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A filter method with a priori and a posteriori parameter choice for the regularization of Cauchy problems for biharmonic equations


In the present paper, we devote our effort to Cauchy boundary value problems for biharmonic equations. In general, the investigated problem is ill-posed. Therefore, we develop a filter method to defeat the ill-posedness of the problem. Explicit convergence rate is established under both a priori and a posteriori parameter choice rules. Finally, a numerical example is presented to illustrate the ill-posedness of the problem as well as the effectiveness of the proposed method.

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  1. 1.

    Andersson, L. E., Elfving, T., Golub, G. H.: Solution of biharmonic equations with application to radar imaging. J. Comput. Appl. Math. 94(2), 153–180 (1998)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Beck, J V, Blackwell, Ben, St Clair, Jr., C.R.: Inverse Heat Conduction Ill-posed Problems. A Wiley-Interscience, New York (1985)

    MATH  Google Scholar 

  3. 3.

    Benrabah, A., Boussetila, N.: Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation. Inverse Probl. Sci. Eng., 1–29 (2018)

  4. 4.

    Doan, V.N., Nguyen, H.T, Vo, A.K., Vo, V.A.: A note on the derivation of filter regularization operators for nonlinear evolution equations. Appl. Anal. 97 (1), 3–12 (2018)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Ehrlich, L. N., Gupta, M. M.: Some difference schemes for the biharmonic equation. SIAM J. Numer. Anal. 12(5), 773–790 (1975)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Eldén, L., Berntsson, F., Regińska, T.: Wavelet and Fourier methods for solving the sideways heat equation. SIAM J. Sci. Comput. 21(6), 2187–2205 (2000)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems, vol. 375. Springer Science & Business Media (1996)

  8. 8.

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

    MathSciNet  Article  Google Scholar 

  9. 9.

    Hào, D.N., Van Duc, N., Lesnic, D.: A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Probl. 25(5), 055002, 27 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Huy, T.N., Kirane, M., Le, L.D., Van Nguyen, T.: Filter regularization for an inverse parabolic problem in several variables. Electron. J Differential Equations, pages Paper 24, 13 (2016)

  11. 11.

    Kal’menov, T., Iskakova, U.: On an ill-posed problem for a biharmonic equation. Filomat 31(4), 1051–1056 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kal’menov, T.S., Sadybekov, M.A., Iskakova, U.A.: On a criterion for the solvability of one ill-posed problem for the biharmonic equation. J. Inverse Ill-Posed Probl. 24(6), 777–783 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Lai, M. C., Liu, H. C.: Fast direct solver for the biharmonic equation on a disk and its application to incompressible flows. Appl Math Comput. 164(3), 679–695 (2005)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Landau, M. D., Lifshits, E. M.: Theory of Elasticity. Pergamon Press, Oxford (1986)

    Google Scholar 

  15. 15.

    Hong, P.L., Minh, T.L., Quan, P.H.: On a three dimensional Cauchy problem for inhomogeneous Helmholtz equation associated with perturbed wave number. J. Comput. Appl. Math. 335, 86–98 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Qian, Z., Chu-Fu, L., Li, Z.-P.: Two regularization methods for a C,auchy problem for the Laplace equation. J. Math. Anal Appl. 338(1), 479–489 (2008)

    MathSciNet  Article  Google Scholar 

  17. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 18.

    Selvadurai, A. P. S.: Partial Differential Equations in Mechanics 2: The biharmonic equation, Poisson’s equation. Springer Science and Business Media (2013)

  19. 19.

    Timoshenko, S., Goodier, J. N.: Theory of Elasticity. McGraw-Hill, New York (1951)

  20. 20.

    Zeb, A., Ingham, D. B., Lesnic, D.: The method of fundamental solutions for a biharmonic inverse boundary determination problem. Comput. Mech. 42(3), 371–379 (2008)

    MathSciNet  Article  Google Scholar 

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Correspondence to Tra Quoc Khanh.

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Luan, T.N., Khieu, T.T. & Khanh, T.Q. A filter method with a priori and a posteriori parameter choice for the regularization of Cauchy problems for biharmonic equations. Numer Algor 86, 1721–1746 (2021).

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  • Biharmonic equation
  • Ill-posed problem
  • Filter method
  • Cauchy boundary value problem

Mathematics Subject Classification 2010

  • Primary 31B30
  • 47A52
  • 65F22
  • 65J20