Class Of Second-order Differential Equations Research Articles

Conditional stability for a class of second-order differential equations

Using two classes of test functions introduced recently by the authors, a criterion for conditional stability of the trivial solution of a second-order semi-linear differential equation is established.

Multiple positive solutions of second-order differential equations

In this paper, by using the coincidence degree method, we obtain some sufficient conditions for the existence of multiple positive periodic solutions of a class of second-order differential equations under the existence of upper and lower solutions.

Explicit bounds for second-order difference equations and a solution to a question of Stević

This note gives explicit, applicable bounds for solutions of a wide class of second-order difference equations with nonconstant coefficients. Among the applications is an affirmative answer to a recent question of Stević.

On the discrete group analysis for solving some classes of Emden-Fowler equations

This paper presents an account of the fundamentals of the discrete group approach for analysis and integration of practical nonlinear ordinary differential equations. In this paper, by means of appropriate transformations of discrete group analysis, some classes of second-order differential equations of Emden-Fowler type is transformed into the other solvable class. For this purpose, we give some applicable relations between the Emden-Fowler and Lane-Emden equations, in order to solve some classes of Emden-Fowler equations. This approach is effective in finding a great number of new integrable equations which thus far could not be integrated using the classical methods.

On a class of difference equations of monotone type

Of concern is the existence and uniqueness of the solution to a class of abstract second-order difference equations. They are the discrete version of some evolution equations which are intensely studied. Some asymptotic behavior results are established. The periodic solutions are also investigated. We use the theory of the maximal monotone operators in Hilbert spaces. An application to a partial differential equation is given.

Estimates of solutions of certain classes of second-order differential equations in a Hilbert space

Linear second-order differential equations of the form in a Hilbert space are studied. Under certain conditions on the (generally speaking, unbounded) operators , , and the correct solubility of the equation in the “energy” space is proved and best possible (in the general case) estimates of the solutions on the half-axis are obtained.

On the second-order differential equation with piecewise constant argument and almost periodic coefficients

In this paper, we study the existence of almost and quasi-periodic solutions to two classes of second-order differential equations. As a corollary, it is shown that periodic and unbounded solutions can coexist for the equation x″( t)+ ω 2 x( t)= bx([ t])+ f( t), which is different from the case: b=0. This phenomena is due to the piecewise constant argument and illustrates a crucial difference between ordinary differential equations and differential equations with piecewise constant argument. The results are extended to nonlinear equations.

An existence result of periodic solutions

Sufficient conditions for the existence of periodic solutions of class of second-order differential equations are obtained.

A novel technique for the solution of second-order difference equations

The class of second-order difference equations that arise in the diffraction of a plane wave at skew incidence on an impedance wedge is investigated. For a typical equation which was solved by the introduction of elliptic integrals, it is shown that the solution can be obtained in a very trivial manner from a convolution-type integral equation and the results are in complete agreement with those previously found.

The geometry of point extensions of a class of second-order differential equations

The geometry of point extensions of a class of second-order differential equations

A Class of Second-Order Differential Equations

A Class of Second-Order Differential Equations

Slow Passage Through a Pitchfork Bifurcation

This paper deals with a class of second-order differential equations with a slowly varying bifurcation parameter. The parameter slowly varies through a critical value corresponding to a transition from a stable equilibrium to one of the two stable branches of an intersecting parabolic curve. The local transition behaviour is described by the second Painleve transcendent. In this study we predict which branch will be followed after passage of the bifurcation point given the initial state. For that purpose, use is made of averaging methods and of asymptotic matching techniques connecting local solutions valid before, during, and after the transition.

Numerov integration for radial wave equations in cylindrical symmetry

Numerov’s method, which is commonly used in numerical integration of a class of second-order differential equations lacking the first derivative, is generalized to cover radial wave equations in cylindrical geometry. The proposed scheme is compared with the Runge–Kutta method and the conventional Numerov method by numerically integrating Bessel’s differential equation.

