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Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation

El-Sayed Adel Abd Elaziz

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20220220

The nonlinear fractional-order cubic-quintic-heptic Duffing problem will be solved through a new numerical approximation technique. The suggested method is based on the Pell-Lucas polynomials’ operational matrix in the fractional and integer orders. The studied problem will be transformed into a nonlinear system of algebraic equations. The numerical expansion containing unknown coefficients will be obtained numerically via applying Newton’s iteration method to the claimed system. Convergence analysis and error estimates for the introduced process will be discussed. Numerical applications will be given to illustrate the applicability and accuracy of the proposed method.

Computing Simulation of the Generalized Duffing Oscillator Basedon EBM and MHPM

Azimi A., Azimi M.

Mechanics and Mechanical Engineering

2016

Vol. 20, nr 4

595--604

This paper is concerned with analytical approximate solutions, to the generalized Duffing oscillation. Modified Homotopy Perturbation Method (MHPM) and Energy Balance Method (EBM) are applied to solve nonlinear equation and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. The general solution can be used to yield the relationship between amplitude and frequency in different strengths of nonlinearity. To verify the accuracy of the present approach, illustrative examples are provided and compared with exact solutions. The procedure yields rapid convergence with respect to the exact solution obtained by numerical integration.