不适定非齐次抛物方程的正则小波解

Jinru Wang, Yuan Zhou
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引用次数: 0

摘要

考虑非齐次问题uxx(x, t) = ut(x, t) + f(x, t), 0≤x <;1, t≥0,其中柯西数据g(t)在x = 1时给出。这是一个不适定的问题,因为边界g(t)上的一个小扰动可以对其解产生很大的改变(如果它存在的话)。在本文中,我们将定义Meyer小波解来获得标度空间Vj中的适定解。我们还将证明在某些条件下,这个正则解收敛于精确解。在以往的论文中,大多数关于误差估计的理论结果都是关于齐次方程的,即f(x, t)≡0。
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Regularized Wavelet Solutions for Ill-posed Nonhomogeneous Parabolic Equations
We consider the nonhomogeneous problem uxx(x, t) = ut(x, t) + f(x, t), 0 ≤ x <; 1, t ≥ 0, where the Cauchy data g(t) is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g(t) can produce a big alteration on its solution (if it exists). In this paper, we shall define a Meyer wavelet solution to obtain well-posed solution in the scaling space Vj. We shall also show that under certain conditions this regularized solution is convergent to the exact solution. In the previous papers, most of the theoretical results concerning the error estimate are about the homogeneous equation, i.e., f(x, t) ≡ 0.
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