Generalized Emden-Fowler Equation Research Articles

Lie Algebra Representation, Conservation Laws and Some Invariant Solutions for a Generalized Emden-Fowler Equation

All generators of the optimal algebra associated with a generalization of the Endem-Fowler equation are showed; some of them allow to give invariant solutions. Variational symmetries and the respective conservation laws are also showed. Finally, a representation of Lie symmetry algebra is showed by groups of matrices.

Generalized Emden–Fowler equations related to constant curvature surfaces and noncentral curl forces

Generalized Emden–Fowler equations related to constant curvature surfaces and noncentral curl forces

Generalized Emden–Fowler equations in noncentral curl forces and first integrals

Noncentral force is an exemplary example of curl force and in general this is not an integrable system. The purpose of this note is twofold. First, we study a different reduction in the noncentral force compared to Berry and Shukla, and this leads to the generalized Emden–Fowler (GEF) equation, which in turn can be mapped to the Thomas–Fermi equation. Second, we compute the first integrals of the integrable standard Emden–Fowler (EF) and the generalized EF equations associated with the reduced noncentral dynamics using old results and new techniques. Finally, we compute the reduction in the nonpolynomial noncentral forces, which also leads to generalized EF equations.

On discrete groups and solvable equations of nonlinear oscillations

The autonomous ordinary differential equation of Lienard is considered. This equation and its generalizations arise in the theory of nonlinear oscillations, dynamical systems, etc. Currently for equations of the Lienard type, the problem of finding exact solutions is relevant. In view of this, the problem is to find connections in Lienard type equations with investigated nonlinear equations. These problems are solved in this article for the Lienard equation with three-parametrical coefficients of polynomial type. We found a transformation that reduces the equation under consideration of the well-studied generalized Emden-Fowler equation. We indicated the method of finding exact solutions of Lienard equations with various parameters. The transformations discrete group of dihedron for the class of generalized Emden-Fauler equations induces a discrete group of transformations for the corresponding class of Lienard equations. Likewise, known solutions of some particular generalized Emden-Fauler equations induce solutions of the Lienard equations. As examples, we found exact solutions in elementary (including polynomials) and special functions of several Lienard equations with certain parameters.

A new method to obtain either first- or second-order reductions for parametric polynomial ODEs

For a n th-order ordinary differential equation that is either polynomial on the dependent variable and its derivatives or polynomial in the derivatives of the dependent variable, a new method to obtain either first- or second-order reductions, respectively, is introduced. The corresponding reduced equations are generalized polynomials in the dependent variable or in its derivative. The method can be applied to equations that may have integer parameters in the exponents of the polynomial of the initial equation. A procedure to determine, in function of the parameters, if the given equation may admit such reductions and to restrict the possible variations of the exponents in the reduced equations is also provided. The introduced method strictly generalizes several methods that have recently appeared in the literature. As applications of the method, this study considers first-order reductions for a generalized Emden–Fowler equation and second-order reductions for the travelling-waves equation of a fifth-order Korteweg–de Vries equation and for a Painlevé–Chazy equation. As far as we know, some of the reported reductions are new. Several Maple programs to determine if a given polynomial equation admits any of the considered reductions and, if this is the case, to determine the reduced equations are also included.

Solvability of a boundary value problem with a Nagumo Condition

ABSTRACTThe aim of this paper is to discuss the existence and localization of solutions for a generalized Emden-Fowler equation involving a conformable derivative and with a Dirichlet boundary condition. Our approach is based on the lower and upper solutions method and Schauder fixed-point theorem under a Nagumo condition.

A complete and partial integrability technique of the Lorenz system

Abstract In this paper we deal with the well-known nonlinear Lorenz system that describes the deterministic chaos phenomenon. We consider an interesting problem with time-varying phenomena in quantum optics. Then we establish from the motion equations the passage to the Lorenz system. Furthermore, we show that the reduction to the third order non linear equation can be performed. Therefore, the obtained differential equation can be analytically solved in some special cases and transformed to Abel, Dufing, Painleve and generalized Emden-Fowler equations. So, a motivating technique that permitted a complete and partial integrability of the Lorenz system is presented.

A family of Emden–Fowler differential equations from a generalized derivative operator

The generalized Emden–Fowler equation was the focus of many research studies over the past years. In this communication, we introduce a generalized derivative operator which led to a family of Emden–Fowler differential equations starting from Lagrangian arguments.

