The new shift perturbation method with applications to vibrational problems

Abstract

Employing variable horizontal and vertical shifts for the unperturbed solution, a new perturbation method is proposed. The new method is named as the shift perturbation method. The existing perturbation methods are interpreted with respect to the horizontal and vertical shifts for the unperturbed solution. The new method is applied to some well-known vibrational problems. The method is capable of producing admissible approximate solutions for a wide range of problems such as the free and forced vibrations of the Duffing equation, equations with quadratic and cubic nonlinearities, variable amplitude nonlinear problems, and boundary layer problems. A variant of the method also produced valid solutions for large perturbation parameters.