Oscillation of second-order Emden–Fowler neutral delay differential equations

We establish some new criteria for the oscillation of second-order Emden–Fowler neutral delay differential equations. We study the case of superlinear and the case of sublinear equations subject to various conditions. The results obtained show that the presence of a neutral term in a differential equation can cause or destroy oscillatory properties. Several examples are provided to illustrate the relevance of new theorems.     

On super-linear Emden–Fowler type differential equations

We study the second order Emden–Fowler type differential equation

(a(t)|x^′|^αsgnx^′)^′+b(t)|x|^βsgnx=0${(a(t){\left|x^{\prime }\right|}^{\alpha }\mathrm{sgn}{x}^{\prime })}^{\prime }+b(t){|x|}^{\beta }\mathrm{sgn}x=0$

in the super-linear case α<β$\alpha <\beta$. Using a Hölder-type inequality, we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions (subdominant, intermediate, and dominant solutions). Jointly with this, sufficient conditions for the existence of globally positive intermediate solutions are established. Some of our results are new also for the Emden–Fowler equation.     

Application of homotopy-perturbation method to Klein–Gordon and sine-Gordon equations

In this paper, the homotopy-perturbation method (HPM) is employed to obtain approximate analytical solutions of the Klein–Gordon and sine-Gordon equations. An efficient way of choosing the initial approximation is presented. Comparisons with the exact solutions, the solutions obtained by the Adomian decomposition method (ADM) and the variational iteration method (VIM) show the potential of HPM in solving nonlinear partial differential equations.     

Homotopy analysis method for singular IVPs of Emden–Fowler type

In this paper, approximate and/or exact analytical solutions of singular initial value problems (IVPs) of the Emden–Fowler type in the second-order ordinary differential equations (ODEs) are obtained by the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. It is shown that the solutions obtained by the Adomian decomposition method (ADM) and the homotopy-perturbation method (HPM) are only special cases of the HAM solutions.     

The Thomas–Fermi Problem and Solutions of the Emden–Fowler Equation

Computational Mathematics and Mathematical Physics - A two-point boundary value problem is considered for the Emden–Fowler equation, which is a singular nonlinear ordinary differential...     

Solutions of singular IVPs of Lane–Emden type by the variational iteration method

In this paper, approximate-exact solutions of a class of Lane–Emden type singular IVPs problems, by the variational iteration method, are presented. The variational iteration method yields solutions in the forms of convergent series with easily calculable terms. The scheme is shown to be highly accurate, and in some cases, yields exact solutions in few iterations.     

On the asymptotic behavior of solutions of Emden–Fowler equations on time scales

Consider the Emden-Fowler dynamic equation $$ x^{\Delta\Delta}(t)+p(t)x^\alpha(t)=0,\:\:\alpha >0 , \qquad \qquad \qquad \qquad (0.1) $$ where ${p\in C_{rd}([t_0,\infty)_{\mathbb{T}},\mathbb{R}), \alpha}$ is the quotient of odd positive integers, and ${\mathbb{T}}$ denotes a time scale which is unbounded above and satisfies an additional condition (C) given below. We prove that if ${\int^\infty_{t_0}t^\alpha |p(t)|\Delta t<\infty}$ (and when ???=?1 we also assume lim _t???? tp(t)??(t)?=?0), then (0.1) has a solution x(t) with the property that $$ \lim_{t\rightarrow\infty} \frac{x(t)}{t}=A\neq 0.$$      

Oscillation criteria for second-order superlinear Emden–Fowler neutral differential equations

We study oscillatory behavior of solutions to a class of second-order superlinear Emden–Fowler neutral differential equations. New oscillation theorems are presented and their efficiency is illustrated.     

Existence and Uniqueness of Solutions of Some Cauchy Problems for the Emden–Fowler Equation

Differential Equations - We consider the Cauchy problem for the Emden–Fowler equation $$y^{prime {}prime }-x^ay^{sigma }=0 $$ with parameters $$ain mathbb {R} $$ and $$sigma &lt;0...     

Oscillation and non-oscillation theorems for superlinear Emden–Fowler equations of the fourth order

We study the oscillatory behavior of solutions of the fourth-order Emden–Fowler equation: (E) y^(iv)+q(t)|y|sgny=0, where >1 and q(t) is a positive continuous function on [t₀,), t₀>0. Our main results Theorem 2 – if (q(t)t^(3+5)/2)0, then equation (E) has oscillatory solutions; Theorem 3 – if lim_tq(t)t^4+(-1)=0, >0, then every solution y(t) of equation (E) is either non-oscillatory or satisfies limsup_tt^-+i|y⁽ⁱ⁾(t)|= for < and i=0,1,2,3,4. These results complement those given by Kura for equation (E) when q(t)<0 and provide analogues to the results of the second-order equation, y+q(t)|y|sgny=0,>1. Mathematics Subject Classification (2000) 34C10, 34C15     

Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane–Emden–Fowler type

The particular motivation of this work is to develop a computational method to calculate exact and analytic approximate solutions to singular strongly nonlinear initial or boundary value problems of Lane–Emden–Fowler type which model many phenomena in mathematical physics and astrophysics. A powerful algorithm is proposed based on the series representation of the solution via suitable base functions. The utilization of such functions converts the solution of a given nonlinear differential equation to the solution of algebraic equations. Error analysis and convergence of the method is presented. Comparisons with the other methods reveal validity, applicability and great potential of the method. Several physical problems are treated to illustrative the good performance and high accuracy of the technique.     

WKB Method for Nonlinear Equations of Emden–Fowler Type

Doklady Mathematics - A method of asymptotic integration based on application of a generalized Prüfer transformation is elaborated for a class of Emden–Fowler type equations, and a...