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)

Abstract

We provide a new mathematical technique leading to the construction of the exact parametric or closed form solutions of the classes of Abel's nonlinear differential equations (ODEs) of the first kind. These solutions are given implicitly in terms of Bessel functions of the first and the second kind (Neumann functions), as well as of the free member of the considered ODE; the parameter being introduced furnishes the order of the above Bessel functions and defines also the desired solutions of the considered ODE as one-parameter family of surfaces. The nonlinear initial or boundary value problems are also investigated. Finally, introducing a relative mathematical methodology, we construct the exact parametric or closed form solutions for several degenerate Abel's equation of the first kind.

Exact and approximate solutions for optical solitary waves in nematic liquid crystals

“…These exact solutions also illustrate the connections between the models governing optical solitary waves in optical media with those ruling reaction-diffusion fronts, Fisher's equation [30], the equation governing a self-gravitating gas in astrophysics, the Lane-Emden equation [31], and the Schrödinger-Newton equations of quantum gravity [24]. The present work has derived a new cylindrically symmetric solution of the Lane-Emden equation in terms of the recent solution of Abel's equation [56,57].…”

“…Recently, an exact solution of Abel's equation of the second kind has been derived [56,57], so that, in principle, solutions of Fisher's equation [30] and the Lane-Emden equation [31] can also be found. The solution as found by Panayotounakos [57] is given as follows. Let us set…”

Stability of singular limit cycles for Abel equations

“…We remark that (1) can always be transformed into the simpler canonical form x = x 3 + C(t) or x = x 2 + C(t)x 3 for some function C (see for instance Sections 2 and 4 of [18] or the classical book [13]). These transformations are very useful for studying, for instance, the integrability of Abel equations.…”

Integrable Abel equations and Vein's Abel equation

“…The case r D 0 has been introduced more than 200 years ago by Abel [1]. Many solvable equations of this type are collected in [2], and other ones can be found in more recent works [3][4][5][6][7][8][9]. The transformation 1=y D w C s converts this equation in the form dy dx D r C .q 2 3rs/y C Â q 1 ds dx 2q 2 s C 3rs 2 Ã y 2 C .p q 1 s C q 2 s 2 rs 3 /y 3 ,…”