Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Hào, DN; Huong, BV; Thanh, PX; Lesnic, D (2014)
Publisher: Taylor & Francis
Languages: English
Types: Article
We consider the problem of identifying a nonlinear heat transfer law at the boundary, or of the temperature-dependent heat transfer coefficient in a parabolic equation from boundary observations. As a practical example, this model applies to the heat transfer coefficient that describes the intensity of heat exchange between a hot wire and the cooling water in which it is placed. We reformulate the inverse problem as a variational one which aims to minimize a misfit functional and prove that it has a solution. We provide a gradient formula for the misfit functional and then use some iterative methods for solving the variational problem. Thorough investigations are made with respect to several initial guesses and amounts of noise in the input data. Numerical results show that the methods are robust, stable and accurate.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • [1] Barbu V., Boundary control problems with nonlinear state equation. SIAM J. Control Optim. 20(1982), 125-143.
    • [2] Casas E., Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35(1997), 1297-1327.
    • [3] Choulli M., On the determination of an unknown boundary function in a parabolic equation. Inverse Problems 15(1999), 659-667.
    • [4] Engl H.W., Fusek P. and Pereverzev S.V., Natural linearization for the identification of nonlinear heat transfer laws. Inverse problems: modeling and simulation. J. Inverse Ill-Posed Problems 13(2005), 567-582.
    • [5] Grever W., A nonlinear parabolic initial-boundary value problem modelling the continuous casting of steel. Z. Angew. Math. Mech. (ZAMM) 78(1998), 109-119.
    • [6] Dinh Nho H`ao, Phan Xuan Thanh and Lesnic D., Determination of heat transfer coefficients in transient heat conduction. Inverse Problems 29(2013), 095020, 21 pp.
    • [7] Janicki M. and Kindermann S., Recovering temperature dependence of heat transfer coefficient in electronic circuits. Inverse Problems Sci. Eng. 17(2009), 1129-1142.
    • [8] Kaiser T. and Tr¨oltzsch F., An inverse problem arising in the steel cooling process. Wiss. Z. Tech. Univ. Karl-Marx-Stadt 29(1987), 212-218.
    • [9] Lesnic D., Onyango T.T.M. and Ingham D.B., The boundary element method for the determination of nonlinear boundary conditions in heat conduction. Mesh Reduction MethodsBEM/MRM XXXI, 45-55, WIT Trans. Model. Simul., 49, WIT Press, Southampton, 2009.
    • [10] Onyango T.T.M., Ingham D.B. and Lesnic D., Reconstruction of boundary condition laws in heat conduction using the boundary element method. Comput. Math. Appl. 57(2009), 153-168.
    • [11] Pilant M. and Rundell W., An iteration method for the determination of an unknown boundary condition in a parabolic initial-boundary value problem, Proc. Edinburgh Math. Soc. 32(1989), 59-71.
    • [12] Raymond J.P. and Zidani H., Pontryagin's principle for state-constrained control problems governed by parabolic equations with unbounded controls. SIAM J. Control Optim. 36(1998), 1853-1879.
    • [13] Raymond J.P. and Zidani H., Hamiltonian-Pontryagin's principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39(1999), 143-177.
    • [14] R¨osch A., Identification of nonlinear heat transfer laws by optimal control. Numer. Funct. Anal. Optim. 15(1994), 417-434.
    • [15] R¨osch A., Fr´echet differentiability of the solution of the heat equation with respect to a nonlinear boundary condition. Z. Anal. Anwendungen 15(1996), 603-618.
    • [16] R¨osch A., Stability estimates for the identification of nonlinear heat transfer laws. Inverse Problems 12(1996), 743-756.
    • [17] R¨osch A., Identification of nonlinear heat transfer laws by means of boundary data. In Progress in Industry (at ECMI 94), 405-412. Wiley-Teubner, 1996.
    • [18] R¨osch A., Second order optimality conditions and stability estimates for the identification of nonlinear heat transfer laws. Control and Estimation of Distributed Parameter Systems (Vorau, 1996), 237-246, Internat. Ser. Numer. Math., 126, Birkh¨auser, Basel, 1998.
    • [19] R¨osch A., A Gauss-Newton method for the identification of nonlinear heat transfer laws. Optimal Control of Complex Structures (Oberwolfach, 2000), 217-230, Internat. Ser. Numer. Math., 139, Birkh¨auser, Basel, 2002.
    • [20] R¨osch A. and Tr¨oltzsch F. An optimal control problem arising from the identification of nonlinear heat transfer laws. Arch. Control Sci. 1(1992), 183-195.
    • [21] Rundell W. and Yin H.-M., A parabolic inverse problem with an unknown boundary condition. J. Differential Equations 86(1990), 234-242.
    • [22] Schmidt E.J.P.G., Boundary control for the heat equation with nonlinear boundary condition. J. Differential Equations 78(1989), 89-121.
    • [23] Tao L.N., Heat conduction with nonlinear boundary condition. Z. Angew. Math. Phys. (ZAMP) 32(1981), 144-155.
    • [24] Tr¨oltzsch F., Optimale Steuerung partieller Differentialgleichungen, Vieweg + Teubner, Wiesbaden, 2005.
  • No related research data.
  • No similar publications.

Share - Bookmark

Cite this article