Abstract
In this paper, we deal with an inverse problem for the biharmonic equation to find an unknown boundary in the plane by using an additional information assumed on the remaining known part of the boundary. As a by-product, we can uniquely determine the solution everywhere in its domain of definition by supposing that the available data have Fourier expansions. The question of the existence and uniqueness of this inverse problem will be investigated, and we will conclude with some analytical examples to ensure the validity of this study.
References
Benrabah A, Boussetila N. Modified nonlocal boundary value problem method for an ill-posed problem 170 for the biharmonic equation. Inv Probs in Sci and Engi. 2018; 27(3): 340-368. https://doi.org/10.1080/17415977.2018.1461859
Ramm AG. A geometrical inverse problem. Inv Prob. 1986; 2(2): 2 L19. [Accessed 2022 Fev 03]. Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.228.4793&rep=rep1&type=pdf
Ramm AG. An inverse problem for biharmonic equation. Internat J Math and Math Sci. 1988; 11(2): 413-415. [Accessed 2022 Fev 03]. Available from: https://downloads.hindawi.com/journals/ijmms/1988/152649.pdf
Hadj A, Saker H. Integral equations method for solving a Biharmonic inverse problem in detection of Robin coefficients. Applied Nume Math. 2021 Feb; 160: 436-450. https://doi.org/10.1016/j.apnum.2020.10.005
Selvadurai APS. Partial Differential Equations in Mechanics 2. Biharmonic Equation, Poisson's Equation Springer. 2000. https://doi.org/10.1007/978-3-662-04006-5
Zeb A, Ingham DB, Lesnic D. The method of fundamental solutions for a biharmonic inverse boundary determination problem. Comput Mech. 2008; 42: 371-379. https://doi.org/10.1007/s00466-008-0246-6
Zeb A, Elliott L, Ingham DB, Lesnic D. A comparison of different methods to solve inverse biharmonic boundary value problems. Numer Meth Engng. 1999;45(12):1791-1806. https://doi.org/10.1002/(SICI)1097-0207(19990830)45:12%3C1791::AID-NME654%3E3.0.CO;2-Z
Tajani C, Kajtih H, Daanoun A. Iterative Method to Solve a Data Completion Problem for biharmonic Equation for Rectangular Domain. Annals West Univ Timi-Math Comp Sci. 2017; 55(1):129-147. https://doi.org/10.1515/awutm-2017-0010
Filippo G, Christoph GH, Guido S. Polyharmonic boundary value problems. A monograph on positivity preserving and nonlinear higher order elliptic equations in bounded domains. Springer-Verlag. 2010.
Sweers G. A survey on boundary conditions for the biharmonic. Compl Variabl Elliptic Equations. 2009; 54(2):79-93. https://doi.org/10.1080/17476930802657640
Lesnic D, Elliott L, Ingham DB. The boundary element solution of the Laplace and biharmonic equations subject to noisy data. Numer Meth Eng. 1998; 43(3) 479-492. https://doi.org/10.1002/(SICI)1097-0207(19981015)43:3%3C479::AID-NME430%3E3.0.CO;2-D
Gan J, Yuan H, Li S, Peng Q, Zhang H. A computing method for bending problem of thin plate on Pasternak foundation. Adv Mech Eng. 2020; 12(7): 1-10. https://doi.org/10.1177%2F1687814020939333
Li J. Application of radial basis meshless methods to direct and inverse biharmonic boundary value 195 problems. Commun Numer Methods Eng. 2005; 21(4):169-182. https://doi.org/10.1002/cnm.736
Blum H, Rannacher R, Leis R. On the boundary value problem of the biharmonic operator on domains with angular corners. Math Meth Appl Sci. 1980; 4(2):556-581. https://doi.org/10.1002/mma.1670020416
Andersson L, Elfving T, Golub GH. Solution of biharmonic equations with application to radar imaging. J Comput Applied Math. 1998; 94(3):153-180. https://doi.org/10.1016/S0377-0427(98)00079-X
Marin L, Lesnic D. The method of fundamental solutions for inverse boundary value problems associated with the two dimentional biharmonic equation. Math Comp Modell. 2005;42(3-4): 261-278. https://doi.org/10.1016/j.mcm.2005.04.004
Lai MC, Lui HC. Fast direct solver of the biharmonic equation on a disk and it's application to incompressible flows. Applied Math Comput. 2005; 164: 679-695. [Accessed 2022 Fev 03]. Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.904.2372&rep=rep1&type=pdf
Doschoris M. Towards a generalization of the separation of variables technique. Methods applications analysis. 2012; 19: 381-402. https://doi.org/10.4310/MAA.2012.V19.N4.A4
Glowinski R, Pironneau O. Numerical methods for the first biharmonic equation and for the two- dimensional stokes problem. SIAM Rev. 1979; 21(2): 167-212. https://doi.org/10.1137/1021028
Granlund S, Marola N. On the problem of unique continuation for the p-Laplace equation. Nonlinear Anal Theo, Meths and Apps. 2014; 101:89-97. https://doi.org/10.1016/j.na.2014.01.020
Kal'menov TS, Iskakova UA. On a boundary value problem for the biharmonic equation. AIP Conference Proceedings. 2015;1676(1) :10.1063-1.4930457. https://doi.org/10.1063/1.4930457
Shapeev V, Golushko S, Bryndin L, Belyaev V. The least squares collocation method for the biharmonic equation in irregular and multiply-connected domains. J Phys Conf Ser. 2019; 1268-012076. http://dx.doi.org/10.1088/1742-6596/1268/1/012076
Rundell W. Recovering an obstacle and a nonlinear conductivity from Cauchy data, Inverse Problems. 2008; 24(5):055015. https://doi.org/10.1088/0266-5611/24/5/055015
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