一维热传导问题时变边界上热通量重构问题

具有Neumann边界条件的抛物方程的初边值问题是偏微分方程研究领域的一类经典问题.正问题是由已知的边界条件和初始条件来求区域温度场的问题.如果边界条件不足,但给出了区域内部的一些额外信息,这样便构成了一类热通量重构的反问题.本文讨论了一维热传导问题时动边界上的热通量重构问题,借助于位势理论方法,引入密度函数,将反问题本质上转化为一类关于密度函数的具有弱奇性核的第一类Volterra积分方程,采用了Tikhonov正则化,在正则化参数的选取上采用了后验的模型函数方法,数值结果验证了反演方法的有效性.

AN INVERSE PROBLEM FOR HEAT FLUX RECONSTRUCTION WITH REGULARIZATION

The initial boundary value problem for parabolic equation with Neumann boundary condition is a kind of classical problem in partial differential equations. The direct problem is to solve the temperature field from given initial and boundary conditions. If the boundary conditions are not specified completely, but with some additional condition given at the interior point of the domain, then it formulates an inverse problem for reconstructing the boundary heat flux, which is ill-posed. In general, the classical solution to this problem may not exist, also the solution may not depend continuously on the input data. So some regularizing method should be applied to get a stable approximate solution. In this paper, we consider a heat inverse problem with moving boundary in 1-dimension. By introducing a density function for potential method, this problem is essentially converted into solving an integral equation of the first kind with respect to the density function, which can be solved by the Tikhonov regularizing method. Model function method is used to determine the regularizing parameter. Numerical results are presented to support our inversion schemes.