Chinese Annals of Mathematics, Series B

, Volume 35, Issue 3, pp 469–482 | Cite as

Two-dimensional parabolic inverse source problem with final overdetermination in reproducing kernel space

  • Wenyan WangEmail author
  • Masahiro Yamamoto
  • Bo Han


A new method of the reproducing kernel Hilbert space is applied to a two-dimensional parabolic inverse source problem with the final overdetermination. The exact and approximate solutions are both obtained in a reproducing kernel space. The approximate solution and its partial derivatives are proved to converge to the exact solution and its partial derivatives, respectively. A technique is proposed to improve some existing methods. Numerical results show that the method is of high precision, and confirm the robustness of our method for reconstructing source parameter.


Inverse source problem Final overdetermination Parabolic equation Reproducing kernel 

2000 MR Subject Classification

35K55 47B32 


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  1. [1]
    Ebel, A. and Davitashvili, T., Air, water and soil quality modelling for risk and impact assessment, Springer, Dordrecht, 2007.CrossRefGoogle Scholar
  2. [2]
    Ling, L., Yamamoto, M., Hon, Y. C. and Takeuchi, T., Identification of source locations in two-dimensional heat equations, Inverse Probl., 22, 2006, 1289–1305.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    Cheng, W., Zhao, L. and Fu, C., Source term identification for an axisymmetric inverse heat conduction problem, Comput. Math. Appl., 59, 2010, 142–148.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    Badia, A. E. and Ha-Duong, T., On an inverse source problem for the heat equation, application to a pollution detection problem, J. Inv. Ill-Posed Problems, 10, 2002, 585–599.CrossRefzbMATHGoogle Scholar
  5. [5]
    Choulli, M. and Yamamoto, M., Conditional stability in determining a heat source, J. Inv. Ill-Posed Problems, 12, 2004, 233–243.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Dehghan, M. and Tatari, M., Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions, Math. Comput. Model., 44, 2006, 1160–1168.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    Sakamoto, K. and Yamamoto, M., Inverse heat source problem from time distributing overdetermination, Appl. Anal., 88, 2009, 735–748.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Cannon, J. R. and Pérez-Esteva, S., Uniqueness and stability of 3D heat sources, Inverse Probl., 7, 1991, 57–62.CrossRefzbMATHGoogle Scholar
  9. [9]
    Choulli, M. and Yamamoto, M., Generic well-posedness of an inverse parabolic problem-the Hölder-space approach, Inverse Probl., 12, 1996, 195–205.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Fan, J. and Nakamura, G., Well-posedness of an inverse problem of Navier-Stokes equations with the final overdetermination, J. Inv. Ill-Posed Problems, 17, 2009, 565–584.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    Choulli, M. and Yamamoto, M., Generic well-posedness of a linear inverse parabolic problem with diffusion parameters, J. Inv. Ill-Posed Problems, 7, 1999, 241–254.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Kamynin, V. L., On the unique solvability of an inverse problem for parabolic equations under a final overdetermination condition, Math. Notes, 73, 2003, 202–211.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    Isakov, V., Inverse parabolic problems with the final overdetermination, Commun. Pure Appl. Math., 44, 1991, 185–209.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    Isakov, V., Inverse problems for partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York, 2006.Google Scholar
  15. [15]
    Yi, Z. and Murio, D. A., Source term identification in 1-D IHCP, Comput. Math. Appl., 47, 2004, 1921–1933.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    Trong, D. D., Long, N. T. and Alain, P. N. D., Nonhomogeneous heat equation: identification and regularization for the inhomogeneous term, J. Math. Anal. Appl., 312, 2005, 93–104.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    Trong, D. D., Quan, P. H. and Alain, P. N. D., Determination of a two-dimensional heat source: uniqueness, regularization and error estimate, J. Comput. Appl. Math., 191, 2006, 50–67.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    Chen, Q. and Liu, J., Solving an inverse parabolic problem by optimization from final measurement data, J. Comput. Appl. Math., 193, 2006, 183–203.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    Choulli, M., An inverse problem for a semilinear parabolic equation, Inverse Probl., 10, 1994, 1123–1132.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    Yang, L., Yu, J. and Deng, Z., An inverse problem of identifying the coefficient of parabolic equation, Appl. Math. Model., 32, 2008, 1984–1995.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [21]
    Takeuchi, T. and Yamamoto, M., Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation, SIAM J. Sci. Comput., 31, 2008, 112–142.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    Hon, Y. C. and Takeuchi, T., Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem, Adv. Comput. Math., 34, 2011, 167–183.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    Cui, M. and Lin, Y., Nonlinear numerical analysis in the reproducing kernel space, Nova Science Publishers, Inc., New York, 2009.zbMATHGoogle Scholar
  24. [24]
    Cui, M. and Geng, F., A computational method for solving one-dimensional variable-coefficient Burgers equation, Appl. Math. Comput., 188, 2007, 1389–1401.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    Zhou, Y., Cui, M. and Lin, Y., Numerical algorithm for parabolic problems with non-classical conditions, J. Comput. Appl. Math., 230, 2009, 770–780.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    Wang, W., Han, B. and Yamamoto, M., Inverse heat problem of determining time-dependent source parameter in reproducing kernel space, Nonlinear Anal. Real, 14, 2013, 875–887.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    Wang, W., Yamamoto, M. and Han, B., Numerical method in reproducing kernel space for an inverse source problem for the fractional diffusion equation, Inverse Probl., 29, 2013, 1–15.zbMATHMathSciNetGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinChina
  2. 2.Department of MathematicsNortheast Forestry UniversityHarbinChina
  3. 3.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan
  4. 4.Department of MathematicsHarbin Institute of TechnologyHarbinChina

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