Numerical Algorithms

, Volume 68, Issue 4, pp 769–790 | Cite as

Estimation of unknown boundary functionsin an inverse heat conduction problem using a mollified marching scheme

  • M. GarshasbiEmail author
  • H. Dastour
Original Paper


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.


Inverse heat conduction Nonlinear boundary condition Mollification Space marching method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cannon, J.R.: The one-Dimensional Heat Equation. Addison-Wesley (1984)Google Scholar
  2. 2.
    Özisik, M.N.: Heat Conduction. John Wiley & Sons INC (1993)Google Scholar
  3. 3.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Wolf, D.H., Incropera, F.P., Viskanta, R.: Jet impingement boiling. Adv. Heat Transf. 23, 1–132 (1993)CrossRefGoogle Scholar
  5. 5.
    Kaiser, T., Troltzsch F.: An inverse problem arising in the steel cooling process. Wissenschaftliche Zeitung TU Karl-Marx-Stadt 29, 212–218 (1987)zbMATHGoogle Scholar
  6. 6.
    Beck, J.V., Blackwell, B.: Inverse Heat Conduction: Ill-Posed Problems. Wiley Interscience, New York (1985)zbMATHGoogle Scholar
  7. 7.
    Cannon, J.R, Zachman, D.: Parameter Determination in Parabolic Differential equation from overspecified boundary data. Int. J. Eng. Sci. 20, 779–788 (1982)CrossRefzbMATHGoogle Scholar
  8. 8.
    Rosch, A.: Identification of nonlinear heat transfer laws by optimal control. Numer. Funct. Anal. Optim. 15, 417–434 (1994)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Rosch, A.: Stability estimates for the identification of nonlinear heat laws. Inverse Probl. 12, 743–756 (1996)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Rosch, A.: A Gauss-Newton method for the identification of non-linear heat transfer laws. Int. Ser. Numer. Math 139, 217–230 (2002)MathSciNetGoogle Scholar
  11. 11.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    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)CrossRefzbMATHGoogle Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    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)MathSciNetGoogle Scholar
  15. 15.
    Murio, D.A.: Mollification and space marching. In: Woodbury, K (ed.) Inverse Engineering Handbook. CRC Press (2002)Google Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    Acosta, C.D., Mejia, C.E.: Stabilization of explicit methods for convection diffusion equations by discrete mollification. Comput. Math. Appl. 55, 368–380 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Acosta, C.D., Mejia, C.E.: Approximate solution of hyperbolic conservation laws by discrete mollification. Appl. Numer. Math. 59, 2256–2265 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of MathematicsIran University of Science and TechnologyTehranIran

Personalised recommendations