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Regularized Solution of the Cauchy Problem for the Biharmonic Equation

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Abstract

In this paper, the Cauchy problem associated with the biharmonic equation is investigated. We prove that in principle, the problem is severely ill-posed in the sense of Hadamard. Therefore, we propose a quasi-boundary value-type regularization method for stabilizing the ill-posed problem. Very sharp convergence estimates are established based on some a priori information on the exact solution. Finally, several numerical examples with random data are provided to show the effectiveness of the proposed method.

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References

  1. Atakhodzhaev, M.A.: Ill-Posed Internal Boundary Value Problems for the Biharmonic Equation. Inverse and Ill-Posed Problems Series. VSP, Utrecht (2002)

    MATH  Google Scholar 

  2. 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)

    Article  MathSciNet  Google Scholar 

  3. Cao, C., Rammaha, M.A., Titi, E.S.: The Navier–Stokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom. Z. Angew. Math. Phys. 50(3), 341–360 (1999)

    Article  MathSciNet  Google Scholar 

  4. Clark, G.W., Oppenheimer, S.F.: Quasi-reversibility methods for non-well-posed problems. Electron. J. Differ. Equ. 1994(8), 1 (1994)

    Google Scholar 

  5. Cheng, W., Ma, Y.J.: A modified quasi-boundary value method for solving the radially symmetric inverse heat conduction problem. Appl. Anal. 96(15), 2505–2515 (2017)

    Article  MathSciNet  Google Scholar 

  6. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Eldén, L., Berntsson, F.: A stability estimate for a Cauchy problem for an elliptic partial differential equation. Inverse Probl. 21(5), 1643 (2005)

    Article  MathSciNet  Google Scholar 

  9. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Springer, Cham (1996)

    Book  Google Scholar 

  10. Feng, X., Ning, W., Qian, Z.: A quasi-boundary-value method for a Cauchy problem of an elliptic equation in multiple dimensions. Inverse Probl. Sci. Eng. 22(7), 1045–1061 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Hieu, P.T., Quan, P.H.: On regularization and error estimates for the Cauchy problem of the modified inhomogeneous Helmholtz equation. J. Inverse Ill-Posed Probl. 24(5), 515–526 (2016)

    Article  MathSciNet  Google Scholar 

  13. Ingham, D.B., Kelmanson, M.A.: Boundary Integral Equation Analysis of Singular, Potential and Biharmonic Problems. Springer, Berlin (1984)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Khanh, T.Q., Hoa, N.V.: On the axisymmetric backward heat equation with non-zero right hand side: regularization and error estimates. J. Comput. Appl. Math. 335, 156–167 (2018)

    Article  MathSciNet  Google Scholar 

  16. Khanh, T.Q., Khieu, T.T., Hoa, N.V.: Stabilizing the nonlinear spherically symmetric backward heat equation via two-parameter regularization method. J. Fixed Point Theory Appl. 19(4), 2461–2481 (2017)

    Article  MathSciNet  Google Scholar 

  17. 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 

  18. Marin, L., Lesnic, D.: The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation. Math. Comput. Modell. 42(3), 261–278 (2005)

    Article  MathSciNet  Google Scholar 

  19. Lesnic, D., Zeb, A.: The method of fundamental solutions for an inverse internal boundary value problem for the biharmonic equation. Int. J. Comput. Methods 06(4), 557–567 (2009)

    Article  MathSciNet  Google Scholar 

  20. Quan, P.H., Trong, D.D., Triet, L.M., Tuan, N.H.: A modified quasi-boundary value method for regularizing of a backward problem with time-dependent coefficient. Inverse Probl. Sci. Eng. 19(3), 409–423 (2011)

    Article  MathSciNet  Google Scholar 

  21. Qian, A., Yang, X., Wu, Y.: Optimal error bound and a quasi-boundary value regularization method for a Cauchy problem of the modified Helmholtz equation. Int. J. Comput. Math. 93(12), 2028–2041 (2016)

