Volterra Integral Operators Research Articles

Spectral Analysis of Linear Models of Viscoelasticity

In this paper, we examine Volterra integrodifferential equations with unbounded operator coefficients in Hilbert spaces. Equations considered are abstract hyperbolic equations perturbed by terms containing Volterra integral operators. These equations can be realized as partial integrodifferential equations that appear in the theory of viscoelasticity (see [2, 5]), as Gurtin–Pipkin integrodifferential equations (see [1, 7]) that describe finite-speed heat transfer in materials with memory. They also appear in averaging problems for multiphase media (Darcy’s law.

Perturbed Integral Operator Equations of Volterra Type with Applications to $${\varvec{p}}$$ p -Laplacian Equations

We consider perturbed Volterra integral operator equations of the form $$\begin{aligned} y(t)=\gamma (t)H(\psi (y))+\lambda \int _0^tf(\tau ,y(\tau ))\,\mathrm{d}\tau , \end{aligned}$$ where $$\psi :{\mathscr {C}}([0,1])\rightarrow {\mathbb {R}}$$ is a linear functional. By utilizing a nonstandard order cone and an associated nonstandard open set, we demonstrate the existence of at least one positive solution to this problem under some more general conditions previously seen in the literature. As a related problem we consider the one-dimensional p-Laplacian equation $$\begin{aligned} (\varphi _{p}(y'(t)))'=-\lambda f(t,y(t)),\quad t\in (0,1) \end{aligned}$$ subject to the nonlocal boundary conditions $$\begin{aligned}&\varphi _p(y'(0))=0\\&y(1)=H(\psi (y)), \end{aligned}$$ where $$\varphi _p(z):=|z|^{p-2}z$$ , for $$p>2$$ , is the one-dimensional p-Laplacian.

Study of functional-differential equations with unbounded operator coefficients

Functional-differential and integro-differential equations with the principal part being an abstract hyperbolic equation perturbed by terms with unbounded variable operator coefficients multiplying variable delays are studied. Additionally, Volterra integral operators are considered. For the equations under study, the well-posedness of initial value problems in Sobolev spaces of vector functions is proved. In the autonomous case, spectral analysis of the operator functions that are the symbols of the indicated equations is performed.

Linear differential-algebraic equations perturbed by Volterra integral operators

We study linear inhomogeneous vector ordinary differential equations of arbitrary order in which the matrix multiplying the highest derivative of the unknown vector function is singular in the domain where the equations are defined. We also study perturbations (not necessarily small) of such equations, which are linear integro-differential equations with a Volterra operator. We obtain sufficient conditions for the solvability of such equations and give representations of their general solutions; solvability and uniqueness conditions are also given for initial value problems for such equations. The influence of small perturbations of the free term and the initial data on the solution is considered. A numerical method is suggested. The results of numerical experiments are given.

Inverse nodal problems for Dirac-type integro-differential operators

The inverse nodal problem for Dirac differential operator perturbated by a Volterra integral operator is studied. We prove that dense subset of the nodal points determines the coefficients of differential part and gives partial information on the coefficients of integral part of the operator. We also provide an algorithm to reconstruct the coefficients of the problem by using the nodal points.

CORDIAL VOLTERRA INTEGRAL EQUATIONS AND SINGULAR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN SPACES OF ANALYTIC FUNCTIONS∗

We study general cordial Volterra integral equations of the second kind and certain singular fractional integro-differential equation in spaces of analytic functions. We characterize properties of the cordial Volterra integral operator in these spaces, including compactness and describe its spectrum. This enables us to obtain conditions under which these equations have a unique analytic solution. We also consider approximate solution of these equations and prove exponential convergence of approximate solutions to the exact solution.

Distributed Fault Detection and Isolation for Interconnected Systems: a Non-Asymptotic Kernel-Based Approach

In this paper, a novel framework is proposed for deadbeat distributed Fault Detection and Isolation (FDI) of large-scale continuous-time LTI dynamic systems. The monitored system is composed of several subsystems which are linearly interconnected with unknown parameterization. Each subsystem is monitored by a local diagnoser based on the measured local output, local inputs and the interconnection variables from the neighboring subsystems. The local FDI decision is based on two non-asymptotic state-parameter estimators using Volterra integral operators which eliminate the effect of the unknown initial conditions so that the estimates converge to the true value in a deadbeat manner and therefore the fault diagnosis can be achieved in finite time. Moreover, the unknown interconnection parameters and the unknown fault parameters are simultaneously estimated. Numerical examples are included to show the effectiveness of the proposed FDI architecture.

The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions

For kernels ν which are positive and integrable we show that the operator g↦Jνg=∫0xν(x−s)g(s)ds on a finite time interval enjoys a regularizing effect when applied to Hölder continuous and Lebesgue functions and a “contractive” effect when applied to Sobolev functions. For Hölder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor N(x)=∫0xν(s)ds. For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of Orlicz integrability. Finally, for functions in Sobolev spaces, we show that the operator Jν “shrinks” the norm of the argument by a factor that, as in the Hölder case, depends on the function N (whereas no regularization result can be obtained).These results can be applied, for instance, to Abel kernels and to the Volterra function I(x)=μ(x,0,−1)=∫0∞xs−1/Γ(s)ds, the latter being relevant for instance in the analysis of the Schrödinger equation with concentrated nonlinearities in R2.

Effective properties of n-coated composite spheres assemblage in an ageing linear viscoelastic framework

The homogenization scheme based on n-coated spherical inclusion morphology, also called Generalized Self-Consistent (GSC) scheme, is derived in an ageing linear viscoelastic framework by using a formal treatment in terms of Volterra integral operator. Estimations of the bulk and shear relaxation functions as well as the creep functions are provided by numerical evaluation of the integral operator. The resulting (semi-)analytical estimations are compared to numerical simulations by Finite Elements Method (FEM) on 2-phase and 3-phase 3D numerical samples, and to Mori-Tanaka estimations in the 2-phase case.

Standard transmutation operators for the one dimensional Schrödinger operator with a locally integrable potential

We study a special class of operators T satisfying the transmutation relation(d2dx2−q)Tu=Td2dx2u in the sense of distributions, where q is a locally integrable function, and u belongs to a suitable space of distributions depending on the smoothness properties of q. A method which allows one to construct a fundamental set of transmutation operators of this class in terms of a single particular transmutation operator is presented. Moreover, following [27], we show that a particular transmutation operator can be represented as a Volterra integral operator of the second kind. We study the boundedness and invertibility properties of the transmutation operators, and use these to obtain a representation for the general distributional solution of the equation d2udx2−qu=λu, λ∈C, in terms of the general solution of the same equation with λ=0.

On some properties of the solution set map to Volterra integral inclusion

For the multivalued Volterra integral equation defined in a Banach space, the set of solutions is proved to be $R_\delta$, without auxiliary conditions imposed in Theorem 6 [J. Math. Anal. Appl. 403 (2013), 643-666]. It is shown that the solution set map, corresponding to this Volterra integral equation, possesses a continuous singlevalued selection; and the image of a convex set under the solution set map is acyclic. The solution set to the Volterra integral inclusion in a separable Banach space and the preimage of this set through the Volterra integral operator are shown to be absolute retracts.

Robust finite-time estimation of biased sinusoidal signals: A volterra operators approach

A novel finite-time convergent estimation technique is proposed for identifying the amplitude, frequency and phase of a biased sinusoidal signal. Resorting to Volterra integral operators with suitably designed kernels, the measured signal is processed yielding a set of auxiliary signals in which the influence of the unknown initial conditions is removed. A second-order sliding mode-based adaptation law–fed by the aforementioned auxiliary signals–is designed for finite-time estimation of the frequency, amplitude, and phase. The worst case behavior of the proposed algorithm in presence of the bounded additive disturbances is fully characterized by Input-to-State Stability arguments. The effectiveness of the estimation technique is evaluated and compared with other existing tools via extensive numerical simulations.

Numerical Analysis of Fractional Neutral Functional Differential Equations Based on Generalized Volterra-Integral Operators

We use waveform relaxation (WR) method to solve numerically fractional neutral functional differential equations and mainly consider the convergence of the numerical method with the help of a generalized Volterra-integral operator associated with the Mittag–Leffler function. We first give some properties of the integral operator. Using the proposed properties, we establish the convergence condition of the numerical method. Finally, we provide a new way to prove the convergence of waveform relaxation method for integer-order neutral functional differential equation, which is a special case of fractional neutral functional differential equation. Compared to the existing proof in the literature, our proof is concise and original.

Control system defined by some integral operator

In the paper we consider a nonlinear control system governed by the Volterra integral operator. Using a version of the global implicit function theorem we prove that the control system under consideration is well-posed and robust, i.e. for any admissible control \(u\) there exists a uniquely defined trajectory \(x_{u}\) which continuously depends on control \(u\) and the operator \(u\mapsto x_{u}\) is continuously differentiable. The novelty of this paper is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken standard assumptions.

Cordial Volterra Integral Equations with Vanishing Delays*

Cordial Volterra integral equations (CVIEs) from some applications models associated with a noncompact cordial Volterra integral operator are discussed in the recent years. A lot of real problems are effected by a delayed history information. In this paper we investigate some properties of cordial Volterra integral operators influenced by a vanishing delay. It is shown that to replicate all eigenfunctions , or , the vanishing delay must be a proportional delay. For such a linear delay, the spectrum, eigenvalues and eigenfunctions of the operators and the existence, uniqueness and solution spaces of solutions are presented. For a nonlinear vanishing delay, we show a necessary and sufficient condition such that the operator is compact, which also yields the existence and uniqueness of solutions to CVIEs with the vanishing delay.

Singular Initial-Value and Boundary-Value Problems for Integrodifferential Equations in Dynamical Insurance Models with Investments

We investigate two insurance mathematical models of the following behavior of an insurance company in the insurance market: the company invests a constant part of the capital in a risk asset (shares) and invests the remaining part in a risk-free asset (a bank account). Changing parameters (characteristics of shares), this strategy is reduced to the case where all the capital is invested in a risk asset. The first model is based on the classical Cramer–Lundberg risk process for the exponential distribution of values of insurance demands (claims). The second one is based on a modification of the classical risk process (the so-called stochastic premium risk process) where both demand values and insurance premium values are assumed to be exponentially distributed. For the infinite-time nonruin probability of an insurance company as a function of its initial capital, singular problems for linear second-order integrodifferential equations arise. These equations are defined on a semiinfinite interval and they have nonintegrable singularities at the origin and at infinity. The first model yields a singular initial-value problem for integrodifferential equations with a Volterra integral operator with constraints. The second one yields more complicated problem for integrodifferential equations with a non-Volterra integral operator with constraints and a nonlocal condition at the origin. We reduce the problems for integrodifferential equations to equivalent singular problems for ordinary differential equations, provide existence and uniqueness theorems for the solutions, describe their properties and long-time behavior, and provide asymptotic representation of solutions in neighborhoods of singular points. We propose efficient algorithms to find numerical solutions and provide the computational results and their economics interpretation.

Nonlocal semilinear integro-differential inclusions via vectorial measures of noncompactness

In this note, we show the existence of mild solutions to a nonlocal semilinear integro-differential problem via a new -valued measure of noncompactness involving a Volterra integral operator. Our existence result extends in a broad sense some recent theorems for integro-differential equations. Then we provide a compactness result on the mild solution set and an application to a reaction–diffusion problem involving a distributed time delay.

Operational Tau method for singular system of Volterra integro-differential equations

The Legendre spectral Tau matrix formulation is proposed to approximate solution of singular system of Volterra integro-differential equations. The existence and uniqueness solution of this system are investigated by means of the υ-smoothing property of a Volterra integral operator and some projectors. The L2−convergence of the numerical method is analyzed. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially. Finally, two numerical examples illustrate the theoretical results.

The Eshelby inclusion problem in ageing linear viscoelasticity

The present paper focuses on the Eshelby inclusion problem which is revisited in the framework of ageing linear viscoelasticity. All known results established in linear viscoelasticity thanks to the correspondence principle are recovered as particular cases of a general solution extended to ageing. A closed form solution is presented for the Hill and Eshelby tensors related to an ellipsoidal inclusion embedded in an infinite anisotropic medium. The case of the isotropic medium is investigated in detail and related solutions are presented for spherical and ellipsoidal inclusions. A numerical procedure which operates in time domain and based on the trapezoidal rule to determine Volterra integral operators allows to efficiently calculate Hill and Eshelby tensors for a wide range of behaviours including in particular time-dependent Poisson ratios. Validation and verification of the new developed solution are presented for the ageing spherical inclusion on the basis of recent published results and it is completed by comparisons with finite element simulations for the general and novel case of an oblate spheroid embedded in an ageing linear viscoelastic matrix.

Numerical solution and structural analysis of two-dimensional integral-algebraic equations

The ź-smoothing property of a one-dimensional Volterra integral operator and some projectors (Liang and Brumer, SIAM J. Numer. Anal. 51, 2238---2259 (2013)) are extended for two-dimensional integral-algebraic equations (TIAEs). Using these concepts, we decompose the given general TIAEs into mixed systems of two-dimensional Volterra integral equations (TVIEs) consisting of second- and first-kind TVIEs. Numerical technique based on the Chebyshev polynomial collocation methods is presented for the solution of the mixed TVIE system. Global convergence results are established and the performance of the numerical scheme is illustrated by means of some test problems.