DOI QR코드

DOI QR Code

NUMERICAL METHODS FOR RECONSTRUCTION OF THE SOURCE TERM OF HEAT EQUATIONS FROM THE FINAL OVERDETERMINATION

  • DENG, YOUJUN ;
  • FANG, XIAOPING ;
  • LI, JING
  • Received : 2014.07.25
  • Published : 2015.09.30

Abstract

This paper deals with the numerical methods for the reconstruction of the source term in a linear parabolic equation from final overdetermination. We assume that the source term has the form f(x)h(t) and h(t) is given, which guarantees the uniqueness of the inverse problem of determining the source term f(x) from final overdetermination. We present the regularization methods for reconstruction of the source term in the whole real line and with Neumann boundary conditions. Moreover, we show the connection of the solutions between the problem with Neumann boundary conditions and the problem with no boundary conditions (on the whole real line) by using the extension method. Numerical experiments are done for the inverse problem with the boundary conditions.

Keywords

linear parabolic equation;source term;inverse problem;numerical methods

References

  1. S. Attaway, Matlab: A Practical Introduction to Programming and Problem Solving, Elsevier, New York, 2009.
  2. J. Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.
  3. F. Berntsson, A spectral method for solving the sideways heat equation, Inverse Problems 15 (1999), no. 4, 891-906. https://doi.org/10.1088/0266-5611/15/4/305
  4. I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, Inverse Problems 11 (1995), no. 4, L11-L16. https://doi.org/10.1088/0266-5611/11/4/001
  5. J. R. Cannon and P. DuChateau, An inverse problem for an unknown source in a heat equation, J. Math. Anal. Appl. 75 (1980), no. 2, 465-485. https://doi.org/10.1016/0022-247X(80)90095-5
  6. J. R. Cannon and P. DuChateau, Structural identification of an unknown source term in a heat equation, Inverse Problems 14 (1998), no. 3, 535-551. https://doi.org/10.1088/0266-5611/14/3/010
  7. M. Choulli, An inverse problem for a semilinear parabolic equation, Inverse Problems 10 (1994), no. 5, 1123-1132. https://doi.org/10.1088/0266-5611/10/5/009
  8. M. Choulli and M. Yamamoto, Generic well-posedness of an inverse parabolic problemthe Holder-space approach, Inverse Problems 12 (1996), no. 3, 195-205. https://doi.org/10.1088/0266-5611/12/3/002
  9. Y. J. Deng and Z. H. Liu, Iteration methods on sideways parabolic equations, Inverse Problems 25 (2009), no. 9, 095004, 14 pp. https://doi.org/10.1088/0266-5611/25/9/095004
  10. Y. J. Deng and Z. H. Liu, New fast iteration for determining surface temperature and heat ux of general sideways parabolic equation, Nonlinear Anal. Real World Appl. 12 (2011), 156-166. https://doi.org/10.1016/j.nonrwa.2010.06.005
  11. L. Elden, F. Berntsson, and T. Reginska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput. 21 (2000), no. 6, 2187-2205. https://doi.org/10.1137/S1064827597331394
  12. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, 1996.
  13. L. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
  14. A. Hasanov, Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach, J. Math. Anal. Appl. 330 (2007), no. 2, 766-779. https://doi.org/10.1016/j.jmaa.2006.08.018
  15. M. I. Ivanchov, The inverse problem of determining the heat source power for a parabolic equation under arbitrary boundary conditions, J. Math. Sci. 88 (1998), no. 3, 432-436. https://doi.org/10.1007/BF02365265
  16. D. W. Kim, J.-E. Lee, and H.-K. Oh, Heat Conduction to Photoresist on Top of Wafer during Post Exposure Bake Process: I. Numerical Approach, Japan. J. Appl. Phys. 47 (2008), 8338-8348. https://doi.org/10.1143/JJAP.47.8338
  17. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, 1996.
  18. L. Ling, M. Yamamoto, Y. C. Hon, etc., Identification of source locations in twodimensional heat equations, Inverse Problems 22 (2006), no. 4, 1289-1305. https://doi.org/10.1088/0266-5611/22/4/011
  19. J. Liu and Y. Deng, A modified landweber iteration for general sideways parabolic equations, Acta Math. Appl. Sin. Engl. Ser. 27 (2011), no. 4, 727-738. https://doi.org/10.1007/s10255-011-0104-8
  20. V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl. 7 (1966), 414-417.
  21. G. Ozkum, A. Demir, S. Erman, E. Korkmaz, and B. Ozgur, On the Inverse Problem of the Fractional Heat-Like Partial Differential Equations: Determination of the Source Function, Adv. Math. Phys. 2013 (2013), Art. ID 476154, 8 pp.
  22. M. Renardy, W. J. Hursa, and J. A. Nohel, Mathematical Problems in Viscoelasticity, Wiley, New York, 1987.
  23. G. M. Vainikko and A. Y. Veretennikov, Iteration Procedures in Ill-Posed Problems, Moscow, Nauka (in Russian) McCormick, S.F., 1986.
  24. R. H. S. Winterton, Heat Transfer, Oxford University Press, Oxford, 1997.
  25. C. Zheng and G. D. Bennett, Applied Contaminant Transport Modelling: Theory and Practice, Van Nostrand Reinhold, New York, 1995.

Acknowledgement

Supported by : NSF