American Institute of Mathematical Sciences

Advanced
February  2008, 2(1): 121-131. doi: 10.3934/ipi.2008.2.121

An inverse boundary value problem for a nonlinear wave equation

Gen Nakamura 1, and  Michiyuki Watanabe 2,

1. 

Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan
2. 

Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Yamazaki 2641, Noda, Chiba, 278-8510, Japan

Received  June 2007 Published  January 2008

An inverse boundary value problem for nonlinear wave equation of divergence form in one space dimension is considered. By assuming the nonlinear term is unknown, we show the linear and quadratic part of this term can be identified from the Dirichlet to Neumann map. Here, the nonlinearity is only in terms of the first derivative with respect to the space variable, and the linear and quadratic parts are defined in terms of this derivative. The identification not only gives the uniqueness but also the reconstruction.
Keywords: Inverse boundary value problems, Nonlinear Wave equations..
Mathematics Subject Classification: Primary: 35R30, 35L70; Secondary: 35L2.
Citation: Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[1]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1
[2]

Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271
[3]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[4]

Runzhang Xu, Mingyou Zhang, Shaohua Chen, Yanbing Yang, Jihong Shen. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5631-5649. doi: 10.3934/dcds.2017244
[5]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380
[6]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[7]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851
[8]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83
[9]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[10]

Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks & Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007
[11]

Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta. One-dimensional nonlinear boundary value problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 179-191. doi: 10.3934/dcdss.2018011
[12]

Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347
[13]

Grey Ballard, John Baxley, Nisrine Libbus. Qualitative behavior and computation of multiple solutions of nonlinear boundary value problems. Communications on Pure & Applied Analysis, 2006, 5 (2) : 251-259. doi: 10.3934/cpaa.2006.5.251
[14]

Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059
[15]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061
[16]

Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107
[17]

Hisashi Morioka. Inverse boundary value problems for discrete Schrödinger operators on the multi-dimensional square lattice. Inverse Problems & Imaging, 2011, 5 (3) : 715-730. doi: 10.3934/ipi.2011.5.715
[18]

Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021094
[19]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
[20]

Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic & Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences