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Year 2012, Volume 2, Issue 2, 195 - 209, 01.12.2012

Abstract

References

  • Abtahi, M., Pourgholi, R. and Shidfar, A., (2011), Existence and uniqueness of solution for a two dimensional nonlinear inverse diffusion problem , Nonlinear Anal., 74(7), 2462-2467.
  • Alifanov, O. M., (1994), Inverse Heat Transfer Problems, Springer, NewYork.
  • Alves, C. J. S., Chen, C. S. and Saler, B., (2002), The method of fundamental solutions for solving Poisson problems, International series on advances in boundary elements, in Boundary Elements XXIV, 13, 67-76.
  • Beck, J. V., Blackwell,B. and St. Clair, C. R., (1985), Inverse Heat Conduction: IllPosed Problems, Wiley-Interscience, NewYork.
  • Cannon, J. R., (1984), The One-Dimensional Heat Equation, Addison Wesley, Reading, MA.
  • Dowding, K. J. and Beck, J. V., (1999), A Sequential Gradient Method for the Inverse Heat Conduction Problems, J. Heat Transfer, 121, 300-306.
  • Molhem, H. and Pourgholi, R., (2008), A numerical algorithm for solving a one-dimensional inverse heat conduction problem, Int. J. Math. Stat., 4(1), 60-63.
  • Pourgholi, R., Azizi, N., Gasimov, Y. S., Aliev, F. and KhalaŞ, H. K., (2009), Removal of Numerical Instability in the Solution of an Inverse Heat Conduction Problem, Commun. Nonlinear Sci. Numer. Simul., 14(6), 2664-2669.
  • Pourgholi, R. and Rostamian, M., (2010), A numerical technique for solving IHCPs using Tikhonov regularization method, Appl. Math. Model., 34(8), 2102-2110.
  • Pourgholi, R., Rostamian, M. and Emamjome, M., (2010), A numerical method for solving a nonlinear inverse parabolic problem, Inverse Probl. Sci. Eng., 18(8), 1151-1164.
  • Shidfar, A. and Azary, H., (1997), Nonlinear Parabolic Problems, Nonlinear Anal., 30(8), 4823-4832. [12] Shidfar, A. and Pourgholi, R., (2006), Numerical approximation of solution of an inverse heat con- duction problem based on Legendre polynomials, Appl. Math. Comput., 175(2), 1366-1374.
  • Shidfar, A., Pourgholi, R. and Ebrahimi, M., (2006), A Numerical Method for Solving of a Nonlinear Inverse Diffusion Problem, Comput. Math. Appl., 52, 1021-1030.
  • Tikhonov, A. N. and Arsenin, V. Y., (1977), Solution of Ill-Posed Problems, V. H. Winston and Sons, Washington, DC.
  • Murio, D. A., (1993), The MolliŞcation Method and the Numerical Solution of Ill-Posed Problems, Wiley-Interscience, NewYork.
  • Yan, L., Yang, F. L. and Fu, C. L., (2009), A meshless method for solving an inverse spacewise- dependent heat source problem, J. Comput. Phys., 228, 123-136.
  • Hansen, P. C., (1998), Rank-DeŞcient and Discrete Ill-Posed Problems, SIAM, Philadelphia.
  • Tikhonov, A. N. and Arsenin, V. Y., (1977), On the solution of ill-posed problems, New York, Wiley. [19] Hansen, P. C., (1992), Analysis of Discrete Ill-posed Problems by Means of the L-curve, SIAM Rev., 34, 561-80.
  • Lawson, C. L. and Hanson, R. J., (1995), Solving Least Squares Problems, Philadelphia, PA: SIAM. First published by Prentice-Hall, 1974.
  • Hon, Y. C. and Wei, T., (2004), A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem., 28, 489-495.

A NUMERICAL SOLUTION OF AN INVERSE PARABOLIC PROBLEM

Year 2012, Volume 2, Issue 2, 195 - 209, 01.12.2012

Abstract

In this paper, we will first study the existence and uniqueness of the solution of an inverse problem for a linear equation with non-linear boundary conditions radiation terms , via an auxiliary problem. Furthermore, a stable numerical algorithm based on the use of the solution to the auxiliary problem as a basis function is proposed. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution. Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data.

