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Exact analytic solutions of the Abel, Emden–Fowler and generalized Emden–Fowler nonlinear ODEs

✍ Scribed by Dimitrios E. Panayotounakos; Dimitrios C. Kravvaritis

Publisher: Elsevier Science
Year: 2006
Tongue: English
Weight: 179 KB
Volume: 7
Category: Article
ISSN: 1468-1218
DOI: 10.1016/j.nonrwa.2005.03.025

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✦ Synopsis

Several basic particular nonlinear ordinary differential equations (ODEs) of the second-order in mathematical physics and nonlinear mechanics are reduced to equivalent equations of the Abel normal form yy x -y = f (x) by means of various admissible functional transformations. These equivalent equations do not admit exact analytic solutions in terms of known (tabulated) functions, since only very special cases of the above type of Abel equation can be solved in parametric form [

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## Abstract Motivated by the study of a two‐dimensional point vortex model, we analyse the following Emden–Fowler type problem with singular potential: where __V__(__x__) = __K__(__x__)/|__x__|^2α^ with α∈(0, 1), 0<__a__⩽__K__(__x__)⩽__b__< + ∞, ∀__x__∈Ω and ∥∇__K__∥~∞~⩽__C__. We first extend var