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The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation

Author

Listed:
  • Adel Elmandouh

    (Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
    Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt)

  • Aqilah Aljuaidan

    (Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia)

  • Mamdouh Elbrolosy

    (Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
    Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt)

Abstract

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3 D representation and a 2 D representation, as well as a 2 D contour plot.

Suggested Citation

  • Adel Elmandouh & Aqilah Aljuaidan & Mamdouh Elbrolosy, 2024. "The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation," Mathematics, MDPI, vol. 12(3), pages 1-15, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:3:p:383-:d:1325870
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    References listed on IDEAS

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    1. He, Chun-Hui & Liu, Chao, 2023. "Variational principle for singular waves," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    2. Yan Wang & Khaled A. Gepreel & Yong-Ju Yang, 2023. "VARIATIONAL PRINCIPLES FOR FRACTAL BOUSSINESQ-LIKE B(m,n) EQUATION," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(07), pages 1-8.
    3. Biswas, Swapan & Ghosh, Uttam & Raut, Santanu, 2023. "Construction of fractional granular model and bright, dark, lump, breather types soliton solutions using Hirota bilinear method," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    4. Tu, Jian-Min & Tian, Shou-Fu & Xu, Mei-Juan & Zhang, Tian-Tian, 2016. "On Lie symmetries, optimal systems and explicit solutions to the Kudryashov–Sinelshchikov equation," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 345-352.
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