On exact rates of decay of solutions of linear systems of Volterra equations with delay

This paper considers the resolvent of a finite-dimensional linear convolution Volterra integral equation. The main results give conditions which ensure that the exact rate of decay of the resolvent can be determined using a positive weight function related to the kernel. The decay rates can be exponential or subexponential. Many other related results on exact rates of exponential and subexponential decay of solutions of Volterra integro-differential equations are given. We also present an application to a linear compartmental system with discrete and continuous lags.     

Long run behaviour of the autocovariance function of ARCH(∞) models

The asymptotic properties of the memory structure of ARCH($\infty$) equations are investigated. This asymptotic analysis is achieved by expressing the autocovariance function of ARCH($\infty$) equations as the solution of a linear Volterra summation equation and analysing the properties of an associated resolvent equation via the admissibility theory of linear Volterra operators. It is shown that the autocovariance function decays subexponentially (or geometrically) if and only if the kernel of the resolvent equation has the same decay property. It is also shown that upper subexponential bounds on the autocovariance function result if and only if similar bounds apply to the kernel.     

Asymptotic behavior for coupled abstract evolution equations with one infinite memory

In this paper, we consider two coupled abstract linear evolution equations with one infinite memory acting on the first equation. Our work is motivated by the recent results of [42], where the authors considered the case of two wave equations with one convolution kernel converging exponentially to zero at infinity, and proved the lack of exponential decay. On the other hand, the authors of [42] proved that the solutions decay polynomially at infinity with a decay rate depending on the regularity of the initial data. Under a boundedness condition on the past history data, we prove that the stability of our abstract system holds for convolution kernels having much weaker decay rates than the exponential one. The general and precise decay estimate of solution we obtain depends on the growth of the convolution kernel at infinity, the regularity of the initial data, and the connection between the operators describing the considered equations. We also present various applications to some distributed coupled systems such as wave-wave, Petrovsky-Petrovsky, wave-Petrovsky, and elasticity-elasticity.     

A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.     

The boundaries of the solutions of the linear Volterra integral equations with convolution kernel

Some boundaries about the solution of the linear Volterra integral equations of the second type with unit source term and positive monotonically increasing convolution kernel were obtained in Ling, 1978 and 1982. A method enabling the expansion of the boundary of the solution function of an equation in this type was developed in I. Özdemir and Ö. F. Temizer, 2002.

In this paper, by using the method in Özdemir and Temizer, it is shown that the boundary of the solution function of an equation in the same form can also be expanded under different conditions than those that they used.

     

Uniform Rates of Decay in Anisotropic Thermo-Viscoelasticity

We consider the anisotropic and inhomogeneous thermo-viscoelastic equation. We prove that the first and second-order energy decay exponentially as time goes to infinity provided the relaxation function also decays exponentially to zero. While if the relaxation functions decay polynomially to zero, then the energy decays also polynomially. That is, the kernel of the convolution defines the rate of decay of the solution.     

Solution of problems on the anti-plane deformation of elastic bodies with corner points by the method of integral equations

The problem of the antiplane deformation of an elastic cylinder with a multiconnected finite or infinite section, bounded by a system of closed curves that can have corner points, is examined. Forces or displacements are given on the whole boundary of the body. The problem is reduced to an integral equation whose kernel has strong stationary singularities at the corner points. Results of an investigation of the solvability of this equation and the smoothness of its solution are presented. A procedure for the numerical solution of the integral equation is described. A space with a prismatic hole of rectangular section or a rigid inclusion subjected to a uniform tangential force at infinity is considered as an example. The generalized stress intensity factors are calculated.     

Nodal Type Bound States for Nonlinear Schrödinger Equations with Decaying Potentials

In this paper, we are concerned with the existence of nodal type bound state for the following stationary nonlinear Schrödinger equation $$-Δu(x)+V(x)u(x)=|u|^{p-1}u, x∈ R^N, N ≥ 3,$$ where 1 < p < (N+2)/(N-2) and the potential V(x) is a positive radial function and may decay to zero at infinity. Under appropriate assumptions on the decay rate of V(x), Souplet and Zhang [1] proved the above equation has a positive bound state. In this paper, we construct a nodal solution with precisely two nodal domains and prove that the above equation has a nodal type bound state under the same conditions on V(x) as in [1].     

Lévy-driven Volterra Equations in Space and Time

We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the Lévy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyze its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.     

Numerical solution of a nonuniquely solvable Volterra integral equation using extrapolation methods

In this work the numerical solution of a Volterra integral equation with a certain weakly singular kernel, depending on a real parameter μ, is considered. Although for certain values of μ this equation possesses an infinite set of solutions, we have been able to prove that Euler's method converges to a particular solution. It is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize, so that general extrapolation algorithms, like the E-algorithm, can be applied to improve the numerical results. This is illustrated by means of some examples.     

