An operational approach to the Emden–Fowler equation

Abstract

Communicated by I. StratisIn this paper, we present a method for solving the Emden-Fowler equation using the Adomian polynomials and the operational calculus introduced by the authors.

Qualitative Properties of the Solutions to the Lane–Emden Equation in the Cylindrical Setup

“…In recent years, the Lane-Emden equation has been widely studied in several versions, although it can be solved in closed form in only a few cases. An approach based on operational calculus, initially introduced by Adomian [13], was outlined by Bengochea et al in some recent works [14][15][16]. In particular, in [14], a procedure is derived which is based on a linear operator acting on the set of all formal series, which turns out to be helpful in solving several kinds of differential equations with variable coefficients, fractional differential equations, and difference equations as well.…”

“…To the best of our knowledge, no previous study has ever captured such a qualitative distinction. The most recent studies, such as [17][18][19], rely on well-structured numerical methods for approximation or on novel techniques that are implemented in wider classes of problems (see [14][15][16]20]). We stress that our attempt is somewhat more specific, because a deep qualitative analysis may lead to new ideas to refine the search for the closed form solutions that are still unknown.…”

Operational Solution to the Nonlinear Klein-Gordon Equation

“…Since our operational methods are based on simple linear algebra they are simpler than those based on the Mikusiński operational calculus [9], and can be used to provide a rigorous foundation of the main results of Heaviside's operational calculus. Several applications of our methods can be found in [10][11][12][13], and some variations in [14,15].…”

“…Several applications of our methods can be found in Refs. [10][11][12][13], and some variations in Refs. [14][15].…”