Abstract
In this article, a one-dimensional inverse heat conduction problem with unknown nonlinear boundary conditions is studied. In many practical heat transfer situations, the heat transfer coefficient depends on the boundary temperature and the dependence has a complicated or unknown structure. For this reason highly nonlinear boundary conditions are imposed involving both the flux and the temperature. A numerical procedure based on the mollification method and the space marching scheme is developed to solve numerically the proposed inverse problem. The stability and convergence of numerical solutions are investigated and the numerical results are presented and discussed for some test problems.
Similar content being viewed by others
References
Cannon, J.R.: The one-Dimensional Heat Equation. Addison-Wesley (1984)
Özisik, M.N.: Heat Conduction. John Wiley & Sons INC (1993)
Saldanha da Gama, R.M.: Simulation of the steady-state energy transfer in rigid bodies, with convective/radiative boundary conditions, employing a minimum principle. J. Comp. Phys. 99, 310–320 (1992)
Wolf, D.H., Incropera, F.P., Viskanta, R.: Jet impingement boiling. Adv. Heat Transf. 23, 1–132 (1993)
Kaiser, T., Troltzsch F.: An inverse problem arising in the steel cooling process. Wissenschaftliche Zeitung TU Karl-Marx-Stadt 29, 212–218 (1987)
Beck, J.V., Blackwell, B.: Inverse Heat Conduction: Ill-Posed Problems. Wiley Interscience, New York (1985)
Cannon, J.R, Zachman, D.: Parameter Determination in Parabolic Differential equation from overspecified boundary data. Int. J. Eng. Sci. 20, 779–788 (1982)
Rosch, A.: Identification of nonlinear heat transfer laws by optimal control. Numer. Funct. Anal. Optim. 15, 417–434 (1994)
Rosch, A.: Stability estimates for the identification of nonlinear heat laws. Inverse Probl. 12, 743–756 (1996)
Rosch, A.: A Gauss-Newton method for the identification of non-linear heat transfer laws. Int. Ser. Numer. Math 139, 217–230 (2002)
Onyango, T.T.M., Ingham, D.B., Lesnic, D.: Reconstruction of boundary condition laws in heat conduction using the boundary element method. Comput. Math. Appl. 57, 153–168 (2009)
Bialecki, R., Divo, E., Kassab, A,J.: Reconstruction of time-dependent boundary heat flux by a BEM based inverse algorithm. Eng. Anal. Bound. Elem. 30, 767–773 (2006)
Garshasbi, M., Damirchi, J., Reihani, P.: Parameter estimation in an inverse initial-boundary value problem of heat equation. J. Adv. Res. Diff. Equ. 2, 49–60 (2010)
Garshasbi, M., Reihani, P., Dastour, H.: A stable numerical solution of a class of semi-linear Cauchy problems. J. Adv. Res. Dyn. Cont. Sys. 4, 56–67 (2012)
Murio, D.A.: Mollification and space marching. In: Woodbury, K (ed.) Inverse Engineering Handbook. CRC Press (2002)
Mejia, C.E., Murio, D.A., Zhan, S.: Some applications of the mollification method. In: Lassonde, M. (ed.) App. Opti. Math Eco., pp. 213–222. Physica-Verlag (2001)
Acosta, C.D., Mejia, C.E.: Stabilization of explicit methods for convection diffusion equations by discrete mollification. Comput. Math. Appl. 55, 368–380 (2008)
Acosta, C.D., Mejia, C.E.: Approximate solution of hyperbolic conservation laws by discrete mollification. Appl. Numer. Math. 59, 2256–2265 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Garshasbi, M., Dastour, H. Estimation of unknown boundary functionsin an inverse heat conduction problem using a mollified marching scheme. Numer Algor 68, 769–790 (2015). https://doi.org/10.1007/s11075-014-9871-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-014-9871-7