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The quasi-boundary value method for identifying the initial value of heat equation on a columnar symmetric domain

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Abstract

In this paper, we consider an inverse problem for determining the initial value of heat equation with inhomogeneous source on a columnar symmetric domain. The quasi-boundary value regularization method is applied to solve this inverse problem. Under the a priori and a posteriori regularization parameter choice rules, the convergence estimates between the regularization solution and the exact solution are given. The numerical examples show this regularization method is effective and stable for dealing with this problem.

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References

  1. Ames, K.A., Clark, G.W., Epperson, J.F., Oppenheimer, S.F.: A comparison of regularizations for an ill-posed problem. Math. Comp. 67(224), 1451–1471 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ames, K.A., Epperson, J.F.: A kernel-based method for the approximate solution of backward parabolic problems. SIAM J. Numer. Anal. 34(4), 1357–1390 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheng, W., Ma, Y.J., Fu, C.L.: Indentifying an unknown source term in radial heat conduction. Inverse Probl. Sci. Eng. 20(3), 335–349 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Denche, M., Bessila, K.: A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 301(2), 419–426 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Feng, X.L., Eldén, L., Fu, C.L.: A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data. J. Inverse Ill-Posed Probl. 18(6), 617–645 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Feng, X.L., Eldén, L.: Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method. Inverse Probl. 30(1), 15005–15021 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Feng, X.L., Eldén, L., Fu, C.L.: Stability and regularization of a backward parabolic PDE with variable coefficients. J. Inverse Ill-Posed Probl. 18, 217–243 (2010)

    MATH  Google Scholar 

  8. Fu, C.L., Xiong, X.T., Qian, Z.: Fourier regularization for a backward heat equation. J. Math. Anal. Appl. 331(1), 472–480 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Han, H., Ingham, D.B., Yuan, Y.: The boundary element method for the solution of the backward heat conduction equation. J. Comput. Phys. 116(2), 292–299 (1995)

    Article  MATH  Google Scholar 

  10. Jourhmane, M., Mera, N.S.: An iterative algorithm for the backward heat conduction problem based on variable relaxation factors. Inverse Probl. Eng. 10(4), 293–308 (2002)

    Article  Google Scholar 

  11. Lattès, R., Lions, J.L.: The method of quasi-reversibility. Applications to partial differential equations, translated from the French edition and edited by Richard Bellman. Modern analytic and computational methods in science and mathematics, vol. 18. American Elsevier Publishing Co., Inc., New York (1969)

    MATH  Google Scholar 

  12. Le, T. M., Pham, Q. H., Dang, T. D., Nguyen, T. H.: A backward parabolic equation with a time-dependent coefficient: regularization and error estimates. J. Comput. Appl. Math. 237, 432–441 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, C.S.: Group preserving scheme for backward heat conduction problems. Int. J. Heat Mass Transf. 47(12-13), 2567–2576 (2004)

    Article  MATH  Google Scholar 

  14. Liu, J.J.: Numerical solution of forward and backward problem for 2-d heat conduction equation. J. Comput. Appl. Math. 145(2), 459–482 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, C.S., Chang, C.W., Chang, J.R.: Past cone dynamics and backward group preserving schemes for backward heat conduction problems. Comput. Model. Eng. Sci. 12(1), 67–81 (2006)

    MathSciNet  MATH  Google Scholar 

  16. Liu, J.J., Yamamoto, M.: A backward problem for the time-fractional diffusion equation. Appl. Anal. 89(11), 1769–1788 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York (1996)

    Book  MATH  Google Scholar 

  18. Mera, N.S.: The method of fundamental solutions for the backward heat conduction problem. Inverse Probl. Sci. Eng. 13(1), 65–78 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mera, N.S., Elliott, L., Ingham, D.B., Lesnic, D.: An iterative boundary element method for solving the one-dimensional backward heat conduction problem. Int. J. Heat Mass Transf. 44(10), 1937–1946 (2001)

    Article  MATH  Google Scholar 

  20. Qian, Z., Fu, C.L., Shi, R.: A modified method for a backward heat conduction problem. Appl. Math. Comput. 185(1), 564–573 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Qin, H.H., Wei, T.: Some filter regularization methods for a backward heat conduction problem. Appl. Math. Comput. 217(24), 10317–10327 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Seidman, T.I.: Optimal filtering for the backward heat equation. SIAM J. Numer. Anal. 33(1), 162–170 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Showalter, R.E.: The final value problem for evolution equations. J. Math. Anal. Appl. 47(3), 563–572 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  24. Showalter, R.E.: Cauchy problem for hyper-parabolic partial differential equations. North-Holland Math. Stud. 110, 421–425 (1985)

    Article  MATH  Google Scholar 

  25. Wang, J.G., Wei, T., Zhou, Y.B.: Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. Appl. Math. Model. 37, 8518–8532 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xiong, X.T., Fu, C.L., Qian, Z.: Two numerical methods for a backward heat conduction equation. Appl. Math. Comput. 179, 370–377 (2006)

    MathSciNet  MATH  Google Scholar 

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Funding

The project is supported by the National Natural Science Foundation of China (No. 11561045), the Doctor Fund of Lan Zhou University of Technology.

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Authors and Affiliations

  1. School of Science, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People’s Republic of China

    Fan Yang, Ya-Ru Sun, Xiao-Xiao Li & Can-Yun Huang

Authors
  1. Fan Yang
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  2. Ya-Ru Sun
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  3. Xiao-Xiao Li
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  4. Can-Yun Huang
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Correspondence to Fan Yang.

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Yang, F., Sun, YR., Li, XX. et al. The quasi-boundary value method for identifying the initial value of heat equation on a columnar symmetric domain. Numer Algor 82, 623–639 (2019). https://doi.org/10.1007/s11075-018-0617-9

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  • DOI: https://doi.org/10.1007/s11075-018-0617-9

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