A mountain pass method for the numerical solution of semilinear wave equations. (English) Zbl 0796.65109
The authors study periodic solutions of the semilinear vibrating string model equation \(u_{tt} - u_{xx} + au^ + -bu^ - = -\sin (\pi x) [1-\gamma \sin (2 \pi t)]\), \(u(0,t) = u(1,t) = 0\), \(u(x,t+1) = u(x,t)\) (here \(u^ + = {1 \over 2} (| u | + u)\), \(u^ - = {1 \over 2} (| u | - u)\). The problem is reformulated as a dual variational problem in such a way that the obvious solution obtained as a solution of the linear problem, is a local minimum. Then additional non-obvious solutions can be found by using a numerical mountain pass algorithm. Three test calculations illustrate the method.
Reviewer: O.Titow (Berlin)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 74K05 | Strings |
| 35L70 | Second-order nonlinear hyperbolic equations |
Keywords:
semilinear wave equations; numerical examples; semilinear vibrating string equation; periodic solutions; dual variational problem; numerical mountain pass algorithmReferences:
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