Scholarly Research Exchange

Scholarly Research Exchange / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 468570 | 6 pages | https://doi.org/10.3814/2009/468570

An Inverse Problem for Parabolic Partial Differential Equations with Nonlinear Conductivity Term

Received28 Dec 2008
Revised19 Jan 2009
Accepted02 Feb 2009
Published04 Mar 2009

Abstract

We consider an inverse problem for partial differential equation with nonlinear conductivity term in one-dimensional space within a finite interval. In the considered problem, a temperature history is unknown in a boundary of domain. The homotopy perturbation technique is used. Moreover, we have presented a numerical example.

References

  1. J. V. Beck, B. Blackwell, and C. J. St. Clair, Jr., Inverse Heat Conduction: Ill-Posed Problems, John Willey & Sons, New York, NY, USA, 1985.
  2. O. M. Alifonov, Inverse Heat Transfer Problems, Springer, Berlin, Germany, 1994.
  3. M. N. Özisik and H. R. B. Orlande, Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis, New York, NY, USA, 2000.
  4. H. M. Park and J. S. Chung, “A sequential method of solving inverse natural convection problems,” Inverse Problems, vol. 18, no. 3, pp. 529–546, 2002. View at: Publisher Site | Google Scholar | MathSciNet
  5. J. R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley, New York, NY, USA, 1984.
  6. A. Shidfar and A. Zakeri, “A numerical method for backward inverse heat conduction problem with two unknown functions,” International Journal of Engineering Science, vol. 304, no. 16, pp. 71–74, 2008. View at: Google Scholar
  7. A. Shidfar and A. Zakeri, “Asymptotic solution for an inverse parabolic problem,” Mathematica Balkanica, vol. 18, no. 3-4, pp. 475–483, 2004. View at: Google Scholar
  8. A. Shidfar, A. Zakeri, and A. Neisi, “A two-dimensional inverse heat conduction problem for estimating heat source,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 10, pp. 1633–1641, 2005. View at: Publisher Site | Google Scholar | MathSciNet
  9. T.-C. Chen, C.-C. Liu, H.-Y. Jang, and P.-C. Tuan, “Inverse estimation of heat flux and temperature in multi-layer gun barrel,” International Journal of Heat and Mass Transfer, vol. 50, no. 11-12, pp. 2060–2068, 2007. View at: Publisher Site | Google Scholar
  10. C.-H. Huang and S.-P. Wang, “A three-dimensional inverse heat conduction problem in estimating surface heat flux by conjugate gradient method,” International Journal of Heat and Mass Transfer, vol. 42, no. 18, pp. 3387–3403, 1999. View at: Publisher Site | Google Scholar
  11. J. Taler and W. Zima, “Solution of inverse heat conduction problems using control volume approach,” International Journal of Heat and Mass Transfer, vol. 42, no. 6, pp. 1123–1140, 1999. View at: Publisher Site | Google Scholar
  12. D. Lesnic, L. Elliott, and D. B. Ingham, “The solution of an inverse heat conduction problem subject to the specification of energies,” International Journal of Heat and Mass Transfer, vol. 41, no. 1, pp. 25–32, 1998. View at: Publisher Site | Google Scholar
  13. 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. View at: Publisher Site | Google Scholar
  14. L. Jinbo and T. Jiang, “Variational iteration method for solving an inverse parabolic equation,” Physics Letters A, vol. 372, no. 20, pp. 3569–3572, 2008. View at: Publisher Site | Google Scholar
  15. J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  18. A. Zakeri, Q. Jannati, and A. Aminataei, “Application of He's homotopy perturbation method for Cauchy problem in one-dimensional nonlinear equation of diffusion,” to appeare in International Journal of Engineering Science. View at: Google Scholar
  19. M. Ghasemi, M. Tavassoli Kajani, and A. Davari, “Numerical solution of two-dimensional nonlinear differential equation by homotopy perturbation method,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 341–345, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  20. A. Yıldırım, “Application of He's homotopy perturbation method for solving the Cauchy reaction-diffusion problem,” Computers & Mathematics with Applications, vol. 57, no. 4, pp. 612–618, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  21. J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  22. J.-H. He, “Recent development of the homotopy perturbation method,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 205–209, 2008. View at: Google Scholar
  23. J.-H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,” International Journal of Modern Physics B, vol. 22, no. 21, pp. 3487–3578, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  24. J. R. Munkres, Topology, Prentice-Hall, Upper Saddle River, NJ, USA, 2nd edition, 2000.
  25. D. D. Ganji, “The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 355, no. 4-5, pp. 337–341, July 2006. View at: Publisher Site | Google Scholar | MathSciNet
  26. D. D. Ganji and M. Rafei, “Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method,” Physics Letters A, vol. 356, no. 2, pp. 131–137, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  27. D. D. Ganji and A. Sadighi, “Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 4, pp. 411–418, 2006. View at: Google Scholar

Copyright © 2009 Ali Zakeri and Q. Jannati. 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.


More related articles

163 Views | 0 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.