Abstract
The initial boundary value problem is considered for the dynamic string equation . Its solution is found by means of an algorithm, the constituent parts of which are the Galerkin method, the modified Crank-Nicolson difference scheme used to perform approximation with respect to spatial and time variables, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. Errors of the three parts of the algorithm are estimated and, as a result, its total error estimate is obtained.
Similar content being viewed by others
References
Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string equation. Trans. Amer. Math. Soc. 348, 305–329 (1996)
Arosio, A., Spagnolo, S.: Global solution of the Cauchy problem for a nonlinear equation. In: Brezis, H., Lions, J.L. (eds.). Nonlinear PDE's and Their Applications. Collège de France Seminar, Pitman, Boston, 1984 pp. 1–26
Attigui, F.: Contrôle et stabilisation de câbles non linéaires, Repport de stage scientifique. Laboratoire des Matériaux et des Structures du Génie Civil, Université Paris XIII, France, 1999 pp. 158
Ball, J.M.: Saddle point analysis for an ordinary differential equation in a Banach space and an application to dynamic bucking of a beam. In: Dickey, R.W. (ed.) Nonlinear Elasticity, Proceedings. Academic Press, New York, 1973 pp. 93–160
Bernstein, S.: Sur une classe d'équations fonctionnelles aux dérivées partielles. Izv. Akad. Nauk SSSR Ser. Mat. 4, 17–26 (1940)
Böhm, M., Toboada, M.: Global existence and regularity of solutions of the nonlinear string equation. IMA Preprint Series No. 1078, Institute for Math. and Its Appl., University of Minnesota (1992)
Carrier, G.F.: On the nonlinear vibration of the elastic string. Quart. Appl Math. 3, 157–165 (1945)
Chipot, M., Rodrigues, J.F.: On a class of nonlinear nonlocal elliptic problems. Model. Math. Anal. Numér. 26, 447–467 (1992)
Christie, I., Sanz-Serna, J.M.: A Galerkin method for a nonlinear integro-differential wave system. Comput. Meth. Appl. Mech. Engrg. 44, 229–237 (1984)
Dickey, R.W.: Infinite systems of nonlinear oscillation equations related to the string. Proc. Amer. Math. Soc. 23, 459–468 (1969)
Dickey, R.W.: Infinite systems of nonlinear oscillation equations with linear damping. SIAM J. Appl. Math. 19, 208–214 (1970)
D'Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992)
D'Ancona, P., Spagnolo, S.: Global existence for the generalized Kirchhoff equation with small data. Arch. Rat. Mech. Anal. 124, 201–219 (1993)
D'Ancona, P., Spagnolo, S.: Nonlinear perturbations of the Kirchhoff equation. Comm. Pure Appl. Math. XLVII, 1005–1029 (1994)
Kirchhoff, G.: Vorlesungen über Mechanik. Teubner, Leipzig, 1883
Liu, I.S., Rincon, M.A.: Effect of moving boundaries on the vibrating elastic string. ELSGMLTM (APNUM): m2, v 1. 162, 1–14, Prn:10/07/2003
Ma, T.F.: Existence results for a model of nonlinear beam on elastic bearings. Appl. Math. Lett. 13, 11–15 (2000)
Ma, T.F., Miranda, E.S., De Souza Cortes, M.B.: A nonlinear differential equation involving of the argument. Archivum Mathematicum (BRNO) Tomus 40, 63–68 (2004)
Medeiros, L.A.: On a new class of nonlinear wave equations. J. Math. Anal. Appl. 69, 252–262 (1979)
Narasimha, R.: Nonlinear vibration of an elastic string. J. Sound Vibration 8, 134–146 (1968)
Newman, W.: Nonlinear string and beam equations, Ph. D. Dissertation, University of Wisconsin-Madison, May, 1993
Nishida, T.: A note on the nonlinear vibrations of the elastic string. Mem. Fac. Engrg. Kyoto Univ. 33, 329–341 (1971)
Nishihara, K.: On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math., v. 7, no. 2, 437–459 (1984)
Perla Menzala, G.: On classical solutions of a quasilinear hyperbolic equation. Nonlinear Anal. 3, 613–627 (1979)
Peradze, J.: Discussions of ``A Galerkin method for a nonlinear integro-differential wave system'' by I. Christie and J.M. Sanz-Serna, Comput. Meth. Appl. Mech.Engrg. 44 (1984), 229–237'', Comput. Meth. Appl. Mech.Engrg. 191, 5249–5250 (2002)
Pohozaev, S.I.: On a class of quasilinear hyperbolic equation. Mat. Sb. 96, 152–165 (1975), translated in Math. USSR Sb. 25, 145–158 (1975)
Rodriguez, P.H.R.: On local strong solutions of a nonlinear partial differential equation. Appl. Anal. 10, 93–104 (1980)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Peradze, J. A numerical algorithm for the nonlinear Kirchhoff string equation. Numer. Math. 102, 311–342 (2005). https://doi.org/10.1007/s00211-005-0642-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-005-0642-1