References

  • Abtahi, M., Pourgholi, R. and Shidfar, A., (2011), Existence and uniqueness of solution for a two dimensional nonlinear inverse diffusion problem , Nonlinear Anal., 74(7), 2462-2467.
  • Alifanov, O. M., (1994), Inverse Heat Transfer Problems, Springer, NewYork.
  • Alves, C. J. S., Chen, C. S. and Saler, B., (2002), The method of fundamental solutions for solving Poisson problems, International series on advances in boundary elements, in Boundary Elements XXIV, 13, 67-76.
  • Beck, J. V., Blackwell,B. and St. Clair, C. R., (1985), Inverse Heat Conduction: IllPosed Problems, Wiley-Interscience, NewYork.
  • Cannon, J. R., (1984), The One-Dimensional Heat Equation, Addison Wesley, Reading, MA.
  • Dowding, K. J. and Beck, J. V., (1999), A Sequential Gradient Method for the Inverse Heat Conduction Problems, J. Heat Transfer, 121, 300-306.
  • Molhem, H. and Pourgholi, R., (2008), A numerical algorithm for solving a one-dimensional inverse heat conduction problem, Int. J. Math. Stat., 4(1), 60-63.
  • Pourgholi, R., Azizi, N., Gasimov, Y. S., Aliev, F. and KhalaŞ, H. K., (2009), Removal of Numerical Instability in the Solution of an Inverse Heat Conduction Problem, Commun. Nonlinear Sci. Numer. Simul., 14(6), 2664-2669.
  • Pourgholi, R. and Rostamian, M., (2010), A numerical technique for solving IHCPs using Tikhonov regularization method, Appl. Math. Model., 34(8), 2102-2110.
  • Pourgholi, R., Rostamian, M. and Emamjome, M., (2010), A numerical method for solving a nonlinear inverse parabolic problem, Inverse Probl. Sci. Eng., 18(8), 1151-1164.
  • Shidfar, A. and Azary, H., (1997), Nonlinear Parabolic Problems, Nonlinear Anal., 30(8), 4823-4832. [12] Shidfar, A. and Pourgholi, R., (2006), Numerical approximation of solution of an inverse heat con- duction problem based on Legendre polynomials, Appl. Math. Comput., 175(2), 1366-1374.
  • Shidfar, A., Pourgholi, R. and Ebrahimi, M., (2006), A Numerical Method for Solving of a Nonlinear Inverse Diffusion Problem, Comput. Math. Appl., 52, 1021-1030.
  • Tikhonov, A. N. and Arsenin, V. Y., (1977), Solution of Ill-Posed Problems, V. H. Winston and Sons, Washington, DC.
  • Murio, D. A., (1993), The MolliŞcation Method and the Numerical Solution of Ill-Posed Problems, Wiley-Interscience, NewYork.
  • Yan, L., Yang, F. L. and Fu, C. L., (2009), A meshless method for solving an inverse spacewise- dependent heat source problem, J. Comput. Phys., 228, 123-136.
  • Hansen, P. C., (1998), Rank-DeŞcient and Discrete Ill-Posed Problems, SIAM, Philadelphia.
  • Tikhonov, A. N. and Arsenin, V. Y., (1977), On the solution of ill-posed problems, New York, Wiley. [19] Hansen, P. C., (1992), Analysis of Discrete Ill-posed Problems by Means of the L-curve, SIAM Rev., 34, 561-80.
  • Lawson, C. L. and Hanson, R. J., (1995), Solving Least Squares Problems, Philadelphia, PA: SIAM. First published by Prentice-Hall, 1974.
  • Hon, Y. C. and Wei, T., (2004), A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem., 28, 489-495.

Details

Primary Language English
Journal Section Research Article
Authors

Reza POURGHOLİ This is me
School of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364, Damghan, Iran


Mortaza ABTAHİ This is me
School of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364, Damghan, Iran


S. Hashem TABASİ This is me
School of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364, Damghan, Iran

Publication Date December 1, 2012
Published in Issue Year 2012, Volume 2, Issue 2

Cite

Bibtex @ { twmsjaem761711, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2012}, volume = {2}, number = {2}, pages = {195 - 209}, title = {A NUMERICAL SOLUTION OF AN INVERSE PARABOLIC PROBLEM}, key = {cite}, author = {Pourgholi, Reza and Abtahi, Mortaza and Tabasi, S. Hashem} }