Abstract
In this paper, we study the inverse problem of identifying unknown sources for nonhomogeneous Biharmonic Equations in bounded domain. It shows that the problem is ill-posed. We use Landweber iterative regularization method to recover the ill-posed solution and give the convergence estimates under the priori and posteriori regularization parameter selection rules. Finally, several numerical examples are given to show the effectiveness and stability of this method.
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References
Andersson LE, Elfving T, Golub GH (1998) Solution of biharmonic equations with application to radar imaging. J Comput Appl Math 94(2):153–180
Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problem. Siam Imaging Sci 2:183–202
Benrabah A, Boussetila N (2018) Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation. Inverse Probl Sci Eng 2:1–29
Bergam A, Chakib A, Nachaoui A et al (2019) Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem. Appl Math Comput 346:865–878
Courant R, Friedrichs K, Lewy H (1928) Uber die partiellen differenzengleichungen der mathematischen physik. Math Ann 100(1):32–74
Engl HW, Hanke M, Neubauer A (1996) Regularization of Inverse Problem. Kluwer Academic, Amsterdam
Feng XL, Eldén L (2014) Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method. Inverse Probl 30(1):015005
Hua QND, Donal O, Van AV et al (2019) Regularization of an initial inverse problem for a biharmonic equation. Adv Difffer Equ 2019:255
Isakov V (2006) Inverse problems for Partial Differential Equations. Springer, New York
Kal’menov T, Iskakova U (2017) On an ill-posed problem for a biharmonic equation. Filomat 31(4):1051–1056
Klann E, Ramlau R (2008) Regularization by fractional filter methods and data smoothing. Inverse Probl. 24(2):025018
Landau MD, Lifshits EM (1986) Theory of Elasticity. Pergamon Press, Oxford
Lesnic D, Elliott L, Ingham DB (1998) The boundary element solution of the Laplace and biharmonic equations subject to noisy data. Int Numer Meth Eng 43:479–492
Li XX, Lei JL, Yang F (2004) An a posteriori Fourier regularization method for identifying the unknown source of the space-fractional diffusion equation. Inequal Appl 2014(1):1–13
Marin L, Lesnic D (2005) The method of fundamential solutions for inverse boundary value problems associated with the two dimensional biharmonic equation. Math Comput Model 42:261–278
Meleshko VV (2003) Selected topics in the history of the two-dimensional biharmonic problem. Appl Mech Rev 56(1):33–85
Monterde J, Ugail H (2006) A general 4th-order PDE method to generate Bézier surfaces from the boundary. Comput Aided Geom D 23(2):208–225
Wang FJ, Fan CM, Hua QS et al (2020) Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equation. Appl Math Comput 364:124658
Wang JG, Wei T, Zhou YB (2013) Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. Appl Math Model 37(18–19):8518–8532
Xiong XT, Fu CL, Li HF (2006) Fourier regularization method of a sideways heat equation for determining surface heat flux. Math Anal Appl 317(1):331–348
Yang F, Fan P, Li XX (2019) Fourier truncation regularization method for a three-dimensional Cauchy problem of the modified Helmholtz equation with perturbed wave number. Mathematics 7(8):705
Yang F, Fu CL (2015) The quasi-reversibility regularization method for identifying the unknown source for time fractional diffusion equation. Appl Math Model 39(5–6):1500–1512
Yang F, Fu CL, Li XX (2014) A mollification regularization method for unknown source in time-fractional diffusion equation. Int J Comput Math 91(7):1516–1534
Yang F, Fu CL, Li XX et al (2014) The Fourier regularization method for identifying the unknown source for the modified Helmholtz equation. Acta Math Sci 34(4):1040–1047
Yang F, Liu X, Li XX (2017) Landweber iteration regularization method for identifying unknown source of the modified Helmholtz equation. Bound Value Probl 2017(1):1–16
Yang F, Ren YP, Li XX (2018) Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain. Inverse Probl Sci Eng 26(8):1109–1129
Yang F, Sun YR, Li XX et al (2019) The quasi-boundary regularization value method for identifying the initial value of heat equation on a columnar symmetric domain. Numer Algorithm 82(2):623–639
Yang F, Wang N, Li XX et al (2019) A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain. Inverse Ill-posed Probl 27(5):609–621
Yang F, Zhang P, Li XX (2019) The truncation method for the Cauchy problem of the inhomogeneous Helmholtz equation. Appl Anal 98(5):991–1004
Yang F, Zhang Y, Li XX (2020) Landweber iteration regularization method for identifying the initial value problem of the time-space fractional diffusion-wave equation. Numer Algor 83:1509–1530
Zhang HW, Qin HH, Wei T (2011) A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation. Int J Comput Math 88(4):839–850
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Yang, F., Wang, QC. & Li, XX. Landweber Iterative Regularization Method for Identifying Unknown Source for the Biharmonic Equation. Iran J Sci Technol Trans Sci 45, 2029–2040 (2021). https://doi.org/10.1007/s40995-021-01189-y
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DOI: https://doi.org/10.1007/s40995-021-01189-y