HOMOCLINIC BIFURCATION IN HIGHER DEGREE SYSTEMS WITH APPLICATION TO RANDOM FLIPPING MODES

Bifurcations from stable homoclinic saddle connections for a class of second-order differential equations of forced van der Pol-Duffing type are investigated using a new method based on a two-parameter perturbation series. The underlying autonomous system is parametrically tuned for homoclinic orbits and then perturbed by small amplitude forcing. These systems exhibit an apparent random flipping about the saddle point. Asymptotic conditions for switching between half-cycles are obtained and compared with numerically integrated solutions for a model system.

Sufficient conditions for periodic solutions to a class of second-order differential equations

where 0 # b E R and g(0, 0,O) = 0, possess nonzero, generalized, even, T-periodic solutions x(t); i.e. x E W2V2( T, T) (defined below) such that x(t + T) = x(t), x( t) = x(t), x(t) is not identically zero and satisfies equation (1) for a.a. t E F?. Equation (1) cannot be “solved” in a useful manner for the highest order derivative. We emphasize that T is not given, but is an unknown quantity to be determined. Our method of solution is to convert the problem to an operator equation in Hilbert space and obtain results using global bifurcation theory. This bifurcation approach produces more that just existence of solutions: we obtain global information on the continuum of T-periodic solutions to equation (1). The idea of solving existence problems of periodic solutions of an autonomous ordinary differential equation around an equilibrium by transformation to a bifurcation problem is due to Krasnosel’skii [2] who worked with compact mappings. Recently, similar problems to equation (1) have been studied by Hetzer [ 11, Mawhin [5], Petryshyn and Yu [7], Webb and Welsh [9] and others. Hetzer’s results we shall discuss in our concluding remarks below; Mawhin invoked his coincidence degree theory and was, therefore, unable to consider the nonlinear term g as a function of the highest order derivative; Petryshyn and Yu studied problems analogous to equation (l), but restricted their results to existence of solutions without considering global properties; Webb and Welsh were more concerned with proving existence and uniqueness of not necessarily periodic solutions of equation (1). Equation (1) was alluded to in [lo] as an application of bifurcation theory, but the details were not given, nor were the hypotheses stated precisely. The main result in this paper (corollary 5) follows as a consequence of a bifurcation theorem for so called A-proper mappings (defined below) proved by the author in [lo]. The usefulness of this A-proper framework is that the resulting theorems apply in a more general setting than earlier bifurcation results which generally demanded that the nonlinear term g generate a

The asymptotic behavior of a class of second-order differential equations with applications to electrical circuit equations

The asymptotic behavior of a class of second-order differential equations with applications to electrical circuit equations

Existence and uniqueness of initial value problems for a class of second-order differential equations

Etude du probleme de la valeur initiale pour les equations differentielles du second ordre de la forme x″(t)=g(x(t), x'(t), x″(t)) qui ne peuvent pas etre resolues pour des derivees d'ordre plus elevees

A class of second-order differential equations and related first-order systems

A class of second-order nonlinear differential equations which arises in several branches of mathematical physics is considered. It is shown that equations of this class may be factorised into first-order equations of 'Riccati type'. Conditions are obtained, on the coefficient functions of the second-order equations, for the first-order equations to be of matrix Riccati form, whose solutions have a finite superposition property. The factorisation into first-order equations is then not unique, and there is an alternative first-order set of equations whose solutions do not have this superposition property. A second-order equation arising in the theory of pellet fusion processes is investigated in detail. Solutions are obtained when the corresponding first-order equations are of matrix Riccati from and shown to be equivalent to solutions derived by alternative methods. Lagrangian systems giving rise to equations of the class are also considered.

Comment on "Solvable Hill equation"

It is shown that the recently published class of Hill equations with exact analytical solutions is in fact an example of a much broader class of second-order differential equations which admit exact WKB-like solutions.

On a class of differential equations derived from plasma statistics

A class of second-order differential equations stemming from an equation of motion for the two-particle spatial correlation function in one-component plasmas is studied. These equations contain an irregular singularity of varying order at the origin. The general form of solution is obtained which, together with the construction of asymptotic series, demonstrates that both solutions to all equations in the class are singular at the origin. Behavior removed from the origin is shown to be oscillatory or exponential depending on specifics of the equations.