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A homogenization method to solve inverse Cauchy–Stefan problems for recovering non-smooth moving boundary, heat flux and initial value
Applied Mathematics in Science and Engineering ( IF 1.3 ) Pub Date : 2021-07-08 , DOI: 10.1080/17415977.2021.1949591
Chein-Shan Liu, Jiang-Ren Chang

In the paper, we solve two Stefan problems. The first problem recovers an unknown moving boundary by specifying the Cauchy boundary conditions on a fixed left-end. The second problem finds a time-dependent heat flux on the left-end, such that a desired moving boundary can be achieved. Then, we solve an inverse Cauchy-Stefan problem, using the over-specified Cauchy boundary conditions on a given moving boundary to recover the solution. Resorting on a homogenization function method, we recast these problems into the ones having homogeneous boundary and initial conditions. Consequently, the approximate solution is obtained by solving a linear system obtained from the collocation method in a reduced domain. For the first Stefan problem the moving boundary can be determined accurately, after solving a nonlinear equation at each discretized time. For the second Stefan problem, we can obtain the required boundary heat flux without needing of iteration. Numerical examples, including non-smooth ones, confirm that the novel methods are simple and robust against large noise. Moreover, the Stefan and inverse Cauchy-Stefan problems are solved without initial conditions.



中文翻译:

求解非光滑运动边界、热通量和初始值的逆 Cauchy-Stefan 问题的均质化方法

在论文中,我们解决了两个 Stefan 问题。第一个问题通过在固定的左端指定柯西边界条件来恢复未知的移动边界。第二个问题在左端找到一个与时间相关的热通量,从而可以实现所需的移动边界。然后,我们解决了一个逆 Cauchy-Stefan 问题,在给定的移动边界上使用过度指定的 Cauchy 边界条件来恢复解。借助同质化函数方法,我们将这些问题重铸为具有同质边界和初始条件的问题。因此,通过在缩减域中求解从搭配方法获得的线性系统来获得近似解。对于第一个 Stefan 问题,在每个离散时间求解非线性方程后,可以准确地确定移动边界。对于第二个 Stefan 问题,我们无需迭代即可获得所需的边界热通量。数值例子,包括非光滑例子,证实了新方法简单且对大噪声具有鲁棒性。此外,Stefan 和逆 Cauchy-Stefan 问题在没有初始条件的情况下求解。

更新日期:2021-07-08
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