The numerical analysis on a Volterra equation with asymptotically periodic solution

We consider order one operational quadrature methods on a certain integro-differential equation of Volterra type on (0,∞), with piecewise linear convolution kernels. The forms of discretization solution are patterned after a continuous one of Hannsgen (1979) [2]. An l¹ remainder stability and an error bound are derived.     

Minimal-speed selection of traveling waves to the Lotka–Volterra competition model

In this paper the minimal-speed determinacy of traveling wave fronts of a two-species competition model of diffusive Lotka–Volterra type is investigated. First, a cooperative system is obtained from the classical Lotka–Volterra competition model. Then, we apply the upper-lower solution technique on the cooperative system to study the traveling waves as well as its minimal-speed selection mechanisms: linear or nonlinear. New types of upper and lower solutions are established. Previous results for the linear speed selection are extended, and novel results on both linear and nonlinear selections are derived.     

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.     

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.     

Survival of three species in a nonautonomous Lotka–Volterra system

In Ahmad and Stamova (2004) [1], the author considers a competitive Lotka–Volterra system of three species with constant interaction coefficients. In this paper, we study a nonautonomous Lotka–Volterra model with one predator and two preys. The explorations involve the persistence, extinction and global asymptotic stability of a positive solution.     

Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics

A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease‐free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease‐free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh‐Volterra type, to be globally asymptotically stable for a special case.     

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.     

An application of the Hurwitz theorem to the root analysis of the characteristic equation

In this work we show that a classical result of A. Hurwitz is still very effective in studying the root analysis of the characteristic equation for a linear functional differential equation. A conjecture was made by Funakubo et al. (2006) [3] regarding the asymptotic stability condition of the zero solution of a linear integro-differential equation of Volterra type. We applied the Hurwitz theorem to the characteristic equation in question and showed the existence of a root with positive real part and solved the conjecture. The Hurwitz theorem is expected to work well for the root analysis in critical cases.     

Existence and global attractivity of positive periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response

In this paper, a Volterra model with mutual interference and Beddington-DeAngelis functional response is investigated, some sufficient conditions which guarantee the existence and global attractivity of positive periodic solution for the system are obtained. Furthermore, our results improve the main results of paper [1].     

On Discrete Superconvergence Properties of Spline Collocation Methods for Nonlinear Volterra Integral Equations

It is shown that the error corresponding to certain spline collocation approximations for nonlinear Volterra integral equations of the second kind is the solution of a nonlinearly perturbed linear Volterra integral equation. On the basis of this result it is possible to derive general estimates for the order of convergence of the spline solution at the underlying mesh points. Extensions of these techniques to other types of Volterra equations are indicated.     

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.     

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.     

Uniform asymptoic stability in linear volterra difference equations

For linear Volterra difference equations of nonconvolution type, uniform asymptotic stability of the zero solution is characterized by the summability of the resolvent matrix. Moreover, the existence of bounded solutions of nonhomogeneous linear Volterra difference equations is studied.     

Asymptotic analysis of a volterra integral equation

A uniformly valid aymptotic solution is obtained for a class of perturbed Volterra integral equations, in which a naive expansion breaks down as t → ∞. The procedure used is an adaption of the formal methodology presented in [1] for the construction of a uniform asymptotic solution to Volterra equations which possess a boundary layer near t = 0.     

The reproducing kernel method for solving the system of the linear Volterra integral equations with variable coefficients

In this paper, we apply the reproducing kernel method to give the exact solution and approximate solution for the system of the linear Volterra integral equations with variable coefficients. Some examples are given, showing its effectiveness and convenience. Finally, the numerical results obtained by the reproducing kernel method are superior to those obtained by other methods in Farshid Mirzaee (2010) [4], Tahmasbi and Fard (2008) [5], Saeed and Ahmed (2008) [8].     

Numerical methods for the Lotka–McKendrick's equation

The Lotka–McKendrick's model is a well-known model which describes the evolution in time of the age structure of a population. In this paper we consider this linear model and discuss a range of methods for its numerical solution. We take advantage of different analytical approaches to the system, to design different numerical methods and compare them with already existing algorithms. In particular we set up some algorithms inspired by the approach based on Volterra integral equations and we also consider a direct approach based on the nonlinear system that describes the evolution of the age profile of the population.     

Necessary and sufficient condition for the unique existence of periodic solution of linear volterra equations

In this paper, we obtained the sufficient and necessary condition for the unique existence of periodic solution of the linear Volterra integro-differential equations of the form $$x'(t) = \int_0^\infty {(dE(s))x(t - s) + f(t)} $$ . We also proved that the mentioned equation has unique periodic solution is a generic property.     

Qualitative analysis of a diffusive prey–predator model

In this work we examine a Lotka–Volterra model with diffusion describing the dynamics of multiple interacting prey and predator species. We show that the solution exists, and is unique, bounded, nonnegative, and globally defined. We also prove the non-existence of nonconstant steady state solutions if certain conditions are satisfied. For the particular case of two prey (e.g., engineered and native, respectively) and one common predator species, by performing a linear stability analysis about the initial native-dominant steady state, we determine under which conditions the engineered species invasion succeeds.     

On the continuity of the Volterra variational derivative

We give a sufficient condition for the continuity of the Volterra variational derivative of a functional with respect to a fixed function. For linear functionals this condition is automatically satisfied, and so the Volterra variational derivative of a linear functional is always continuous.