    Article  MathSciNet  Google Scholar 

  22. Showalter, R.E.: Cauchy problem for hyper-parabolic partial differential equations. In: Eydeland, A. (ed.) Trends in the Theory and Practice of Non-linear Analysis. Elsevier, Amsterdam (1983)

    Google Scholar 

  23. Schaefer, P.: On the Cauchy problem for the nonlinear biharmonic equation. J. Math. Anal. Appl. 36(3), 660–673 (1971)

    Article  MathSciNet  Google Scholar 

  24. Schaefer, P.: Bounds in the Cauchy problem for a fourth order quasi-linear equation. SIAM J. Appl. Math. 21(1), 44–50 (1971)

    Article  MathSciNet  Google Scholar 

  25. Schaefer, P.: On existence in the Cauchy problem for the biharmonic equation. Compos. Math. 28, 203–207 (1974)

    Article  MathSciNet  Google Scholar 

  26. Selvadurai, A.P.S.: Partial Differential Equations in Mechanics 2: The Biharmonic Equation, Poissons Equation. Springer, Cham (2013)

    Google Scholar 

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

    MATH  Google Scholar 

  28. Trong, D.D., Quan, P.H., Khanh, T.V., Tuan, N.H.: A nonlinear case of the 1-D backward heat problem: regularization and error estimate. Z. Anal. Anwend. 26(2), 231–245 (2007)

    Article  MathSciNet  Google Scholar 

  29. Trong, D.D., Tuan, N.H.: A nonhomogeneous backward heat problem: regularization and error estimates. Electron. J. Differ. Equ. 2008(33), 114 (2008)

    MathSciNet  MATH  Google Scholar 

  30. Trong, D.D., Duy, B.T., Minh, M.N.: Backward heat equations with locally lipschitz source. Appl. Anal. 94(10), 2023–2036 (2015)

    Article  MathSciNet  Google Scholar 

  31. 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(18–19), 8518–8532 (2013)

    Article  MathSciNet  Google Scholar 

  32. Wei, T., Wang, J.G.: A modified quasi-boundary value method for the backward time-fractional diffusion problem. ESAIM Math. Model. Numer. Anal. 48(2), 603–621 (2014)

    Article  MathSciNet  Google Scholar 

  33. Wei, T., Wang, J.G.: A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation. Appl. Numer. Math. 78, 95–111 (2014)

    Article  MathSciNet  Google Scholar 

  34. 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)

    Article  MathSciNet  Google Scholar 

  35. Zhang, H.W.: Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation. Int. J. Comput. Math. 89(12), 1689–1703 (2012)

    Article  MathSciNet  Google Scholar 

  36. Zhang, H.W., Wei, T.: An improved non-local boundary value problem method for a Cauchy problem of the Laplace equation. Numer. Algorithms 59(2), 249–269 (2012)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Computational Science and Technology, Ho Chi Minh City, Vietnam

    Tran Nhat Luan & Tran Thi Khieu

  2. Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City National University, No. 227 Nguyen Van Cu Street, Ward 4, District 5, Ho Chi Minh City, Vietnam

    Tran Thi Khieu

  3. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Tra Quoc Khanh

  4. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Tra Quoc Khanh

Authors
  1. Tran Nhat Luan
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  2. Tran Thi Khieu
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  3. Tra Quoc Khanh
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Correspondence to Tra Quoc Khanh.

Additional information

Norhashidah Hj. Mohd. Ali.

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Luan, T.N., Khieu, T.T. & Khanh, T.Q. Regularized Solution of the Cauchy Problem for the Biharmonic Equation. Bull. Malays. Math. Sci. Soc. 43, 757–782 (2020). https://doi.org/10.1007/s40840-018-00711-7

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  • DOI: https://doi.org/10.1007/s40840-018-00711-7

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