Finite difference approximate solutions for the strongly damped extensible beam equations. (English) Zbl 1026.74079

Summary: We obtain approximate solutions for strongly damped extensible beam equations using the method of lines and an implicit finite difference scheme. Existence and stability of the corresponding methods are studied, and error estimates are also obtained. Energy conservation and decay properties of finite difference approximate solutions are shown using the discrete energy method. Numerical results are also given in order to check the properties of analytical solutions.


74S20 Finite difference methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[2] Avrin, J. D., Energy convergence results for strongly damped nonlinear wave equations, Math. Z, 196, 7-12 (1987) · Zbl 0603.35065
[3] Ball, J. M., Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl, 42, 61-90 (1973) · Zbl 0254.73042
[4] Ball, J. M., Stability theory for an extensible beam, J. Diff. Eq, 14, 399-418 (1973) · Zbl 0247.73054
[7] Choo, S. M.; Chung, S. K., \(L^2\)-error estimate for the strongly damped extensible beam equations, Appl. Math. Lett, 11, 101-107 (1998) · Zbl 0965.74061
[9] Dickey, R. W., Free vibrations and dynamic buckling of the extensible beam, J. Math. Appl. Anal, 29, 443-454 (1970) · Zbl 0187.04803
[11] Geveci, T.; Christie, I., The convergence of a Galerkin approximation scheme for an extensible beam, \(M^2\) AN, 23, 597-613 (1989) · Zbl 0727.73093
[12] Woinowsky-Krieger, S., The effect of axial force on the vibration of hinged bars, J. Appl. Mech, 17, 35-36 (1950) · Zbl 0036.13302
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.