Generalized Emden-Fowler Equation Research Articles

Boundary value problems of discrete generalized Emden-Fowler equation

By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established.

On boundary value problems for a discrete generalized Emden–Fowler equation

In this paper, some sufficient conditions for the existence of solutions to the boundary value problems of a class of second order difference equation are obtained by using the critical point theory.

Exact analytic solutions of nonlinear boundary value problems in fluid mechanics (Blasius equations)

In this work it is shown that, by a series of admissible functional transformations, the generalized Blasius equation in fluids can be exactly reduced to a three-term generalized Emden–Fowler equation. Furthermore, the restricted in axisymmetric flows and simplified forms of this equation can be reduced to (i) two-term generalized Emden–Fowler equations; (ii) generalized Emden–Fowler equations; (iii) Emden–Fowler equations of the normal form; and (iv) Abel equations of the second kind. By means of a recently developed mathematical solution methodology (Panayotounakos, Fifth Greek Congress on Mechanics, Xania, Crete, 22–25 June 2004, Hellas, Greece), we provide exact analytic solutions for the simplified as well as for the restricted forms of the above-mentioned Blasius equations. Thus, it is proved that important, unsolvable in exact form problems in nonlinear fluid dynamics now can be analytically solved.

On Proper and Entire Solutions of a Generalized Emden-Fowler Equation

On Proper and Entire Solutions of a Generalized Emden-Fowler Equation

On an extension of the Fubini theorem and its applications in ODEs

The paper deals with the change of integration for integrals∫0∞b(t)∫0tA(s)dsmdt,∫0∞A(t)∫t∞b(s)ds1/mdt,where m>0 and A,b are continuous nonnegative functions on [0,∞). There are given applications for the second-order differential nonlinear equations(a(t)|x′|αsgnx′)′+b(t)|x|βsgnx=0,where 0<β⩽α and a,b are continuous positive functions on [0,∞). The first application brings comparison results for oscillation of two generalized Emden–Fowler equations, the second one shows the similarities and discrepancies for nonoscillatory half-linear and linear differential equations.

Integral averages and oscillation criteria of second-order nonlinear differential equations

The purpose of this paper is to present some new sufficient criteria for the oscillation of all solutions of the second-order nonlinear differential equation ( a ( t ) ψ ( x ) x ′ ) ′ + q ( t ) f ( x ) = 0 , t ≥ t 0 > 0 , ( * ) where α C 1 ([ t 0, ∞)) is a positive function, qC([ t 0, ∞)) has no restriction on its sign, Ψ C(R), ƒ C 1( R) are such that ƒ′() ≥ 0 and Ψ() > 0, xƒ() > 0 for ≠ 0. These criteria are obtained by using a general class of the parameter functions in the averaging techniques and represent extension, as well as improvement of known oscillation criteria of several authors for the generalized Emden-Fowler equation.

On an extension of the Fubini theorem and its applications in ODEs

The paper deals with the change of integration for integrals ∫ 0 ∞ b(t) ∫ 0 t A(s) ds m dt, ∫ 0 ∞ A(t) ∫ t ∞ b(s) ds 1/m dt, where m>0 and A, b are continuous nonnegative functions on [0,∞). There are given applications for the second-order differential nonlinear equations (a(t)|x′| α sgn x′)′+b(t)|x| β sgn x=0, where 0< β⩽ α and a, b are continuous positive functions on [0,∞). The first application brings comparison results for oscillation of two generalized Emden–Fowler equations, the second one shows the similarities and discrepancies for nonoscillatory half-linear and linear differential equations.

Adomian decomposition method for a reliable treatment of the Emden–Fowler equation

In this paper, we present a reliable algorithm to determine new exact and new approximate solutions of the generalized Emden–Fowler equation. The algorithm rests mainly on Adomian decomposition method with an alternative framework designed to overcome the difficulty of the singular point at x=0. The proposed framework reveals quite a number of remarkable features of the scheme used here.

Numerical solution of a singular boundary value problem for a generalized Emden–Fowler equation

In this paper we shall deal with an equation of the form y″(x)=−g(x)x p y(x) q, where p and q are real parameters satisfying p>−2, q<−1 and g is a positive and continuous function on [0,1]. We shall search for positive solutions which satisfy the boundary conditions: y(0)=y(1)=0. The initial nonlinear problem is transformed into a sequence of linear ones, each one of them is approximated by a finite difference scheme. Asymptotic expansions of the error are obtained and numerical examples are then analysed.

Positive solutions of generalized Emden-Fowler equation

In this paper, first we obtain a necessary and sufficient condition for the existence of positive solution to a singular impulsive Dirichlet boundary value problem, then we discuss the existence of...

Positive solutions of generalized Emden–Fowler equation

In this paper, first we obtain a necessary and sufficient condition for the existence of positive solution to a singular impulsive Dirichlet boundary value problem, then we discuss the existence of multiple solutions to a singular impulsive BVP. In the last section of this paper, to illustrate the application of our main results, we discuss the existence of positive radial solutions to a semilinear elliptic radial solutions problem.

