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Free Boundary Determination in Nonlinear Diffusion

Published online by Cambridge University Press:  28 May 2015

M. S. Hussein ,
D. Lesnic  and
M. Ivanchov
M. S. Hussein*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq
D. Lesnic*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
M. Ivanchov*
Affiliation:
Faculty of Mechanics and Mathematics, Department of Differential Equations, Ivan Franko National University of Lviv, 1, Universytetska str., Lviv, 79000, Ukraine
*
Corresponding author. Email Address: mmmsh@leeds.ac.uk
Corresponding author. Email Address: amt51d@maths.leeds.ac.uk
Corresponding author. Email Address: ivanchov@franko.lviv.ua
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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

35R3535K5549N4565M06Nonlinear diffusionfree boundary problemfinite difference method
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 4 , November 2013 , pp. 295 - 310
Copyright
Copyright © Global-Science Press 2013

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