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Solving Nonlinear Second-Order Differential Equations through the Attached Flow Method

Author

Listed:
  • Carmen Ionescu

    (Department of Physics, University of Craiova, 13 A.I.Cuza, 200585 Craiova, Romania)

  • Radu Constantinescu

    (Department of Physics, University of Craiova, 13 A.I.Cuza, 200585 Craiova, Romania)

Abstract

The paper considers a simple and well-known method for reducing the differentiability order of an ordinary differential equation, defining the first derivative as a function that will become the new variable. Practically, we attach to the initial equation a supplementary one, very similar to the flow equation from the dynamical systems. This is why we name it as the “attached flow equation”. Despite its apparent simplicity, the approach asks for a closer investigation because the reduced equation in the flow variable could be difficult to integrate. To overcome this difficulty, the paper considers a class of second-order differential equations, proposing a decomposition of the free term in two parts and formulating rules, based on a specific balancing procedure, on how to choose the flow. These are the main novelties of the approach that will be illustrated by solving important equations from the theory of solitons as those arising in the Chafee–Infante, Fisher, or Benjamin–Bona–Mahony models.

Suggested Citation

  • Carmen Ionescu & Radu Constantinescu, 2022. "Solving Nonlinear Second-Order Differential Equations through the Attached Flow Method," Mathematics, MDPI, vol. 10(15), pages 1-14, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2811-:d:883060
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    References listed on IDEAS

    as
    1. Dimitrios E. Panayotounakos & Theodoros I. Zarmpoutis, 2011. "Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2011, pages 1-13, October.
    2. Guicheng Shen & Yunchuan Sun & Yongping Xiong, 2013. "New Travelling-Wave Solutions for Dodd-Bullough Equation," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-5, July.
    3. Rubén Caballero & Alexandre N. Carvalho & Pedro Marín-Rubio & José Valero, 2021. "About the Structure of Attractors for a Nonlocal Chafee-Infante Problem," Mathematics, MDPI, vol. 9(4), pages 1-36, February.
    4. Vigirdas Mackevičius & Gabrielė Mongirdaitė, 2022. "Weak Approximations of the Wright–Fisher Process," Mathematics, MDPI, vol. 10(1), pages 1-20, January.
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