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International Journal of Differential Equations
Volume 2017, Article ID 1467049, 17 pages
https://doi.org/10.1155/2017/1467049
Research Article

Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential

Laboratoire de Mathématiques, Informatique et Sciences de l’Ingénieur (MISI), Université Hassan 1er, 26000 Settat, Morocco

Correspondence should be addressed to B. Khouiti; rf.oohay@b_itiuohk

Received 28 September 2016; Revised 13 March 2017; Accepted 19 March 2017; Published 14 June 2017

Academic Editor: Jen-Chih Yao

Copyright © 2017 K. Atifi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A hybrid algorithm and regularization method are proposed, for the first time, to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements. The evolution of the heat is given by a degenerate parabolic equation with singular potential. This problem can be formulated in a least-squares framework, an iterative procedure which minimizes the difference between the given measurements and the value at sensor locations of a reconstructed field. The mathematical model leads to a nonconvex minimization problem. To solve it, we prove the existence of at least one solution of problem and we propose two approaches: the first is based on a Tikhonov regularization, while the second approach is based on a hybrid genetic algorithm (married genetic with descent method type gradient). Some numerical experiments are given.