Determination of temperature distribution and thermal flux for two-dimensional inhomogeneous sideways heat equations

Abstract

We investigate an inverse and ill-posed problem for the two-dimensional inhomogeneous heat equation in the presence of a general source term. The goal here consists of recovering not only the temperature distribution but also the thermal flux from the measure data. With the appearance of the general source term, this model gets far worse than its homogeneous counterpart. Based on an analysis of the instability caused in the solution, we propose a kernel regularization scheme to stabilize the investigated problem. The Hölder-type convergence estimates are achieved under some appropriate a priori assumptions. We further numerically demonstrate the theoretical results by proposing a robust algorithm based on the 2-D fast Fourier transform. The numerical outputs exemplify the feasibility and efficiency of the proposed method.

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References

  1. 1.

    Beck, J.V., Blackwell, B., Clair, C.R.S., Clair, C.R.S., Clair, C.S., Artûhin, E.A., Pavlovc, I.I: Inverse Heat Conduction: Ill-Posed Problems. Wiley-Interscience publication. Wiley (1985)

  2. 2.

    Carasso, A.: Determining surface temperatures from interior observations. SIAM J. Appl. Math. 42(3), 558–574 (1982)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Egger, H, Yi, H, Marquardt, W, Mhamdi, A: Efficient solution of a three-dimensional inverse heat conduction problem in pool boiling. Inverse Probl. 25 (9), 095006,19 (2009)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Eldén, L.: Approximations for a Cauchy problem for the heat equation. Inverse Probl. 3(2), 263–273 (1987)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Eldén, L.: Modified equations for approximating the solution of a Cauchy problem for the heat equation. In: Inverse and Ill-Posed Problems (Sankt Wolfgang, 1986), Volume 4 of Notes Rep. Math. Sci. Engrg., pp 345–350. Academic Press, Boston (1987)

  6. 6.

    Fu, C.-L., Xiong, X.-T., Fu, P.: Fourier regularization method for solving the surface heat flux from interior observations. Math Comput. Modelling 42(5–6), 489–498 (2005)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Fu, C., Qiu, C.: Wavelet and error estimation of surface heat flux. J. Comput. Appl Math. 150(1), 143–155 (2003)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Hào, D.N., Reinhardt, H.-J., Schneider, A.: Numerical solution to a sideways parabolic equation. Internat. J. Numer. Methods Engrg. 50(5), 1253–1267 (2001)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Hào, D.N., Schneider, A., Reinhardt, H.-J.: Regularization of a non-characteristic Cauchy problem for a parabolic equation. Inverse Probl. 11(6), 1247–1263 (1995)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Liu, S., Feng, L.: A revised Tikhonov regularization method for a cauchy problem of two-dimensional heat conduction equation. Math. Probl. Eng. 1216357, 8 (2018)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Qian, Z., Fu, C.-L., Xiong, X.-T.: A modified method for determining the surface heat flux of IHCP. Inverse Probl. Sci Eng. 15(3), 249–265 (2007)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Qian, Z.: Regularization methods for a Cauchy problem for a parabolic equation in multiple dimensions. J. Inverse Ill-Posed Probl. 17(9), 891–911 (2009)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Qian, Z.: A new generalized Tikhonov method based on filtering idea for stable analytic continuation. Inverse Probl. Sci. Eng. 26(3), 362–375 (2018)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Qian, Z., Feng, X.: A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation. Appl Anal. 96(10), 1656–1668 (2017)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Qian, Z, Fu, C.-L.: Regularization strategies for a two-dimensional inverse heat conduction problem. Inverse Probl. 23(3), 1053–1068 (2007)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Qiu, C.-Y., Fu, C.-L., Zhu, Y.-B.: Wavelets and regularization of the sideways heat equation. Comput. Math Appl. 46(5–6), 821–829 (2003)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Quan, P.H., Trong, D.D., Dinh Alain, P.N.: Sinc approximation of the heat flux on the boundary of a two-dimensional finite slab. Numer. Funct. Anal Optim. 27(5–6), 685–695 (2006)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Regińska, T.: Application of wavelet shrinkage to solving the sideways heat equation. BIT 41(5, suppl.), 1101–1110 (2001). BIT 40th Anniversary Meeting

    MathSciNet  Google Scholar 

  19. 19.

    Regińska, T., Eldén, L.: Solving the sideways heat equation by a wavelet-Galerkin method. Inverse Probl. 13(4), 1093–1106 (1997)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Sørli, K, Skaar, I.M.: Monitoring the wear-line of a melting furnace. Port Ludlow, WA, USA. In: 3rd International Conference on Inverse Problems in Engineering, ASME, Inverse Problems in Engineering, Theory and Practice (1999)

  21. 21.

    Sørli, K, Skaar, IM: Sensitivity analysis for thermal design and monitoring problems of refractories. Norway. Proceedings of CHT-04, CTH-04-132. ICHMT International Symposium on Advances in Computational Heat Transfer (2004)

  22. 22.

    Seidman, T.I., Eldén, L.: An “optimal filtering” method for the sideways heat equation. Inverse Probl. 6(4), 681–696 (1990)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Woodfield, P, Monde, M, Mitsutake, Y: Implementation of an analytical two-dimensional inverse heat conduction technique to practical problems. 49, 187–197, 01 (2006)

  24. 24.

    Xiong, X., Zhou, Q., Hon, Y.C.: An inverse problem for fractional diffusion equation in 2-dimensional case: Stability analysis and regularization. J. Math. Anal. Appl. 393(1), 185–199 (2012)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Zhao, J., Liu, S., Liu, T.: A modified kernel method for solving Cauchy problem of two-dimensional heat conduction equation. Adv. Appl. Math Mech. 7(1), 31–42 (2015)

    MathSciNet  Google Scholar 

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Funding

The work of the first-name author, Tran Nhat Luan, is supported by the Institute for Computational Science and Technology (ICST), Ho Chi Minh City, and Department of Science and Technology, Ho Chi Minh City (under the grant no. 456/QÐ-KHCNTT).

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Correspondence to Tra Quoc Khanh.

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Communicated by: Lothar Reichel

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Luan, T.N., Khanh, T.Q. Determination of temperature distribution and thermal flux for two-dimensional inhomogeneous sideways heat equations. Adv Comput Math 46, 54 (2020). https://doi.org/10.1007/s10444-020-09796-w

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Keywords

  • Two-dimensional sideways heat equation
  • Kernel regularization method
  • Stable estimate
  • Hölder-type error estimates

Mathematics subject classification (2010)

  • 35K05
  • 35Q79
  • 35R25
  • 35R30
  • 80A23