Laplace–Pade Parametric Homotopy Perturbation Method for Uncertain Nonlinear Oscillators

Nonlinear oscillators have wide applicability in science and engineering problems. In this paper, nonlinear oscillator having initial conditions varying over fuzzy numbers has been initially taken into consideration. Here, the fuzziness in the uncertain nonlinear oscillators has been handled using parametric form. Using parametric form in terms of r-cut, the nonlinear uncertain differential equations are reduced to parametric differential equations. Then, based on classical homotopy perturbation method (HPM), a parametric homotopy perturbation method (PHPM) is proposed to compute solution enclosure of such uncertain nonlinear differential equations. A sufficient convergence condition of parametric solution obtained using PHPM is also proved. Further, a parametric Laplace–Pade approximation is incorporated in PHPM for retaining the periodic characteristic of nonlinear oscillators throughout the domain. The efficiency of Laplace–Pade PHPM has been verified for uncertain Duffing oscillator. Finally, Laplace–Pade PHPM is also applied to solve other uncertain nonlinear oscillator, viz., Rayleigh oscillator, with respect to fuzzy parameters.