Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T08:32:56.697Z Has data issue: false hasContentIssue false
  • English
  • Français

Identifiability for Linearized Sine-GordonEquation

Published online by Cambridge University Press:  28 January 2013

J. Ha  and
S. Gutman
J. Ha
Affiliation:
School of Liberal Arts, Korea University of Technology and Education Cheonan 330-708, South Korea
S. Gutman*
Affiliation:
Department of Mathematics, University of Oklahoma Norman, Oklahoma 73019, USA
*
∗ Corresponding author. E-mail: [email protected]
Get access

Abstract

The paper presents theoretical and numerical results on the identifiability, i.e. theunique identification for the one-dimensional sine-Gordon equation. The identifiabilityfor nonlinear sine-Gordon equation remains an open question. In this paper we establishthe identifiability for a linearized sine-Gordon problem. Our method consists of a carefulanalysis of the Laplace and Fourier transforms of the observation of the system, conductedat a single point. Numerical results based on the best fit to data method confirm that theidentification is unique for a wide choice of initial approximations for the sought testparameters. Numerical results compare the identification for the nonlinear and thelinearized problems.

Keywords

Identificationidentifiabilitysine-Gordon equation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 106 - 121
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Bishop, A. R., Fesser, K., Lomdahl, P. S.. Influence of solitons in the initial state on chaos in the driven damped sine-Gordon system. Physica D, 7 (1983), 259–279. CrossRefGoogle Scholar
N. Dunford, J. Schwartz. Linear operators, Part I. Wiley & Sons, New York, NY. Reprint of the 1958 original, 1988.
L. C. Evans. Partial differential equations. Graduate studies in Mathematics, vol. 19. American Mathematical Society, Providence, R.I, 1998.
Gutman, S., Ha, J.. Identifiability of piecewise constant conductivity in a heat conduction process. SIAM J. Control and Optimization, 46 (2) (2007), 694–713. CrossRefGoogle Scholar
Gutman, S., Ha, J.. Parameter identifiability for heat conduction with a boundary input. Math. Comp. Simul., 79 (2009), 2192–2210. CrossRefGoogle Scholar
Ha, J., Gutman, S.. Optimal parameters for a damped sine-Gordon equation. J. Korean Math. Soc., 46 (2009), 1105–1117. CrossRefGoogle Scholar
Ha, J., Nakagiri, S.. Identification of constant parameters in perturbed sine-Gordon equations. J. Korean Math. Soc., 43(5) (2006), 931–950. CrossRefGoogle Scholar
V. Isakov. Inverse problems for partial differential equations. Second edition. Applied Mathematical Sciences, vol. 127. Springer, New York, 2006.
Kitamura, S., Nakagiri, S.. Identifiability of spatially-varying and constant parameters in distributed systems of parabolic type. SIAM J. Control and Optimization, 15 (1977), 785–802. CrossRefGoogle Scholar
M. Levi. Beating modes in the Josephson junction. Chaos in Nonlinear Dynamical Systems, J. Chandra (Ed.) (1984), SIAM, Philadelphia.
J. L. Lions. Optimal control of systems governed by partial differential equations. Springer-Verlag, New York, 1971.
A. C. Metaxas, R. J. Meredith. Industrial microwave heating. Peter Peregrinus, London, 1993.
Nakagiri, S.. Review of Japanese work of the last 10 years on identifiability in distributed parameter systems. Inverse Problems, 9(2) (1993), 143–191. CrossRefGoogle Scholar
Ortega, R., Robles-Perez, A. M.. A maximum principle for periodic solutions of the telegraph equation. J. Math. Anal. Appl., 221 (1998), 625–651. CrossRefGoogle Scholar
Pierce, A.. Unique identification of eigenvalues and coefficients in a parabolic problem. SIAM J. Control and Optimization, 17(4) (1979), 494–499. CrossRefGoogle Scholar
W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical Recepies in FORTRAN, 2nd ed. Cambridge University Press, Cambridge, 1992.
N. F. Smyth, A. L. Worthy. Soliton evolution and radiation loss for the sine-Gordon equation. Physical Reviews E, (1999), 2330–2336.
M. Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, vol. 115. Springer, New York, 2011.
R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. Applied Mathematical Sciences, vol. 68, Springer-Verlag, 1997.
K. Yosida. Functional Analysis, 6th ed. Springer-Verlag, 1980.