Development of alternative perturbation methods for solving strong nonlinear dynamics problems

In this work, a new perturbation method, Multiple Scales Lindstedt Poincare method, (MSLP) producing valid solutions for strong nonlinear systems is developed. The new method is an integration of Method of Multiple Scales (MS) and Lindstedt-Poincare (LP) Method.The outline of the new method and the guidelines are depicted first. Then new method is applied to linear damped vibration equation, Duffing equation, damped cubic nonlinear equation and an equation with quadratic and cubic nonlinearities. Solutions of multiple scales method and MSLP method are contrasted with the numerical solutions. MSLP method produced solutions with good agreement with the numerical solutions for strongly nonlinear systems.Forced vibrations are considered next. The new method is applied to forced vibrations of cubic nonlinear equation with damping. MSLP solutions are contrasted with multiple scales method and numerical simulations. For strongly nonlinear systems, frequency response curves of MSLP method and numerical solutions are in good agreement.Finally, partial differential equations are considered. Using the general operator notation, the new method is applied to quadratic and cubic nonlinear partial differential equation. Solutions of Multiple Scales method and MSLP method are contrasted.