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In this paper, we discuss the application of an iterative collocation method based on the use of Lagrange polynomials for the numerical solution of a class of nonlinear third-kind Volterra integral equations. The approximate solution is given by explicit formulas. The error analysis of the proposed numerical method is studied theoretically. Some numerical examples are given to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution of a two‐dimensional partial Volterra integro‐differential equation (2D‐PVIDE) is provided by extending the Taylor collocation scheme of one‐dimensional Volterra integral equations. The method is based on the use of Taylor polynomials in two‐dimensional, while the approximate solution is given by using explicit schemes. The method is proved to be high‐order convergent with respect to the maximum norm. Some numerical examples are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The present paper deals with the numerical solution for a general form of a system of nonlinear Volterra delay integro-differential equations (VDIDEs). The main purpose of this work is to provide a current numerical method based on the use of continuous collocation Taylor polynomials for the numerical solution of nonlinear VDIDEs systems. It is shown that this method is convergent. Numerical results will be presented to prove the validity and effectiveness of this convergent algorithm. We apply models to the COVID-19 epidemic in China, Spain, and Italy and one for the Predator–Prey model in mathematical ecology. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide a new direct numerical method for high-order linear Volterra integro-differential equations (VIDEs). An algorithm based on the use of Taylor polynomials is developed for the numerical solution of high-order linear VIDEs. It is shown that this algorithm is convergent. Numerical results are presented and comparisons are made with well-known numerical methods to prove the effectiveness of the presented algorithm. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of a singular boundary value problem on the half line for a second order nonlinear ordinary differential equation. Due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain, the proposed approaches are complex and require a considerable computational effort. In the present paper, we describe an alternative approach, based on the reduction of the original problem to an integro-differential equation. Though the problem is posed on the half-line, we just need to approximate the solution on a finite interval. By analyzing the behavior of the numerical approximation on this interval, we identify the solution that satisfies the prescribed boundary condition. Although the numerical algorithm is much simpler than the ones proposed before, it provides accurate approximations. We illustrate the proposed methods with some numerical examples. [ABSTRACT FROM AUTHOR]

This article concerns nonlinear functional integral equations of product type. The first two equations set on a the positive half-axis encompass different classes of nonlinear integral equations and may involve the product of finitely many integral functions. The existence of integrable solutions is based on improved versions of Krasnoselskii's fixed point theorem combined with techniques of measure of weak noncompactness and some elements from functional analysis. The third one is an integro-differential equation set on a bounded interval, for which the existence of absolutely continuous solutions is provided. Examples show the applicability of our results. [ABSTRACT FROM AUTHOR]

In this article, we consider a model for the spread of certain infectious disease governed by a delay integro-differential equation. We obtain the existence and the uniqueness of a positive periodic solution, by using Perov's fixed point theorem in generalized metric spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of delay integro-differential equations. The main purpose of this work is to provide a new numerical approach based on the use of continuous collocation Taylor polynomials for the numerical solution of delay integro-differential equations. It is shown that this method is convergent. Numerical illustrations confirm our theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A numerical method based on the use of Taylor polynomials is proposed to construct a collocation solution $u\in S_{m-1}^{(-1)}(\Pi _{N})$ for approximating the solution of delay integral equations. It is shown that this method is convergent. Some numerical examples are given to show the validity of the presented method. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with nonlinear integral equations with two constant delays. According to the behavior of the data functions, existence and uniqueness results of a measurable solution, an exponentially stable solution, a bounded solution and an integrable solution are provided. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide a numerical approach for two-dimensional Volterra integral equations (2D-VIEs). An algorithm based on the use of Taylor polynomials is developed to construct a collocation solution v ∈ S p − 1 , p − 1 (− 1) (Π N , M) for approximating the solution of 2D-VIEs. It is shown that this algorithm is convergent. Some numerical examples are included to demonstrate the accuracy of the presented method. [ABSTRACT FROM AUTHOR]