Collocation methods for second-order volterra integro-differential equations

We study the numerical solution of second-order Volterra integro-differential equations by means of collocation techniques in certain polynomial spline spaces. Suitable discretization of the resulting collocation equation yields implicit methods which may be viewed as extensions of m-stage implicit Runge-Kutta-Nyström methods for initial-value problems of second-order ordinary differential equations to second-order integro-differential equations. The attainable order of (local) convergence of these methods is analyzed in detail.     

Periodic boundary value problems for second order impulsive integro-differential equations of volterra type

The existence of solutions of periodic boundary value problems for second order impulsive integro-differential equations of Volterra type is investigated. By using the method of upper and lower solutions, it is proved that the problem in whi h impulses occur at fixed times has a solution. Some impulsive integro-differential inequalities related to such problem are also established.     

Well-posed solvability of volterra integro-differential equations in Hilbert space

We study the well-posed solvability of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are an abstract form of linear partial integro-differential equations that arise in the theory of viscoelasticity and have a series of other important applications. We obtain results on the wellposed solvability of the considered integro-differential equations in weighted Sobolev spaces of vector functions defined on the positive half-line and ranging in a Hilbert space.     

On the numerical solution of linear and nonlinear volterra integral and integro-differential equations

Sinc bases are developed to approximate the solutions of linear and nonlinear Volterra integral and integro-differential equations. Properties of these sinc bases and some operational matrices are first presented. These properties are then used to reduce the integral and integro-differential equations to systems of linear or nonlinear algebraic equations. Numerical examples illustrate the pertinent features of the method and its applicability to a large variety of problems. The examples include convolution type, singular as well as singularly-perturbed problems.     

Hyperbolic integro-differential equations of convolution type

Making use of the theory of Wiener-Hopf operators in the scale of abstract Krein spaces, we prove existence and uniqueness of unbounded solutions for the linear hyperbolic integrodifferential equation (P_o). We extend herewith results obtained in [8] for hyperbolic evolution equations, where the convolution integral was absent. The method utilizes Dunford's functional calculus and permits thus a constructive existence proof for solutions exhibiting an exponential growth rate when time increases. Our approach bases upon the fundamental hypothesis that the spectrum of the time-independent mapping -A shows a parabolic condensation along the negative real axis. This condition completely determines the admissible geometry of the spectral set of the convolution integral operator, and a fortiori the magnitude of the exponential growth rate. The theory works in arbitrary reflexive Banach spaces.     

Stability of discrete volterra equations of hammerstein type

Stability conditions for Volterra equations with discrete time are obtained using direct Liapunov method, without usual assumption of the summability of the series of the coeffcients. Using such conditions, the stability of some numerical methods for second kind Volterra integral equation is analyzed.     

Some problems for multidimensional integro-differential equations of hyperbolic type

We prove the well-posedness of the Cauchy, Goursat, and Darboux problems for multidimensional in-tegro-differential equations of the hyperbolic type encountered in biology.     

Factorization of convolution type vector integro-differential equations

The paper investigates a class of convolution type integro-differential equations for vector-functions. Some special factorization methods are used to prove the solvability in several functional spaces.     

Numerical methods of solving nonlinear integral equations of volterra type

Applying continued fractions. we propose numerical methods of solving nonlinear Volterra integral equations of second kind. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 127–132     

Finite element analysis of parabolic integro-differential equations of Kirchhoff type

The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.     

Asymptotic stability versus exponential stability in linear volterra difference equations of convolution type

It is shown that uniform asymptotic stability does not imply exponential stability in linear Volterra difference equations. However, if the kernel of the equation decays exponentially. then both concepts are equivalent as in the case of ordinary difference equations.     

Fast and precise spectral method for solving pantograph type Volterra integro-differential equations

Numerical Algorithms - This paper focuses on studying a general form of pantograph type Volterra integro-differential equations (PVIDEs). We apply a new collocation spectral approach, based on...     

Application of partial averaging for a class of integro-differential equations of fredholm type

The Cauchy problem for a system of integro-differential equations of a standard type is considered. The authors justify a method for partial averaging of the considered problem for asymptotically large interval of variation of the independent variable.     

Integral boundary value problems for first order impulsive integro-differential equations of mixed type

This paper considers existence of solutions for a class of first order impulsive differential equation with integral boundary value conditions. We present a new comparison theorem and show that the monotone iterative technique coupled with lower and upper solutions is still valid. The results we obtained generalize and improve many recent results.     

Exact reachability for second-order integro-differential equations

In this Note we analyze a reachability problem for an integro-differential equation by using a harmonic analysis approach. To cite this article: P. Loreti, D. Sforza, C. R. Acad. Sci. Paris, Ser. I 347 (2009).