Abstract
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.
Similar content being viewed by others
References
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)
Beck, J V, Blackwell, Ben, St Clair, Jr., C.R.: Inverse Heat Conduction Ill-posed Problems. A Wiley-Interscience, New York (1985)
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)
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)
Ehrlich, L. N., Gupta, M. M.: Some difference schemes for the biharmonic equation. SIAM J. Numer. Anal. 12(5), 773–790 (1975)
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)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems, vol. 375. Springer Science & Business Media (1996)
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)
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)
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)
Kal’menov, T., Iskakova, U.: On an ill-posed problem for a biharmonic equation. Filomat 31(4), 1051–1056 (2017)
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)
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)
Landau, M. D., Lifshits, E. M.: Theory of Elasticity. Pergamon Press, Oxford (1986)
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)
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)
Seidman, T.I.: Optimal filtering for the backward heat equation. SIAM J. Numer. Anal. 33(1), 162–170 (1996)
Selvadurai, A. P. S.: Partial Differential Equations in Mechanics 2: The biharmonic equation, Poisson’s equation. Springer Science and Business Media (2013)
Timoshenko, S., Goodier, J. N.: Theory of Elasticity. McGraw-Hill, New York (1951)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s11075-020-00951-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-020-00951-4