On the global solvability of the Cauchy problem for an infinite system of differential equations

We consider the Cauchy problem for the semi-infinite Volterra lattice with an asymptotically 2-periodic initial condition. We prove the global solvability of the problem in some class.     

The Cauchy problem for a system of two partial differential equations of infinite order

Starting from the generalized scheme of separation of variables, we propose a new effective method of constructing the solution of the Cauchy problem for a system of two partial differential equations, in general of infinite order with respect to the spatial variable. We consider the example of the Cauchy problem for the system of Lamé equations in the case of a two-dimensional strain.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 204–210.     

Operator part of infinite system of differential equations

Let T be a differential operator in a Hilbert space generated by a first-order infinite system of an ordinary linear differential expression, which is subject to infinitely many boundary conditions. We solve completely for y from Ty = g. The main tools involved are operator parts of closed linear manifolds, which are closely related to generalized inverses.     

On the stability problem for functional differential equations with infinite delay

We study the stability of functional differential equations with infinite delay, using the Lyapunov functional of constant sign with a derivative of constant sign. Limit equations are constructed in a special phase space. We establish a theorem on localization of a positive limit set and theorems on the stability and the asymptotic stability. The results are illustrated by examples.     

Compatible differential constraints to an infinite chain of transport equations for cumulants

We study an infinite chain of transport equations for cumulants which appears in modeling the dynamics of a momentumless turbulent planar wake. The method of compatible differential constraints for formulating its integrability properties is applied. The compatibility conditions obtained make it possible to realize a reduction of the original infinite chain of transport equations and to present an algorithm for calculating cumulants of arbitrary order.     

Necessary conditions for an infinite time optimal control problem

In this paper, we deal with an optimal control problem with infinite transfer time. Using methods which involve local conditional stability, we obtain necessary conditions for optimality under the assumption that the right-hand side of the state equation is Fréchet-differentiable at every point of the optimal solution, and under some weak assumptions about the asymptotical behaviour of the set of perturbations of the solution. The results are illustrated in a specific case considered by Pontryagain et al.     

Time-dependent analysis for a queue modeled by an infinite system of partial differential equations

By studying the spectrum of the underlying operator corresponding to the exhaustive-service M/G/1 queueing model with single vacations we prove that the time-dependent solution of the model strongly converges to its steady-state solution.     

Well-posedness of an infinite system of partial differential equations modelling parasitic infection in an age-structured host

We study a deterministic model for the dynamics of a population infected by macroparasites. The model consists of an infinite system of partial differential equations, with initial and boundary conditions; the system is transformed in an abstract Cauchy problem on a suitable Banach space, and existence and uniqueness of the solution are obtained through multiplicative perturbation of a linear C₀-semigroup. Positivity and boundedness are proved using the specific form of the equations.     

The Jacobian problem as a system of ordinary differential equations

LetP=x ⁿ +P _n?1(y)x ^n?1+…+P ₀(y),Q=x ^m +Q _m?2(y)x ^m?2+…+Q ₀(y) belong toK[x, y], whereK is a field of characteristic zero. The main result of this paper is the following: Assume thatP _x Q _y ?P _y Q _x =1. Then:*

1. K[Q _m?2(y), …,Q ₀(y)]=K[y],
2. K[P, Q]=K[x, y] ifQ=x ^m +Q _k (y)x ^k +Q _r (y)x ^r

     

Integral equations related to the study of an inverse coefficient problem for a system of partial differential equations

We consider an inverse coefficient problem for a linear system of partial differential equations. The values of one solution component on a given curve are used as additional information for determining the unknown coefficient. The proof of the uniqueness of the solution of the inverse problem is based on the analysis of the unique solvability of a homogeneous integral equation of the first kind. The existence of a solution of the inverse problem is proved by reduction to a system of nonlinear integral equations.     

Solution of an infinite system of algebraic equations

A system of infinite algebraic equations is solved explicitly using a Carleman-type problem.     

The variational stability of an optimal control problem for Volterra-type equations

We study the variational stability of an optimal control problem for a Volterra-type nonlinear functional-operator equation. This means that for this optimal control problem (P _? ) with a parameter ? we study how its minimum value min(P _? ) and its set of minimizers argmin(P _? ) depend on ?. We illustrate the use of the variational stability theorem with a series of particular problems.