Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2014, Special Issue (2013), Article ID 529618, 7 pages.

Inverse Estimates for Nonhomogeneous Backward Heat Problems

Tao Min, Weimin Fu, and Qiang Huang

Full-text: Open access

Abstract

We investigate the inverse problem in the nonhomogeneous heat equation involving the recovery of the initial temperature from measurements of the final temperature. This problem is known as the backward heat problem and is severely ill-posed. We show that this problem can be converted into the first Fredholm integral equation, and an algorithm of inversion is given using Tikhonov's regularization method. The genetic algorithm for obtaining the regularization parameter is presented. We also present numerical computations that verify the accuracy of our approximation.

Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2013), Article ID 529618, 7 pages.

Dates
First available in Project Euclid: 1 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1412177539

Digital Object Identifier
doi:10.1155/2014/529618

Mathematical Reviews number (MathSciNet)
MR3178961

Zentralblatt MATH identifier
07010669

Citation

Min, Tao; Fu, Weimin; Huang, Qiang. Inverse Estimates for Nonhomogeneous Backward Heat Problems. J. Appl. Math. 2014, Special Issue (2013), Article ID 529618, 7 pages. doi:10.1155/2014/529618. https://projecteuclid.org/euclid.jam/1412177539


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References

  • A. Shidfar, G. R. Karamali, and J. Damirchi, “An inverse heat conduction problem with a nonlinear source term,” Nonlinear Analysis, Theory, Methods and Applications, vol. 65, no. 3, pp. 615–621, 2006.
  • A. Shidfar and R. Pourgholi, “Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomials,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1366–1374, 2006.
  • J. Shi and J. Wang, “Inverse problem of estimating space and time dependent hot surface heat flux in transient transpiration cooling process,” International Journal of Thermal Sciences, vol. 48, no. 7, pp. 1398–1404, 2009.
  • B. T. Johansson and D. Lesnic, “A procedure for determining a spacewise dependent heat source and the initial temperature,” Applicable Analysis, vol. 87, pp. 265–276, 2008.
  • M. Ebrahimian, R. Pourgholi, M. Emamjome, and P. Reihani, “A numerical solution of an inverse parabolic problem with unknown boundary conditions,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 228–234, 2007.
  • R. Model and U. Hammerschmidt, “Numerical methods for the determination of thermal properties by means of transient measurements,” in Proceedings of the 6th International Conference on Advanced Computational Methods in Heat Transfer, pp. 407–416, WIT Press, June 2000.
  • R. Lattes and J. L. Lions, Methode de Quasi-Reversibilite et Applications, Dunod, Paris, France, 1967.
  • H. Gajewski and K. Zacharias, “Zur regularisierung einer Klasse nichtkorrekter probleme bei evolutionsgleichungen,” Journal of Mathematical Analysis and Applications, vol. 38, no. 3, pp. 784–789, 1972.
  • G. Nakamura, S. Saitoh, and A. Syarif, “Representations of initial heat distributions by means of their heat distributions as functions of time,” Inverse Problems, vol. 15, no. 5, pp. 1255–1261, 1999.
  • N. Al-Khalidy, “On the solution of parabolic and hyperbolic inverse heat conduction problems,” International Journal of Heat and Mass Transfer, vol. 41, no. 23, pp. 3731–3740, 1998.
  • J. V. Beck, B. Blackwell, and C. R. S. Clair, Inverse Heat Conduction: Ill-Poed Problems, Wiley Interscience, New York, NY, USA, 1985.
  • E. Hensel, Inverse Theory and Applications for Engineers, Prentice Hall, 1991.
  • D. T. Dang and H. T. Nguyen, “Regularization and error estimates for nonhomogeneous backward heat problems,” Electronic Journal of Differential Equations, vol. 2006, pp. 1–10, 2006.
  • D. D. Trong, P. H. Quan, T. V. Khanh, and N. H. Tuan, “A nonlinear case of the 1-D Backward heat problem: regularizai ion and error estimate,” Zeitschrift fur Analysis und ihre Anwendung, vol. 26, no. 2, pp. 231–245, 2007.
  • A. Hasanov, P. Duchateau, and B. Pektaş, “An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation,” Journal of Inverse and Ill-Posed Problems, vol. 14, no. 5, pp. 435–463, 2006.
  • Y.-F. Wang and T.-Y. Xiao, “Fast realization algorithms for determining regularization parameters in linear inverse problems,” Inverse Problems, vol. 17, no. 2, pp. 281–291, 2001.
  • G. D. Li and Y. F. Wang, “A regularizing trust region algorithm for nonlinear ill-posed problems,” Inverse Problems in Science and Engineering, vol. 14, no. 8, pp. 859–872, 2006.
  • V. A. Morozov, “Choice of a parameter for the solution of functional equations by the regularization method,” Soviet Mathematics: Doklady, vol. 8, pp. 1000–1003, 1967.
  • V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, Berlin, Germany, 1984.
  • D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, Mass, USA, 1989.
  • P. Hajela, “Genetic search. An approach to the nonconvex optimization problem,” AIAA Journal, vol. 28, no. 7, pp. 1205–1210, 1990.
  • J. S. Arora, O. A. Elwakeil, A. I. Chahande, and C. C. Hsieh, “Global optimization methods for engineering applications: a review,” Structural Optimization, vol. 9, no. 3-4, pp. 137–159, 1995. \endinput