Blow-up Solution for $4$-dimensional Generalized Emden-Fowler Equation with Exponential Nonlinearity

Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of generalized Emden-Fowler equation with exponential nonlinearity in fourth-dimensional given by \[ \begin{cases} \Delta (a(x) \Delta u) - V(x) \operatorname{div}(a(x) \nabla u) = \rho^{4} a(x) e^{u} & \textrm{in $\Omega \subset \mathbb{R}^{4}$}, u = \Delta u = 0 & \textrm{on $\partial \Omega$}. \end{cases} \] The leading part $\Delta$ is, usually, called Laplacian operator. The potential $V(x)$ belongs to $L^{\infty}_{\operatorname{loc}}(\mathbb{R}^{4})$ it is smooth and bounded and $a = a(x)$ is a given smooth function over $\overline{\Omega}$, called the Schrodinger wave function. Namely, we are still looking for solutions which concentrate at the points $x^j \in \Omega$, $j = 1,\ldots,m$ as the parameter $\rho$ tends to $0$. We find sufficient conditions under which, as $\rho$ tend to $0$, there exists an explicit class of solutions which admit a concentration behavior with a prescribed bubble profile around some given $m$-points in $\Omega$, for any given integer $m$. These are the so-called singular limits. The candidate $m$-points of concentration must be nondegenerate (in essential way) critical points of a suitable finite dimensional functional explicitly and the higher order Green's function with respect to the imposed boundary conditions.

Some regularity results to the generalized Emden–Fowler equation with irregular data

We deal with the generalized Emden–Fowler equation f″(x) + g(x)f−θ(x) = 0, where belongs to Lp((a,b)). We obtain a priori estimates for the solutions, as well as information about their asymptotic behavior near boundary points. As a tool, we derive new nonlinear variants of first‐order and second‐order Poincaré inequalities, which are based on strongly nonlinear multiplicative inequalities obtained recently by first author and Peszek. Copyright © 2014 John Wiley & Sons, Ltd.

Oscillation behavior of a class of new generalized Emden-Fowler equations

In this paper, we analyze a class of new generalized Emden-Fowler equations. By using the generalized Riccati transformation and specific analytical skills, new oscillation criteria are obtained which generalize and improve some known results.

Oscillation and asymptotic analysis on a new generalized Emden–Fowler equation

In this work, we analyze the new generalized Emden–Fowler equation with neutral type delays:r(t)|Z′(t)|α-1Z′(t)′+q(t)|x(σ(t))|β-1x(σ(t))=0,where Z(t)=x(t)+p(t)x(τ(t)),α⩾β>0. By use of averaging technique and specific analytical skills, some easily-accessible oscillation and asymptotic criteria are established, which have extended the results in the cited literature.

Group classification and exact solutions of a generalized Emden–Fowler equation

The aim of this work is to perform a complete symmetry classification of a generalized Emden-Fowler equation. The various forms of this equation are extensively studied in the literature and they have applications in astrophysical and physiological phenomena. The classical approach of group classification and the procedure based upon the Lie algebras of low dimension are employed for classification. Exact solutions of the invariant equations are derived.

Multiple solutions for boundary value problems of a discrete generalized Emden–Fowler equation

By using critical point theory, some new sufficient conditions for the existence of at least 2 N distinct solutions to boundary value problems of a discrete generalized Emden–Fowler equation are obtained.

Solution of the generalized Emden–Fowler equations by the hybrid functions method

In this paper, we present a numerical algorithm for solving the generalized Emden–Fowler equations, which have many applications in mathematical physics and astrophysics. The method is based on hybrid functions approximations. The properties of hybrid functions, which consist of block-pulse functions and Lagrange interpolating polynomials, are presented. These properties are then utilized to reduce the computation of the generalized Emden–Fowler equations to a system of nonlinear equations. The method is easy to implement and yields very accurate results.

On variational methods to a generalized Emden–Fowler equation

In the present paper, the variational principle to the boundary value problems for a generalized Emden-Fowler equation is given and some existence results of solutions are obtained by using the critical point theory.

Existence of Solutions to Boundary Value Problems for the Discrete Generalized Emden-Fowler Equation

We consider the existence of solutions to boundary value problems for the discrete generalized Emden-Fowler equation. By means of the minimax methods in the critical point theory, some new results are obtained. Two examples are also given to illustrate the main results.

Positive solutions of the generalized Emden–Fowler equation in Hölder spaces

A singular sublinear BVP related to the Emden–Fowler equation is considered. Existence, nonexistence, and regularity of positive solutions in Hölder spaces is obtained.

Analytical-numerical investigation of a singular boundary value problem for a generalized Emden–Fowler equation

In this paper we are concerned about a singular boundary value problem for a quasilinear second-order ordinary differential equation, involving the one-dimensional p -laplacian. Asymptotic expansions of the one-parameter families of solutions, satisfying the prescribed boundary conditions, are obtained in the neighborhood of the singular points and this enables us to compute numerical solutions using stable shooting methods.

Transformation between a generalized Emden–Fowler equation and the first Painlevé transcendent

AbstractWe use the generalized Sundman transformation to obtain a relation between a generalized Emden–Fowler equation and the first Painlevé transcendent. Copyright © 2007 John Wiley & Sons, Ltd.