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Numerical mountain pass solutions of a suspension bridge equation. (English) Zbl 0877.35126

The author develops a new global numerical method to find the large amplitude oscillations of a nonlinear model for suspension bridges. The central idea of the method is to find the solutions in a straightforward global way, without requiring good initial guesses as to where solutions might be. The framework of the algorithm is based on a work by Choi and McKenna. The mountain pass algorithm is a constructive implementation of the mountain pass theorems of Ambrosetti, Rabinowitz, and Ekeland.

The basic idea of the method is quite intuitive. On a finite-dimensional approximation subspace, take a piecewise linear path joining the local minimum and a point whose image is lower. After calculating the maximum of the functional along the path, the path is deformed by pushing the point at which the maximum is located in the direction of the steepest descent. One repeats this step, stopping only when the critical point is reached. The algorithm is extremely robust and global in nature and it was tested in the cases where the mathematical proof guarantees the existence of large amplitude periodic solutions and when the solutions can be reduced to solutions of ordinary differential equations. Some solutions are obtained where there are no proofs of existence.

35Q72Other PDE from mechanics (MSC2000)
65N99Numerical methods for BVP of PDE
35G20General theory of nonlinear higher-order PDE
35A15Variational methods (PDE)