IMSc Webinar

Study of nonlinear systems using improved homotopy perturbation methods

Sagar Zephania C F

IIITDM Kancheepuram

The homotopy perturbation method (HPM) has been found to produce approximate analytical solutions in a simple way but with better accuracy than those obtained from some of the established approximation methods for some physically relevant anharmonic oscillators, including the autonomous conservative oscillator (ACO). Although HPM is found to be better than other approximation methods, deviation of HPM solution from the numerical solution is observed for ACO. The expansion of frequency and an auxiliary parameter () are incorporated into the homotopy perturbation method (HPM) framework to improve the accuracy by retaining its simplicity. Laplace transform is used to make the calculation simpler. This improved HPM (LH) is simple but provides highly accurate results for the autonomous conservative oscillator in comparison to classical HPM. An expansion of is considered in LH (LHh). The new method (LHh) is applied to find the approximate analytical solution for anharmonic oscillators with polynomial restoring forces in the generalized form for symmetric and asymmetric problems. A better accuracy (by order of magnitude) is achieved in LHh than in HPM and its recently modified versions. The improved HPM (LH) is applied to find out the analytical approximate soliton solutions for the Korteweg-de Vries-Burgers (KdVB) equation, which is a (1+1)-dimensional problem. Simple and compact analytical expressions for the solutions are obtained not only for the leading order but also for the higher-order approximations, which mimic the profiles of the exact results.

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