Multiple Positive Solutions of Semilinear Differential Equations with Singularities

The existence of positive solutions of a second order differential equation of the form (formula here) with the separated boundary conditions: α z (0) − β z ′(0) = 0 and γ z (1)+δ z ′(1) = 0 has proved to be important in physics and applied mathematics. For example, the Thomas–Fermi equation, where f = z 3/2 and g = t −1/2 (see [ 12 , 13 , 24 ]), so g has a singularity at 0, was developed in studies of atomic structures (see for example, [ 24 ]) and atomic calculations [ 6 ]. The separated boundary conditions are obtained from the usual Thomas–Fermi boundary conditions by a change of variable and a normalization (see [ 22 , 24 ]). The generalized Emden–Fowler equation, where f = z p , p > 0 and g is continuous (see [ 24 , 28 ]) arises in the fields of gas dynamics, nuclear physics, chemically reacting systems [ 28 ] and in the study of multipole toroidal plasmas [ 4 ]. In most of these applications, the physical interest lies in the existence and uniqueness of positive solutions.

Integral averages and oscillation of second-order nonlinear differential equations

We present some criteria for the oscillation of the second-order nonlinear differential equation ▪ where a ϵ C 1([ t 0, ∞)) is a nonnegative function, q ϵ C ([ t 0, ∞)) are allowed to change sign on [ t 0, ∞), ψ, f ϵ C 1 ▪, ψ( x) > 0, xf( x) > 0, f′( x) ≥ 0 for x ≠ 0. These criteria are obtained by using a general class of the parameter functions H( t,s) in the averaging techniques and represent extension, as well as improvement of known oscillation criteria of Philos and Purnaras for the generalized Emden-Fowler equation.

A NONOSCILLATION THEOREM FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DECAYING COEFFICIENTS

The purpose of this paper is to give sufficient conditions for all nontrivial solutions of the nonlinear differential equation x″ +a(t)g(x) = 0 to be nonoscillatory. Here, g(x) satisfies the sign condition xg(x) &gt; 0 if x ≠ 0, but is not assumed to be monotone increasing. This differential equation includes the generalized Emden–Fowler equation as a special case. Our main result extends some nonoscillation theorems for the generalized Emden–Fowler equation. Proof is given by means of some Liapunov functions and phase-plane analysis.

On the Dynamics of the Generalized Emden–Fowler Equation

Abstract We present some recent results dealing with the qualitative behavior of solutions of the quasilinear second order differential equation where a, b are positive continuous real functions, and Φ j (u) = |u| j–2 u, j &gt; 1. Following a classification of solutions, proposed in the linear case in [Marini and Zezza, J. Diff. Equat. 28: 1–17, 1978], we divide all the solutions of (∗) with respect to their asymptotic behavior into two classes. The existence and uniqueness is considered: a topological approach is employed and a fixed point result for operators associated to boundary value problems on a half-line is used. In addition, some asymptotic estimates are presented and the convergence of solutions to zero as t → ∞ is studied.

Generalized Lie Symmetries and the Integrability of Generalized Emden–Fowler Equations

Generalized Lie symmetries and the integrability of generalized Emden–Fowler equations (GEFEs) are considered. It is shown that the constraint which the variable-coefficient functions must satisfy for the GEFEs to have infinite-dimensional symmetry algebras is precisely the same as this in order that the equation may be transformed into the integrable Emden–Fowler equation. fiom the nature of the symmetry vector fields one can write down the integrals of motion for the above systems. The structure of the symmetry algebras is also presented.

Conditions for the existence of nonoscillatory solutions of a second-order nonlinear differential equation

For the nonlinear differential equation $$y + p(t)|y|^\sigma |y|^\lambda {\text{ sign }}y = 0$$ whereσ>−1, λ<1, andp: [a, +∞[→]0, +∞[ is a continuous function, we establish sufficient criteria for the nonoscillation of all regular solutions; in the case of the classical generalized Emden-Fowler equation (λ=0) these criteria supplement well-known results.

Asymptotic expansions and numerical approximation of nonlinear degenerate boundary-value problems

In the present paper we are concerned with boundary-value problems (BVP) for the generalized Emden–Fowler equations. Asymptotic expansions of the solution are obtained near the endpoints. We use a finite-difference scheme to approximate the solution and the convergence is accelerated by means of extrapolation methods. A variable substitution is introduced to diminish the effect of the singularity at the origin. Numerical results, obtained by different methods, are presented for two particular cases.

Polynomial conservation laws of the generalized Emden-Fowler equation

In this paper we consider the polynomial conservation laws of fourth degree with respect to x of the generalized Emden-Fowler's equation x ̈ + bt −1x + λx αt k = 0 . It is demonstrated that the existence of conservation laws depends upon the solution of a system of partial differential equations, usually termed the generalized Killing's equations. The general form of the fourth degree conservation laws of the Emden-Fowler equation is given and some concrete examples are discussed.

Integration ofD-dimensional cosmological models with two factor spaces by reduction to the generalized Emden-Fowler equation

The D-dimensional cosmological model on the manifold $M = R \times M_{1} \times M_{2}$ describing the evolution of 2 Einsteinian factor spaces, $M_1$ and $M_2$, in the presence of multicomponent perfect fluid source is considered. The barotropic equation of state for mass-energy densities and the pressures of the components is assumed in each space. When the number of the non Ricci-flat factor spaces and the number of the perfect fluid components are both equal to 2, the Einstein equations for the model are reduced to the generalized Emden-Fowler (second-order ordinary differential) equation, which has been recently investigated by Zaitsev and Polyanin within discrete-group analysis. Using the integrable classes of this equation one generates the integrable cosmological models. The corresponding metrics are presented. The method is demonstrated for the special model with Ricci-flat spaces $M_1,M_2$ and the 2-component perfect fluid source.