Volume 3, Issue 4
Free Boundary Determination in Nonlinear Diffusion

M. S. Hussein, D. Lesnic & M. Ivanchov

East Asian J. Appl. Math., 3 (2013), pp. 295-310.

Published online: 2018-02

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  • Abstract

Free boundary problems with nonlinear diffusion occur in various applications, such as solidification over a mould with dissimilar nonlinear thermal properties and saturated or unsaturated absorption in the soil beneath a pond. In this article, we consider a novel inverse problem where a free boundary is determined from the mass/energy specification in a well-posed one-dimensional nonlinear diffusion problem, and a stability estimate is established. The problem is recast as a nonlinear least-squares minimisation problem, which is solved numerically using the lsqnonlin routine from the MATLAB toolbox. Accurate and stable numerical solutions are achieved. For noisy data, instability is manifest in the derivative of the moving free surface, but not in the free surface itself nor in the concentration or temperature.

  • Keywords

Nonlinear diffusion, free boundary problem, finite difference method.

  • AMS Subject Headings

35R35, 35K55, 49N45, 65M06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-3-295, author = {}, title = {Free Boundary Determination in Nonlinear Diffusion}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {3}, number = {4}, pages = {295--310}, abstract = {

Free boundary problems with nonlinear diffusion occur in various applications, such as solidification over a mould with dissimilar nonlinear thermal properties and saturated or unsaturated absorption in the soil beneath a pond. In this article, we consider a novel inverse problem where a free boundary is determined from the mass/energy specification in a well-posed one-dimensional nonlinear diffusion problem, and a stability estimate is established. The problem is recast as a nonlinear least-squares minimisation problem, which is solved numerically using the lsqnonlin routine from the MATLAB toolbox. Accurate and stable numerical solutions are achieved. For noisy data, instability is manifest in the derivative of the moving free surface, but not in the free surface itself nor in the concentration or temperature.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.100913.061113a}, url = {http://global-sci.org/intro/article_detail/eajam/10859.html} }
TY - JOUR T1 - Free Boundary Determination in Nonlinear Diffusion JO - East Asian Journal on Applied Mathematics VL - 4 SP - 295 EP - 310 PY - 2018 DA - 2018/02 SN - 3 DO - http://doi.org/10.4208/eajam.100913.061113a UR - https://global-sci.org/intro/article_detail/eajam/10859.html KW - Nonlinear diffusion, free boundary problem, finite difference method. AB -

Free boundary problems with nonlinear diffusion occur in various applications, such as solidification over a mould with dissimilar nonlinear thermal properties and saturated or unsaturated absorption in the soil beneath a pond. In this article, we consider a novel inverse problem where a free boundary is determined from the mass/energy specification in a well-posed one-dimensional nonlinear diffusion problem, and a stability estimate is established. The problem is recast as a nonlinear least-squares minimisation problem, which is solved numerically using the lsqnonlin routine from the MATLAB toolbox. Accurate and stable numerical solutions are achieved. For noisy data, instability is manifest in the derivative of the moving free surface, but not in the free surface itself nor in the concentration or temperature.

M. S. Hussein, D. Lesnic & M. Ivanchov. (1970). Free Boundary Determination in Nonlinear Diffusion. East Asian Journal on Applied Mathematics. 3 (4). 295-310. doi:10.4208/eajam.100913.061113a
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