Convolution equations and nonlinear functional equations

The survey is devoted to applications of nonlinear integral equations to linear convolution equations, their discrete analogues, and also the connection of these equations with problems of radiative transfer, in particular, with the Ambartsumyan equations.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 175–244, 1984.     

Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations

The combined quasi-neutral and non-relativistic limit of compressible Navier-Stokes-Maxwell equations for plasmas is studied. For well-prepared initial data, it is shown that the smooth solution of compressible Navier-Stokes-Maxwell equations converges to the smooth solution of incompressible Navier-Stokes equations by introducing new modulated energy functional.     

Oscillation of solutions of second-order linear differential equations and corresponding difference equations

In this paper, we present conditions ensuring that solutions of linear second-order differential equations oscillate, provided solutions of corresponding difference equations oscillate. We also establish the converse result, namely, when oscillation of solutions of difference equations implies oscillation of solutions of corresponding differential equations.     

On difference Riccati equations and second order linear difference equations

In this paper, we treat difference Riccati equations and second order linear difference equations in the complex plane. We give surveys of basic properties of these equations which are analogues in the differential case. We are concerned with the growth and value distributions of transcendental meromorphic solutions of these equations. Some examples are given.     

Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = (p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations

In this paper, by using the improved Riccati equations method, we obtain several types of exact traveling wave solutions of breaking soliton equations and Whitham-Broer-Kaup equations. These explicit exact solutions contain solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions. The method employed here can also be applied to solve more nonlinear evolution equations.     

Measure functional differential equations and functional dynamic equations on time scales

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging.     

Prolongation of solutions of measure differential equations and dynamic equations on time scales

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations.     

Dynamic equations on time scales and generalized ordinary differential equations

The aim of this paper is to show that dynamic equations on time scales can be treated in the framework of generalized ordinary differential equations as introduced by J. Kurzweil. We also use some known results for generalized ordinary differential equations to obtain new theorems related to stability and continuous dependence on parameters for dynamic equations on time scales.     

Hamilton equations for infinite-dimensional systems, and their variational equations

For systems with infinitely many degrees of freedom, we establish a relationship between the solutions and first integrals of noncanonical Hamilton equations, their variational equations, and the adjoint variational equations.     

Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations

A direct link between a Vlasov equation and the equations of motion of a rotating fluid with an effective pressure depending only on a pseudo-density is illustrated. In this direct link, the resulting fluid equations necessarily appear in flux conservative form when there are no topographical and rotational terms. In contrast, multilayer isopycnic and isentropic equations used in atmosphere and ocean dynamics, in the absence of topographical and rotational terms, cannot be brought into a conservative flux form, and, hence, cannot be derived directly from the Vlasov equations. Another route is explored, therefore: deriving the Hamiltonian formulation of the two-layer isopycnic and isentropic equations as a restriction from a Hamiltonian formulation of two decoupled Vlasov equations. The work is motivated by our search for energy-preserving or even Hamiltonian (kinetic) numerical schemes.     

A bibliography of graph equations

Graph equations are equations in which the unknowns are graphs. Many problems and results in graph theory can be formulated in terms of graph equations. Here we offer a classification and a large bibliography of graph equations.     

Some differences between difference equations and differential equations

In this expository paper we discuss some instances in which analogues which might be expected between the behavior of solutions of differential equations and difference equations fail to hold, focussing particularly on questions related to boundedness and oscillation.     

Compressible navier-stokes-poisson equations

This is a survey paper on the study of compressible Navier-Stokes-Poisson equations. The emphasis is on the long time behavior of global solutions to multi-dimensional compressible Navier-Stokes-Poisson equations, and the optimal decay rates for both unipolar and bipolar compressible Navier-Stokes-Poisson equations are discussed.     

Some nonlinear equations reducible to diffusion-type equations

The problem of the reducibility of a system of second-order quasi-linear parabolic differential equations to diffusion-type equations is considered. An effective solution algorithm is suggested for this problem in the nondegenerate case. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 26–37, April, 2000.     

Second-order type-changing evolution equations with first-order intermediate equations

This paper presents a partial classification for C^∞ type-changing symplectic Monge-Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems.     

Impulsive nonlocal differential equations through differential equations on time scales

We propose a non-standard approach to impulsive differential equations in Banach spaces by embedding this type of problems into differential (dynamic) problems on time scales. We give an existence result for dynamic equations and, as a consequence, we obtain an existence result for impulsive differential equations.     

The equations of compatibility of deformations

Nine equations of compatibility of deformations are obtained in which, unlike the classical Saint-Venant compatibility equations, only first derivatives with respect to the coordinates occur. It is proved that, of these nine equations, only six are independent. It is shown that the classical compatibility equations can be obtained from these equations.     

Integrability of Riccati equations and stationary Korteweg-de vries equations

By using the Lie infinitesimal method, we establish the correspondence between the integrability of a one-parameter family of Riccati equations and the hierarchy of the higher Korteweg-de Vries equations. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal. Vol. 51, No. 6, pp. 856–860, June, 1999.