Back
 JAMP  Vol.2 No.2 , January 2014
Determination of an Unknown Source in the Heat Equation by the Method of Tikhonov Regularization in Hilbert Scales
Abstract: In this paper, we consider the problem for determining an unknown source in the heat equation. The Tikhonov regularization method in Hilbert scales is presented to deal with ill-posedness of the problem and error estimates are obtained with a posteriori choice rule to find the regularization parameter. The smoothness parameter and the a priori bound of exact solution are not needed for the choice rule. Numerical tests show that the proposed method is effective and stable.
Cite this paper: Zhao, Z. , Xie, O. , Meng, Z. and You, L. (2014) Determination of an Unknown Source in the Heat Equation by the Method of Tikhonov Regularization in Hilbert Scales. Journal of Applied Mathematics and Physics, 2, 10-17. doi: 10.4236/jamp.2014.22002.
References

[1]   F. F. Dou, C. L. Fu and F. L. Yang, “Optimal Error Bound and Fourier Regularization for Identifying an Unknown Source in the Heat Equation,” Journal of Computational and Applied Mathematics, Vol. 230, No. 2, 2009, pp. 728-737.
http://dx.doi.org/10.1016/j.cam.2009.01.008

[2]   J. R. Cannon and P. Du Chateau, “Structural Identification of an Unknown Source term in a Heat Equation,” Inverse Problems, Vol. 14, No. 3, 1998, pp. 535-551. http://dx.doi.org/10.1088/0266-5611/14/3/010

[3]   J. R. Cannon and S. Perez-Esteva, “Uniqueness and Stability of 3d Heat Sources,” Inverse Problems, Vol. 7, No. 1, 1991, pp. 57-62. http://dx.doi.org/10.1088/0266-5611/7/1/006

[4]   M. Choulli and M. Yamamoto, “Conditional Stability in Determining a Heat Source,” Journal of Inverse and Ill-Posed Problems, Vol. 12, No. 3, 2004, pp. 233-243. http://dx.doi.org/10.1515/1569394042215856

[5]   A. El Badia and T. Ha-Duong, “On an Inverse Source Problem for the Heat Equation. Application to a Pollution Detection Problem,” Journal of Inverse and Ill-Posed Problems, Vol. 10, No. 6, 2002, pp. 585-600.
http://dx.doi.org/10.1515/jiip.2002.10.6.585

[6]   G. S. Li, “Data Compatibility and Conditional Stability for an Inverse Source Problem in the Heat Equation,” Applied Mathematics and Computation, Vol. 173, No. 1, 2006, pp. 566-581. http://dx.doi.org/10.1016/j.amc.2005.04.053

[7]   M. Yamamoto, “Conditional Stability in Determination of Force Terms of Heat Equations in a Rectangle,” Mathematical and Computer Modelling, Vol. 18, No. 1, 1993, pp. 79-88. http://dx.doi.org/10.1016/0895-7177(93)90081-9

[8]   A. A. Burykin and A. M. Denisov, “Determination of the Unknown Sources in the Heat-Conduction Equation,” Computational Mathematics and Modeling, Vol. 8, No. 4, 1997, pp. 309-313. http://dx.doi.org/10.1016/0895-7177(93)90081-9

[9]   A. Farcas and D. Lesnic, “The Boundary-Element Method for the Determination of a Heat Source Dependent on One Variable,” Journal of Engineering Mathematics, Vol. 54, No. 4, 2006, pp. 375-388. http://dx.doi.org/10.1007/s10665-005-9023-0

[10]   L. Ling, M. Yamamoto, Y. C. Hon and T. Takeuchi, “Identification of Source Locations in Two-Dimensional Heat Equations,” Inverse Problems, Vol. 22, No. 4, 2006, pp. 1289-1305. http://dx.doi.org/10.1088/0266-5611/22/4/011

[11]   H. M. Park and J. S. Chung, “A Sequential Method of Solving Inverse Natural Convection Problems,” Inverse Problems, Vol. 18, No. 3, 2002, pp. 529-546. http://dx.doi.org/10.1088/0266-5611/18/3/302

[12]   V. S. Ryaben’kii, S. V. Tsynkov and S. V. Utyuzhnikov, “Inverse Source Problem and Active Shielding for Composite Domains,” Applied Mathematics Letters, Vol. 20, No. 5, 2007, pp. 511-515. http://dx.doi.org/10.1016/j.aml.2006.05.019

[13]   L. Yan, C. L. Fu and F. L. Yang, “The Method of Fundamental Solutions for the Inverse Heat Source Problem,” Engineering Analysis with Boundary Elements, Vol. 32, No. 3, 2008, pp. 216-222. http://dx.doi.org/10.1016/j.enganabound.2007.08.002

[14]   L. Yan, F. L. Yang and C. L. Fu, “A Meshless Method for Solving an Inverse Spacewise-Dependent Heat Source Problem,” Journal of Computational Physics, Vol. 228, No. 1, 2009, pp. 123-136. http://dx.doi.org/10.1016/j.jcp.2008.09.001

[15]   F. Yang, “The Truncation Method for Identifying an Unknown Source in the Poisson Equation,” Applied Mathematics and Computation, Vol. 22, 2011, pp. 9334-9339. http://dx.doi.org/10.1016/j.amc.2011.04.017

[16]   F. Yang and C. L. Fu, “The Modified Regularization Method for Identifying the Unknown Source on Poisson Equation,” Applied Mathematical Modelling, Vol. 36, No. 2, 2012, pp. 756-763. http://dx.doi.org/10.1016/j.apm.2011.07.008

[17]   Z. Yi and D. A. Murio, “Source Term Identification in 1-D IHCP,” Computers & Mathematics with Applications, Vol. 47, No. 12, 2004, pp. 1921-1933. http://dx.doi.org/10.1016/j.camwa.2002.11.025

[18]   Z. Y. Zhao and L. You, “A Modified Tikhonov Regularization Method for Identifying an Unknown Source in the Heat Equation,” Acta Mathematica Scientia, 2012.

[19]   A. Neubauer, “An a Posteriori Parameter Choice for Tikhonov Regularization in Hilbert Scales Leading to Optimal Convergence Rates,” SIAM Journal on Numerical Analysis, Vol. 25, No. 6, 1988, pp. 1313-1326. http://dx.doi.org/10.1137/0725074

[20]   H. W. Engl, M. Hanke and A. Neubauer, “Regularization of Inverse Problems,” Springer, Netherlands, 1996.
http://dx.doi.org/10.1137/0725074

[21]   F. Natterer, “Error Bounds for Tikhonov Regularization in Hilbert Scales,” Applicable Analysis, Vol. 18, No. 1-2, 1984, pp. 29-37. http://dx.doi.org/10.1080/00036818408839508

 
 
Top