Abstract
We are concerned with the wave propagation in a homogeneous 2D or 3D membrane Ω of finite size. We assume that either the membrane is initially at rest or we know its initial shape (but not necessarily both) and its boundary is subject to a known boundary force. We address the question of estimating the needed time-dependent body force to exert on the membrane to reach a desired state at a given final time T. As an additional information, we ask for the displacement on the boundary. We consider the displacement either at a single point of the boundary or on the whole boundary. First, we show the uniqueness of solution of these inverse problems under natural conditions on the final time T. If, in addition, the displacement on the whole boundary is only time dependent (which means that the boundary moves with a constant speed), this condition on T is removed if Ω satisfies Schiffer’s property. Second, we derive a conditional Hölder stability inequality for estimating such a time-dependent force. Third, we propose a numerical procedure based on the application of the satisfier function along with the standard Fourier expansion of the solution to the problems. Numerical tests are given to illustrate the applicability of the proposed procedure.
Acknowledgements
The authors would like to thank the anonymous reviewers for their constructive comments and suggestions which have helped to improve the quality of the paper.
References
[1] M. Bellassoued and M. Yamamoto, Inverse source problem for the wave equation, Proceedings of the Conference on Differential and Difference Equations and Applications, Hindawi Publishing, New York (2006), 149–158. Search in Google Scholar
[2] J. R. Cannon and D. R. Dunninger, Determination of an unknown forcing function in a hyperbolic equation from overspecified data, Ann. Mat. Pura Appl. (4) 85 (1970), 49–62. 10.1007/BF02413529Search in Google Scholar
[3] G. Chavent, G. Papanicolaou, P. Sacks and W. W. Symes, Inverse Problems in Wave Propagation, IMA Vol. Math. Appl. 90, Springer, New York, 1997. 10.1007/978-1-4612-1878-4Search in Google Scholar
[4] R. Dalmasso, A note on the Schiffer conjecture, Hokkaido Math. J. 28 (1999), no. 2, 373–383. 10.14492/hokmj/1351001220Search in Google Scholar
[5]
J. Deng,
Some results on the Schiffer conjecture in
[6] H. W. Engl, O. Scherzer and M. Yamamoto, Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary data, Inverse Problems 10 (1994), no. 6, 1253–1276. 10.1088/0266-5611/10/6/006Search in Google Scholar
[7]
P. C. Hansen,
Analysis of discrete ill-posed problems by means of the
[8] A. Hazanee, M. I. Ismailov, D. Lesnic and N. B. Kerimov, An inverse time-dependent source problem for the heat equation, Appl. Numer. Math. 69 (2013), 13–33. 10.1016/j.apnum.2013.02.004Search in Google Scholar
[9] S. O. Hussein and D. Lesnic, Determination of a space-dependent force function in the one-dimensional wave equation, Electron. J. Bound. Elem. 12 (2014), no. 1, 1–26. 10.14713/ejbe.v12i1.1838Search in Google Scholar
[10] S. O. Hussein and D. Lesnic, Determination of forcing functions in the wave equation. Part II: The time-dependent case, J. Engrg. Math. 96 (2016), 135–153. 10.1007/s10665-015-9786-xSearch in Google Scholar
[11] V. A. Il’in and E. I. Moiseev, Optimization of boundary control of string vibrations by an elastic force on an arbitrary sufficiently large time interval, Differ. Equ. 42 (2006), 1775–1786. 10.1134/S0012266106120123Search in Google Scholar
[12] V. A. Il’in and E. I. Moiseev, Optimal boundary control by an elastic force at one end being force for an arbitrary sufficiently large time interval, Differ. Equ. 43 (2007), 1697–1704. 10.1134/S0012266107120099Search in Google Scholar
