The Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition

We examine the Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition. The question is addressed of existence of a solution with the same asymptotics at infinity as the initial condition. We demonstrate that the method of the inverse scattering problem is applicable to this problem.     

Computation of periodic orbits in three‐dimensional Lotka‐Volterra systems

This paper deals with an adaptation of the Poincaré‐Lindstedt method for the determination of periodic orbits in three‐dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three‐dimensional Lotka‐Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.     

Analysis and solution of the Cauchy problem for linear Volterra integro-differential equations with functional delay

We study how to transform Cauchy problems for Volterra integro-differential equations with functional delays to resolving Volterra integral equations with conventional argument by using a modification of a function of flexible structure. We show that such a transformation is possible for all linear Volterra integro-differential equations of retarded type. There exists a unique solution of the resolving equation provided that the kernels and the right-hand side are bounded in the closed square. The presence of parameters in the expression for the function of flexible structure permits one to choose these parameters in an optimal way in the course of the solution of the problem so as to represent the solution in closed form or, if this is difficult, optimize an approximate solution method. The accuracy of the approximate solutions is estimated.     

The method of volterra integral equations in contact problems for thin-walled structural elements

We reduce the solution of contact problems in the interaction of rigid bodies (dies) with thin-walled elements (one-dimensional problems) to Volterra integral equations. We study the effect of the model describing the stress-strain state of plates on the type of integral equations and the structure of their solutions. It is shown that taking account of reducing turns the problem into a Volterra integral equation of second kind, which has a unique solution that is continuous and agrees quite well with the results obtained from the three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra integral equation of first kind that has a unique continuous solution; but for dies without corners the Herz condition does not hold (p(a) ≠ 0), and the contact pressure assumes its maximal value at the end of the zone of contact. For thin-walled elements, whose state can be described by the classical Kirchhoff-Love theory, the integral equation of the problem (a Volterra equation of first kind) has a solution in the class of distributions. The contact pressure is reduced to concentrated reactions at the extreme points of the contact zone. We give a comparative analysis of the solutions in all the cases just listed (forces, normal displacements, contact pressures). Three figures, 1 table. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 96–103. Original article submitted March 15, 1997.     

Characterizations of Positive Linear Volterra Integro-differential Systems

We first give a criterion for positivity of the solution semigroup of linear Volterra integro-differential systems. Then, we offer some explicit conditions under which the solution of a positive linear Volterra system is exponentially stable or (robustly) lies in L²[0,+∞). The first and last author are supported by the Japan Society for Promotion of Science (JSPS) ID No. P 05049.     

On the perturbation of Volterra integro-differential equations

In this work, we will prove that every solution of a perturbed Volterra integro-differential equation can be approximated by a solution of the Volterra integro-differential equation.     

Integrable boundary-value problem for the Volterra chain on the half-axis

We study quasiperiodic (finite-gap) solutions of the Volterra chain satisfying an integrable boundary condition on the semiaxis. From the set of general finite-gap solutions, only those corresponding to the boundary-value problem are singled out, the relevant condition being expressed as a system of algebraic equations.     

Reconstruction of a convolution operator from the right-hand side on the real half-axis

We study a Volterra convolution integral equation of the first kind on a semi-infinite interval. Under some rather natural constraints on the kernel and the right-hand side of the Volterra integral equation (the kernel has bounded support, while the support of the right-hand side may be unbounded), it is possible to reconstruct the integral operator of the equation (i.e., the solution and the kernel of the integral operator) from the right-hand side of the equation. The uniqueness theorem is proved, the necessary and sufficient conditions for solvability are found, and the explicit formulas for the solution and the kernel are obtained.     

Floquet theory for a Volterra integro-dynamic system

In this paper, we study periodic linear Volterra systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We discuss the relationship between the solution of the Volterra integro-dynamic system and the limiting equation of the corresponding system. We also develop integrability conditions of the resolvent of Volterra integro-dynamic systems.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Global stability of a two-species piecewise linear volterra ecosystem

This paper extends the results of Goh [1], and Takeuchi and Adachi [2,3] concerning the generalized linear Volterra model. We introduce a piecewise linear Volterra model for a two- species system. The solution of the steady-state problem is then shown to be equivalent to finding the solution to the Generalized Linear Complementarity Problem. We show when this nonnegative equilibrium is unique and globally asymptotically stable in the sense of Goh [1].     

On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces

The main purpose of this paper is to deal with almost automorphic and asymptotically almost automorphic solutions of the initial value problem as well as the nonlinear Volterra integral equation in Banach spaces. We obtain a collection of existence results of such solutions to these equations. We investigate also a topological structure of such solution sets. Moreover, we prove Aronszajn-type theorems for solutions of the initial value problem as well as the nonlinear Volterra integral equation, defined on the whole real line.     

Generalized Dirichlet‐to‐Neumann Map in Time‐Dependent Domains

We study the heat, linear Schrödinger (LS), and linear KdV equations in the domain l(t) < x < ∞ , 0 < t < T , with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.     

Classical solution of the first mixed problem for the Klein-Gordon-Fock equation in a half-strip

We consider a classical solution of the first boundary value problem for the Klein-Gordon-Fock equation in a half-strip in the one-dimensional case. We prove the existence and uniqueness of the classical solution under certain smoothness conditions and matching conditions for the given functions. To solve the problem, one should solve Volterra integral equations of the second kind.     

Observer design for open and closed trophic chains

Monitoring of ecological systems is one of the major issues in ecosystem research. The concepts and methodology of mathematical systems theory provide useful tools to face this problem. In many cases, state monitoring of a complex ecological system consists in observation (measurement) of certain state variables, and the whole state process has to be determined from the observed data. The solution proposed in the paper is the design of an observer system, which makes it possible to approximately recover the state process from its partial observation. Such systems-theoretical approach has been applied before by the authors to Lotka–Volterra type population systems. In the present paper this methodology is extended to a non-Lotka–Volterra type trophic chain of resource–producer–primary consumer type and numerical examples for different observation situations are also presented.     

Solvability of the Goursat problem in quadratures

We find new variants of Goursat problem solution in quadratures on the basis of the combination of the Riemann method and the cascade integration. The results are applied to two Volterra equations with particular integrals.     

Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels

Numerical Algorithms - We discuss the numerical solution to a class of weakly singular Volterra integral equations in this paper. Firstly, the fractional Lagrange interpolation is applied to deal...     

Galerkin information,the hyperbolic cross,and the complexity of operator equations

We obtain an exact power order of the complexity of the approximate solution of a certain class of operator equations in a Hibert space. We show that the optimal power order is realized by an algorithm that uses Galerkin information associated with the hyperbolic cross. As a corollary we derive an exact power order of the complexity of the approximate solution of Volterra integral equations whose kernels and free terms belong to Sobolev classes.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 639–648, May, 1991.