In this paper we consider the nonlinear parabolic integro-differential equation with initial and boundary conditions. We develop the method of generalized quasilinearization to generate linear iterates that converge quadratically to the unique solution of the nonlinear parabolic integro-differential equation. For this purpose, we establish comparison results for the parabolic integro-differenti...

A class of singular integro-differential equations in Lebesgue spaces are studied. There are many applications of the singular integro-differential equations discussed in this paper. An example in modeling the stress distribution of an elastic medium with holes is discussed in the paper. Direct numerical schemes using a collocation method and a mechanical quadrature rule designed for the singul...

Abstract: The aim of study is to solve parabolic integro-differential equation with a weakly singular kernel. Problems involving partial integro-differential equations arise in fluid dynamics, viscoelasticity, engineering, mathematical biology, financial mathematics and other areas. Many mathematical formulations of physical phenomena contain integro-differential equations. Integro-differential...

the spline collocation method  is employed to solve a system of linear and nonlinear fredholm and volterra integro-differential equations. the solutions are collocated by cubic b-spline and the integrand is approximated by the newton-cotes formula. we obtain the unique solution for linear and nonlinear system $(nn+3n)times(nn+3n)$ of integro-differential equations. this approximation reduces th...

In this paper, Itô stochastic integro-differential equations are considered. By establishing an L -operator integro-differential inequality and using the properties of M -cone and stochastic analysis technique, we obtain some new sufficient conditions ensuring the exponential p -dissipativity of the stochastic integro-differential equations. An example is also discussed to illustrate the effici...

Integro-differential polynomials are a novel generalization of the well-known differential polynomials extensively used in differential algebra [17]. They were introduced in [29] as a kind of universal extensions of integro-differential algebras and have recently been applied in a confluence proof [34] for the rewrite system underlying the so-called “integro-differential operators”. In this pap...

In our symbolic approach to boundary problems for linear ordinary differential equations we use the algebra of integro-differential operators as an algebraic analogue of differential, integral and boundary operators (Section 2). They allow to express the problem statement (differential equation and boundary conditions) as well as the solution operator (an integral operator called “Green’s opera...

In this paper we study the existence of classical solutions for a class of abstract neutral integro-differential equation with unbounded delay. A concrete application to partial neutral integro-differential equations is considered.

In this paper we consider a risk model with two classes of insurance risks in the presence of a multi-layer dividend startegy. We assume that the two claim counting processes are, respectively, Poisson and Sparre Andersen with generalized Erlang(2) claim inter-arrival times. We derive an integro-differential equation system for the Gerber-Shiu functions for surplus-dependent premium rates and a...

In this paper, we study algorithmic aspects of linear ordinary integro-differential operators with polynomial coefficients. Even though this algebra is not noetherian and has zero divisors, Bavula recently proved that it is coherent, which allows one to develop an algebraic systems theory. For an algorithmic approach to linear systems theory of integro-differential equations with boundary condi...