[13] V. A. Il’in and E. I. Moiseev, Optimization boundary control by displacements at two ends of a string for an arbitrary sufficiently large time interval, Differ. Equ. 43 (2007), 1569–1584. 10.1134/S0012266107110122Search in Google Scholar
[14] V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Sci. 127, Springer, New York, 1998. 10.1007/978-1-4899-0030-2Search in Google Scholar
[15] M. I. Ismailov, F. Kanca and D. Lesnic, Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions, Appl. Math. Comput. 218 (2011), no. 8, 4138–4146. 10.1016/j.amc.2011.09.044Search in Google Scholar
[16] B. T. Johansson, D. Lesnic and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan problem, Appl. Math. Model. 35 (2011), no. 9, 4367–4378. 10.1016/j.apm.2011.03.005Search in Google Scholar
[17] B. T. Johansson, D. Lesnic and T. Reeve, A method of fundamental solutions for two-dimensional heat conduction, Int. J. Comput. Math. 88 (2011), no. 8, 1697–1713. 10.1080/00207160.2010.522233Search in Google Scholar
[18] B. T. Johansson, D. Lesnic and T. Reeve, Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions, Inverse Probl. Sci. Eng. 19 (2011), no. 5, 659–677. 10.1080/17415977.2011.579610Search in Google Scholar
[19] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 2nd ed., Appl. Math. Sci. 120, Springer, New York, 2011. 10.1007/978-1-4419-8474-6Search in Google Scholar
[20] M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems 8 (1992), no. 4, 575–596. 10.1515/9783110745481Search in Google Scholar
[21] E. I. Moiseev and A. A. Kholomeeva, Optimal boundary control by displacement at one end of a string under a given elastic force at the other end, Proc. Steklov Inst. Math. 276 (2012), no. S1, S153–S160. 10.1134/S0081543812020125Search in Google Scholar
[22] E. I. Moiseev and A. A. Kholomeeva, Solvability of the mixed problem for the wave equation with a dynamic boundary condition (in Russian), Differ. Equ. 48 (2012), no. 10, 1392–1397, translation in Differ. Uravn. 4(8) (2012), no. 10, 1412–1417. 10.1134/S0012266112100096Search in Google Scholar
[23] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monogr. Textb. Pure Appl. Math. 231, Marcel Dekker, New York, 2000. 10.1201/9781482292985Search in Google Scholar
[24] K. Rashedi, H. Adibi and M. Dehghan, Application of the Ritz–Galerkin method for recovering the spacewise-coefficients in the wave equation, Comput. Math. Appl. 65 (2013), no. 12, 1990–2008. 10.1016/j.camwa.2013.04.005Search in Google Scholar
[25] K. Rashedi, H. Adibi and M. Dehghan, Determination of space-time-dependent heat source in a parabolic inverse problem via the Ritz–Galerkin technique, Inverse Probl. Sci. Eng. 22 (2014), no. 7, 1077–1108. 10.1080/17415977.2013.854354Search in Google Scholar
[26] K. Rashedi, H. Adibi, J. A. Rad and K. Parand, Application of meshfree methods for solving the inverse one-dimensional Stefan problem, Eng. Anal. Bound. Elem. 40 (2014), 1–21. 10.1016/j.enganabound.2013.10.013Search in Google Scholar
[27] K. Rashedi and M. Sini, Stable recovery of the time-dependent source term from one measurement for the wave equation, Inverse Problems 31 (2015), no. 10, Article ID 105011. 10.1088/0266-5611/31/10/105011Search in Google Scholar
[28] S. A. Williams, Analyticity of the boundary for Lipschitz domains without the Pompeiu property, Indiana Univ. Math. J. 30 (1981), no. 3, 357–369. 10.1512/iumj.1981.30.30028Search in Google Scholar
[29] M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems 11 (1995), no. 2, 481–496. 10.1088/0266-5611/11/2/013Search in Google Scholar
[30] L. Zalcman, A bibliographic survey of the Pompeiu problem, Approximation by Solutions of Partial Differential Equations (Hanstholm 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 365, Kluwer Academic Publisher, Dordrecht (1992), 185–194. 10.1007/978-94-011-2436-2_17Search in Google Scholar
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