Chinese Annals of Mathematics, Series B

, Volume 35, Issue 3, pp 469–482

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

• Wenyan Wang
• Masahiro Yamamoto
• Bo Han
Article

## Abstract

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.

## Keywords

Inverse source problem Final overdetermination Parabolic equation Reproducing kernel

35K55 47B32

## References

1. [1]
Ebel, A. and Davitashvili, T., Air, water and soil quality modelling for risk and impact assessment, Springer, Dordrecht, 2007.
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.
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.
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.
5. [5]
Choulli, M. and Yamamoto, M., Conditional stability in determining a heat source, J. Inv. Ill-Posed Problems, 12, 2004, 233–243.
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.
7. [7]
Sakamoto, K. and Yamamoto, M., Inverse heat source problem from time distributing overdetermination, Appl. Anal., 88, 2009, 735–748.
8. [8]
Cannon, J. R. and Pérez-Esteva, S., Uniqueness and stability of 3D heat sources, Inverse Probl., 7, 1991, 57–62.
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.
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.
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.
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.
13. [13]
Isakov, V., Inverse parabolic problems with the final overdetermination, Commun. Pure Appl. Math., 44, 1991, 185–209.
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.
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.
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.
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.
19. [19]
Choulli, M., An inverse problem for a semilinear parabolic equation, Inverse Probl., 10, 1994, 1123–1132.
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.
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.
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.
23. [23]
Cui, M. and Lin, Y., Nonlinear numerical analysis in the reproducing kernel space, Nova Science Publishers, Inc., New York, 2009.
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.
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.
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.
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.