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Ansatz and Averaging Methods for Modeling the (Un)Conserved Complex Duffing Oscillators

Author

Listed:
  • Weaam Alhejaili

    (Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia)

  • Alvaro H. Salas

    (FIZMAKO Research Group, Department of Mathematics and Statistics, Universidad Nacional de Colombia, Manizales 170001, Colombia)

  • Samir A. El-Tantawy

    (Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt
    Research Center for Physics (RCP), Department of Physics, Faculty of Science and Arts, Al-Mikhwah, Al-Baha University, Al-Baha 1988, Saudi Arabia)

Abstract

In this study, both the ansatz and averaging methods are carried out for analyzing the complex Duffing oscillators including the undamped/conserved complex Duffing oscillator (CDO) and the damped/unconserved CDO to obtain some approximate analytical solutions. To analyze the conserved CDO, it is reduced to two decoupled conserved Duffing oscillators. After that, the exact solution of the conserved Duffing oscillator is employed to derive an approximation of the conserved CDO in terms of the Jacobi elliptic function. To analyze the damped CDO, two methodologies are considered. For the first methodology, the damped CDO is reduced to two decoupled damped Duffing oscillators, and the ansatz method is devoted to analyzing the damped Duffing oscillator. Accordingly, an approximation of the damped CDO in terms of trigonometric functions is obtained. In the second methodology, the averaging method is applied directly to the damped CDO to derive an approximation in terms of trigonometric functions. All the obtained solutions are compared with the fourth-order Runge–Kutta (RK4) numerical approximations. This study may help many researchers interested in the field of plasma physics to interpret their laboratory and observations results.

Suggested Citation

  • Weaam Alhejaili & Alvaro H. Salas & Samir A. El-Tantawy, 2023. "Ansatz and Averaging Methods for Modeling the (Un)Conserved Complex Duffing Oscillators," Mathematics, MDPI, vol. 11(9), pages 1-12, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2007-:d:1131078
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    References listed on IDEAS

    as
    1. El-Tantawy, S.A. & Salas, Alvaro H. & Alyousef, Haifa A. & Alharthi, M.R., 2022. "Novel approximations to a nonplanar nonlinear Schrödinger equation and modeling nonplanar rogue waves/breathers in a complex plasma," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    2. Weaam Alhejaili & Alvaro H. Salas & Samir A. El-Tantawy, 2022. "Novel Approximations to the (Un)forced Pendulum–Cart System: Ansatz and KBM Methods," Mathematics, MDPI, vol. 10(16), pages 1-12, August.
    3. Selina Akter & Harun-Or-Roshid & N. F. M. Noor & Mohammad Mirzazadeh, 2022. "New Solitons and Multishock Wave Structures for the Conformable Space Fractional Burger and Time Fractional Sharma-Tasso-Olver Models," Advances in Mathematical Physics, Hindawi, vol. 2022, pages 1-19, May.
    4. Cveticanin, L., 2001. "Analytic approach for the solution of the complex-valued strong non-linear differential equation of Duffing type," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 297(3), pages 348-360.
    5. Weaam Alhejaili & Alvaro H. Salas & Samir A. El-Tantawy, 2022. "Analytical and Numerical Study on Forced and Damped Complex Duffing Oscillators," Mathematics, MDPI, vol. 10(23), pages 1-13, November.
    6. Mahmoud, Gamal M., 1998. "Approximate solutions of a class of complex nonlinear dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 253(1), pages 211-222.
    7. Cveticanin, L., 2003. "Analytic solution of the system of two coupled differential equations with the fifth-order non-linearity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 317(1), pages 83-94.
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