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

Carmen Ionescu and Radu Constantinescu ()
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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

Mathematics, 2022, vol. 10, issue 15, 1-14

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.

Keywords: nonlinear differential equations; attached flow; Chafee–Infante equation; Fisher equation; Benjamin–Bona–Mahony equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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