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.
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.
O. M. Alifonov, Inverse Heat Transfer Problems, Springer, Berlin, Germany, 1994.
M. N. Özisik and H. R. B. Orlande, Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis, New York, NY, USA, 2000.
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 | MathSciNetJ. R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley, New York, NY, USA, 1984.
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 ScholarA. 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 ScholarA. 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 | MathSciNetT.-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 ScholarC.-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 ScholarJ. 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 ScholarD. 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 ScholarN. 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 ScholarL. 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 ScholarJ.-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 | MathSciNetJ.-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 | MathSciNetJ.-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 | MathSciNetA. 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 ScholarM. 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 | MathSciNetA. 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 MATHJ.-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 | MathSciNetJ.-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 ScholarJ.-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 MATHJ. R. Munkres, Topology, Prentice-Hall, Upper Saddle River, NJ, USA, 2nd edition, 2000.
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 | MathSciNetD. 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 | MathSciNetD. 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