Class Of Second-order Equations Research Articles

Rate of Convergence to Equilibrium and Łojasiewicz-Type Estimates

A well known result states that the Łojasiewicz gradient inequality implies some estimates of the rate of convergence to equilibrium for solutions of gradient systems. We generalize this result to gradient-like systems satisfying certain angle condition and Kurdyka–Łojasiewicz inequality and to even more general situation. We apply the results to a broad class of second order equations with damping.

On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.

The electromagnetic coupling in Kemmer-Duffin-Petiau theory

We analyze the electromagnetic coupling in the Kemmer-Duffin-Petiau (KDP) equation. Since the KDP equation which describes spin-0 and spin-1 bosons is of Dirac type, we examine some analogies with and differences from the Dirac equation. The main difference with the Dirac equation is that the KDP equation contains redundant components. We will show that as a result certain interaction terms in the Hamilton form of the KDP equation do not have a physical meaning and will not affect the calculation of physical observables. We point out that a second order KDP equation derived by Kemmer as an analogy to the second order Dirac equation is of limited physical applicability as (i) it belongs to a class of second order equations which can be derived from the original KDP equation and (ii) it lacks a back-transformation which would allow one to obtain solutions of the KDP equation out of solutions of the second order equation.

An abstract Lax-Phillips scattering theory for a class of second-order equations

An abstract Lax-Phillips scattering theory for a class of second-order equations

The symmetry approach to higher-dimensional nonlinear equations

Conditions for the existence of recursion operators are studied for arbitrary-order scalar partial differential equations with two independent variables. The coefficients of the operator are successfully determined by those of the terms with higher degree. For a class of second-order equations, the operators with degree one are completely determined.

The class of second order equations which Riemannian geometry can be applied to

The class of second order equations which Riemannian geometry can be applied to

Differential equations for rational conformal characters

The characters of a rational conformal field theory are the solutions of a modular invariant differential equation. We cast this equation into a form in which the independent variable is not the modular parameter τ, but rather a modular function of τ, successively the Picard λ-function and the modular invariant function j. This enables us to determine the modular transformations of the characters from the monodromy of the solutions of the equation. The requirement that the characters form a unitary representation of the modular group results in restrictions on h and c. Knowledge of the modular transformations also allows us to find the relative normalization of the characters in the one-loop partition function, which specifies the absolute multiplicities of states in the conformal field theory. We carry this out explicitly for a certain class of second order equations in which the characters can be written in terms of hypergeometric functions.

Multilevel continuation techniques for nonlinear boundary value problems with parameter dependence

A new technique is described to continue along solution branches of parameter- dependent nonlinear boundary value problems in order to obtain accurate multigrid solutions at specified points. In particular, the implementation for a general class of second order equations in divergence form in the program PLTMG is described and analyzed. This method has proven to be very robust and efficient, as is illustrated by several examples.

On periodic solutions of several classes Riccati's equation and second-order differential equations

In this paper, the problem on periodic solutions of several classes of Riccati's equation with periodic coefficients is discussed, and the conditions, under which several classes of secondorder equations with periodic coefficients have periodic solutions, are given.

Equipartition of Energy for a Class of Second Order Equations

Equipartition of Energy for a Class of Second Order Equations

Equipartition of energy for a class of second order equations

We consider the Cauchy problem for a class of second order equations of the form ( d / d t − A 2 ) ( d / d t − A 1 ) u ( t ) = 0 (d/dt - {A_2})(d/dt - {A_1})u(t) = 0 in a Hilbert space H. A d’Alembert type solution formula is presented and we give a suitable definition of energy. Also, we derive a necessary and sufficient condition for the asymptotic equipartition of energy (Kinetic and Potential) to hold. These results generalize corresponding results for the abstract wave equation ( d 2 / d t 2 + A 2 ) u ( t ) = 0 ({d^2}/d{t^2} + {A^2})u(t) = 0 .