The approximate solving methods for the cubic Duffing equation based on the Jacobi elliptic functions

摘要

In this paper various analytical asymptotic techniques for solving the strictly strong non-linear Duffing equation are investigated. The basic function used in the methods is the Jacobi elliptic one. The following methods are emphasized: (1) the elliptic harmonic balance method, (2) the elliptic Galerkin method (the weighted residual method), (3) the straightforward expansion method, (4) the elliptic Lindstedt-Poincare method (parameter-expanding method), (5) the elliptic Krylov-Bogolubov method (the parameter perturbation method), (6) homotopy perturbation method and (7) homotopy analysis method. The methods are tested on the Duffing equation which contains the additional quadratic non-linear term. The obtained approximate analytical solutions are compared with each other and with numerical 'exact' ones. It is shown that the analytical results exhibit good agreement with the numerical integration solutions even for moderate values of the system parameters. Besides, the methods give much accurate solutions in comparison to the previous one based on the trigonometric functions.