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.     

Solvability of vector integro-differential equations of convolution type on the semiaxis

The paper deals with vector integro-differential equation of convolution type that have the form

$ - \frac{{d^2 f_i }}{{dx^2 }} + a_i f_i (x) = g_i (x) + \sum\limits_{j = 1}^N {\int\limits_0^\infty {K_{ij} (x - t)f_j (t)dt, } i = 1,2, \ldots ,N,} $

where \(\vec f\) = (f ₁, f ₂, ..., f _N ) ^T is the unknown vector-function, a _i are nonnegative numbers, \(\vec g\) = (g ₁, g ₂, ..., g _N ) ^T ? L ₁ ^×N (0,+∞) ≡ L ₁ (0,+∞) × ... × L (0,+∞) is the independent term of the equation with nonnegative components and 0 ≤ K _ij ? L ₁ (?∞,+∞), i, j = 1, 2, …,N are the kernel-functions. These equations have significant applications in the wave non-local interaction theory. Using some special factorization methods, solvability of the system is proved in different functional spaces.

     

Periodic solutions of Quasilinear Hyperbolic integro-differential equations of second order

We study a periodic boundary-value problem for a quasilinear integro-differential equation with the d’Alembert operator on the left-hand side and a nonlinear integral operator on the right-hand side. We establish conditions under which the uniqueness theorems are true.     

Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels

Variable stepsize algorithms for the numerical solution of nonlinear Volterra integral and integro-differential equations of convolution type are described. These algorithms are based on an embedded pair of Runge–Kutta methods of order p=5 and p=4 proposed by Dormand and Prince with interpolation of uniform order p=4. They require O(N) number of kernel evaluations, where N is the number of steps. The cost of the algorithms can be further reduced for equations that have rapidly vanishing convolution kernels, by using waveform relaxation iterations after computing the numerical approximation by variable stepsize algorithm on some initial interval. AMS subject classification (2000)  65R20, 45L10, 93C22     

On nonlinear Volterra equations of convolution type

Weakly nonlinear and strongly nonlinear convolution-type Volterra equations u(x) = (K * ϕ(u))(x) are studied in new classes of weakly synchronous and quasiconcave functions f under assumptions less restrictive than the classical ones. Existence and uniqueness theorems, as well as theorems on the absence of solutions, are proved. Smoothness issues for solutions of both weakly nonlinear and strongly nonlinear equations are considered. An integral inequality is obtained for the weight function of a metric ensuring that a nonlinear operator is a contraction, and a number of other results are obtained.     

Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions

In this paper, we use operational matrices of piecewise constant orthogonal functions on the interval [0,1)$[0,1)$ to solve Volterra integral and integro-differential equations of convolution type without solving any system. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by operational matrices. Numerical examples show that the approximate solutions have a good degree of accuracy.     

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.     

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.     

About stability of a difference analogue of a nonlinear integro-differential equation of convolution type

A nonlinear integro-differential equation of convolution type with order of nonlinearity more than one and a stable trivial solution is considered. The integral in this equation has an exponential kernel and polynomial integrand. The difference analogue of the equation considered is constructed in the form of a difference equation with continuous time and it is shown that this difference analogue preserves the properties of stability of his original.     

On multi-dimensional integral equations of convolution type with shift

The criterion of invertibility or Fredholmness of some multi-dimensional integral equations with Carleman type shifts are given. The investigation is based on some Banach space approach to equations with an involutive operator. A modified version of this approach is also presented in the paper.This approach is applied to multi-dimensional convolution type equations when the kernels may be integrable or of singular Calderon-Zygmund-Mikhlin type and shift generated by a linear transformation in the Euclidean space satisfying the generalized Carleman condition. The convolution type equations are also specially considered in the two-dimensional case in a sector on the plane symmetric with respect to one of the axes and the corresponding reflection shift. Another application deals with multi-dimensional equations with homogeneous kernels and the shift .     

Study on Sobolev type Hilfer fractional integro-differential equations with delay

In this paper, we deal with a class of nonlinear Sobolev type fractional integro-differential equations with delay using Hilfer fractional derivative, which generalized the famous Riemann–Liouville fractional derivative. The definition of mild solutions for studied problem was given based on an operator family generated by the operator pair (A, B) and probability density function. Combining with the techniques of fractional calculus, measure of noncompactness and fixed point theorem, we obtain new existence result of mild solutions with two new characteristic solution operators and the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition. The results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.     

Numerical iterative method for Volterra equations of the convolution type

The objective of this paper is to present an algorithm from which a rapidly convergent solution is obtained for Volterra integral equations of Hammerstein type. Such equations are often encountered when describing the response of viscoelastic materials where the time dependency of the material properties is often expressed in the form of a convolution integral. Frequently, singularity is encountered and often ignored when dealing with the constitutive equations of viscoelastic materials. In this paper, the singularity is incorporated in the solution and the iterative scheme used to solve the equation converges within six iterations to a typical toleration error of 10^?5. The algorithm is applied to the strain response of a polymer under impulsive (constant) loading and the results show excellent correlation between the experimental and analytical solution. Copyright © 2010 John Wiley & Sons, Ltd.     

Lp-solutions of singular integro-differential equations

We study a variety of scalar integro-differential equations with singular kernels including linear, nonlinear, and resolvent equations. The first result involves a type of existence theorem which uses a fixed point mapping defined by the integro-differential equation itself and produces a unique solution with a continuous derivative in a very simple way. We then construct a Liapunov functional yielding qualitative properties of solutions. The work answers questions raised by Volterra in 1928, by Levin in 1963, and by Grimmer and Seifert in 1975. Previous results had produced bounded solutions from bounded perturbations. Our results mainly concern integrable solutions from integrable perturbations.     

Asymptotic stability of two classes of second-order integro-differential equations of Volterra type

We study the asymptotic stability of linear homogeneous second-order integrodifferential equations of Volterra type on a half-line for the case in which the corresponding linear homogeneous differential equation is asymptotically unstable. The exponential stability of these equations in the same setting is considered as well. Illustrative examples are given.     

On the approximate solution of weakly singular integro-differential equations of Volterra type

Recently, the convergence rate of the collocation method for integral and integro-differential equations with weakly singular kernels has been studied in a series of papers [1–7]. The present paper belongs to the same series. We analyze the possibility of constructing approximate solutions of high-order accuracy on a uniform or almost uniform grid for weakly singular integro-differential equations of Volterra type.Translated from Differentsialnye Uravneniya, Vol. 40, No. 9, 2004, pp. 1271–1279.Original Russian Text Copyright © 2004 by Pedas.     

Oscillations of linear integro-differential equations

Sufficient conditions which guarantee that certain linear integro-differential equation cannot have a positive solution are established.