A quasi-linear evolution equation and the method of Galerkin
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- by R. W. Dickey
- Proc. Amer. Math. Soc. 37 (1973), 149-156
- DOI: https://doi.org/10.1090/S0002-9939-1973-0308620-3
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Abstract:
In this paper it is shown that under specified conditions on the initial data a certain infinite coupled system of ordinary differential equations has a solution satisfying an auxiliary convergence condition. The infinite system discussed is essentially the Galerkin expansion of the solution to a given quasi-linear wave equation. The results obtained suffice to prove the existence of a solution to this wave equation.References
- R. Narasimha, Non-linear vibrations of an elastic string, J. Sound. Vib. 8 (1968), 134-146.
- S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech. 17 (1950), 35–36. MR 34202
- R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl. 29 (1970), 443–454. MR 253617, DOI 10.1016/0022-247X(70)90094-6
- R. W. Dickey, Infinite systems of nonlinear oscillation equations related to the string, Proc. Amer. Math. Soc. 23 (1969), 459–468. MR 247189, DOI 10.1090/S0002-9939-1969-0247189-8
- J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61–90. MR 319440, DOI 10.1016/0022-247X(73)90121-2
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 69338
- Norman J. Zabusky, Exact solution for the vibrations of a nonlinear continuous model string, J. Mathematical Phys. 3 (1962), 1028–1039. MR 146545, DOI 10.1063/1.1724290
- Peter D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys. 5 (1964), 611–613. MR 165243, DOI 10.1063/1.1704154
- R. C. MacCamy and V. J. Mizel, Existence and nonexistence in the large of solutions of quasilinear wave equations, Arch. Rational Mech. Anal. 25 (1967), 299–320. MR 216165, DOI 10.1007/BF00250932
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 149-156
- MSC: Primary 35Q99
- DOI: https://doi.org/10.1090/S0002-9939-1973-0308620-3
- MathSciNet review: 0308620