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Mathematical Problems in Engineering
Volume 2009, Article ID 510934, 14 pages
Research Article

Parameter Estimation for Partial Differential Equations by Collage-Based Numerical Approximation

1College of Basic Sciences, Huazhong Agricultural University, Wuhan 430070, China
2College of Engineering and Technology, Huazhong Agricultural University, Wuhan 430070, China

Received 14 December 2008; Revised 13 April 2009; Accepted 30 April 2009

Academic Editor: Slimane Adjerid

Copyright © 2009 Xiaoyan Deng and Qingxi Liao. 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.


The inverse problem of using measurements to estimate unknown parameters of a system often arises in engineering practice and scientific research. This paper proposes a Collage-based parameter inversion framework for a class of partial differential equations. The Collage method is used to convert the parameter estimation inverse problem into a minimization problem of a function of several variables after the partial differential equation is approximated by a differential dynamical system. Then numerical schemes for solving this minimization problem are proposed, including grid approximation and ant colony optimization. The proposed schemes are applied to a parameter estimation problem for the Belousov-Zhabotinskii equation, and the results show that the proposed approximation method is efficient for both linear and nonlinear partial differential equations with respect to unknown parameters. At worst, the presented method provides an excellent starting point for traditional inversion methods that must first select a good starting point.