Decay estimates for second order evolution equations with memory

This paper develops a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms. Depending on the properties of convolution kernels at infinity, we show that the energy of a mild solution decays exponentially or polynomially as t→+∞. Our approach is based on integral inequalities and multiplier techniques.These decay results can be applied to various partial differential equations. We discuss three examples: a semilinear viscoelastic wave equation, a linear anisotropic elasticity model, and a Petrovsky type system.     

The exact solution of a class of Volterra integral equation with weakly singular kernel

In this paper, the weakly singular Volterra integral equations with an infinite set of solutions are investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation. The aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory. Applying weighted integral, we provide a new definition for reproducing kernel space and obtain reproducing kernel function. Using the good properties of reproducing kernel function, the only smooth solution is exactly expressed in the form of series. The n-term approximate solution is obtained by truncating the series. Meanwhile, we prove that the derivative of approximation converges to the derivative of exact solution uniformly. The final numerical examples compared with other methods show that the method is efficient.     

Existence results for an indefinite unbounded perturbation of a resonant Schrödinger equation

We study existence results for a nonlinear Schrödinger equation at resonance. The nonlinearity is assumed to change sign, be unbounded but sublinear with a power like growth at infinity. Under a suitable coercivity assumption on the primitive of the nonlinear term on the kernel of the Schrödinger operator, we prove the existence of at least one solution.     

On the sample path properties of mixed Poisson processes

The mixed Poisson process has been widely used in financial engineering for modeling arrival of events that cluster in time, as it has strictly stationary and positively correlated increments. However, we show that, surprisingly, the sample autocovariance and autocorrelation of the increments of a mixed Poisson process converge to zero almost surely as the sample size goes to infinity. Consequently, the sample autocovariance or autocorrelation cannot be used in the method of moments for parameter estimation of mixed Poisson processes.     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

Nonlocal diffusion,a Mittag-Leffler function and a two-dimensional Volterra integral equation

In this paper we consider a particular class of two-dimensional singular Volterra integral equations. Firstly we show that these integral equations can indeed arise in practice by considering a diffusion problem with an output flux which is nonlocal in time; this problem is shown to admit an analytic solution in the form of an integral. More crucially, the problem can be re-characterized as an integral equation of this particular class. This example then provides motivation for a more general study: an analytic solution is obtained for the case when the kernel and the forcing function are both unity. This analytic solution, in the form of a series solution, is a variant of the Mittag-Leffler function. As a consequence it is an entire function. A Gronwall lemma is obtained. This then permits a general existence and uniqueness theorem to be proved.     

Well-posedness of a non-local abstract Cauchy problem with a singular integral

A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some boundedness and smoothness conditions on the coefficient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with a singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with in ow boundary conditions.     

EXTRAPOLATION AND A-POSTERIORI ERROR ESTIMATORS OF PETROV-GALERKIN METHODS FOR NON-LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

1. IntroductionThe pmpme of tabs Paper is to show that the ~ardson edrapolation can be used toenhance the nUmerical solutions generated by a cab of Petrov-Gaierkin lhate element methodsfor the nonlinear VOlterra integrO-chrential equation (VIDE):where j = j(t,y): I x R --+ R and k = k(t,8,g): D x R - R (with D:= {(t,8): 0 S & S t ST}) denote given hmctions.Throughout tab paperl it will always be assumed that the problem (1.1) possesses a piquesolution y E C'(I), namely, the given hmc…     

Conditions at infinity for boltzmann's equation.

We consider here realistic conditions at infinity for solutions of the Boltzmann's equation, such as a pure Maxwellian equilibrium at infinity possibly with suitable boundary conditions on an exterior domain, different Maxwellian equilibria at +∞ and -;∞ in a tube-like situation and more generally conditions at infinity obtained from a fixed solution. In order to adapt the recent global existence and compactness results due to R.J. DiPerna and the author, we have to obtain some local a priori estimates on the mass, kinetic energy and entropy. And this is precisely what we achieve here by two different and new methods. The first one consists in using the relative entropy of solutions with respect to a fixed, possibly local, Maxwellian. This method allows to treat general collision kernels with angular cut-off and some of the conditions at infinity mentioned above. The second method is based upon a L¹ estimate and an extension of the entropy identity which uses a truncated H-functional. This method requires a “uniform integrability” condition on the collision kernel but allows to consider the most general conditions at infinity.     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.