A homogenization method to solve inverse Cauchy–Stefan problems for recovering non-smooth moving boundary, heat flux and initial value. Issue 13 (23rd December 2021)

• Record Type: Journal Article
• Authors: Liu, Chein-Shan; Chang, Jiang-Ren
• Is Part Of: Inverse problems in science and engineering. Volume 29:Issue 13(2021)
• Page Start: 2772
• Page End: 2803
• Publication Date: 2021-12-23
• Subjects: Inverse Cauchy–Stefan problems -- moving boundary identification -- heat flux identification -- homogenization function -- homogenization method -- space-time boundary shape function method; Stefan problem -- Inverse Cauchy–Stefan problems -- Homogenization function; Engineering mathematics -- Periodicals; Inverse problems (Differential equations) -- Periodicals; 620.001515357
• DOI: 10.1080/17415977.2021.1949591
• Languages: English
• ISSNs: 1741-5977

Abstract

Abstract : 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.