Exact solutions to some classes of nonlinear integral, integro-functional, and integro-differential equations (original) (raw)

On the solution of a mixed nonlinear integral equation

Applied Mathematics and Computation, 2011

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method,

Three Methods to Solve Two Classes of Integral Equations of the Second Kind

2019

abstract: Three methods to solve two classes of integral equations of the second kind are introduced in this paper. Firstly, two Kantorovich methods are proposed and examined to numerically solving an integral equation appearing from mathematical modeling in biology. We use a sequence of orthogonal finite rank projections. The first method is based on general grid projections. The second one is established by using the shifted Legendre polynomials. We present a new convergence analysis results and we prove the associated theorems. Secondly, a new Nyström method is introduced for solving Fredholm integral equation of the second kind.

Linear Integral Equations

Graduate Texts in Contemporary Physics

Integral equations have proved itself as one of the most important branches of mathematics. In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Green's function, Fredholm's theory, and Maxwell's equations. Integral equations can be divided into three main classes: linear integral equations, non-linear integral equations and singular integral equations. The present book is based on lectures given by the author to students of various colleges studying mathematics. In designing this book the author tried to select the most important mathematical facts and present them so that the reader could acquire the necessary mathematical conception and apply mathematics to other branches. This book consists of six chapters. Chapter 1, Integral Equations in Mathematics. Chapter 2 contains the resolvent kernel and the Neumann series. In chapter 3 , we will discuss Fredholm's Equations with degenerate kernels. Chapter 4 presents Volterra Integral Equation. Chapter 5 contains the classical Fredholm's theory. Chapter 6, contains Symmetric Integral Equations-Hermitian Kernels.

On a class of Urysohn–Stieltjes quadratic integral equations and their applications

Journal of Computational and Applied Mathematics, 2000

We investigate a class of nonlinear quadratic integral equation of Urysohn-Stieltjes type. Such a class contains a number of classical nonlinear quadratic integral equations such as the Chandrasekhar quadratic integral equation. We consider the solvability of the equations in question in the space of continuous functions under very general assumptions. Several particular cases of the assumed hypotheses are discussed and numerous applications are indicated.

A new analytical solution procedure for nonlinear integral equations

Mathematical and Computer Modelling, 2012

The chief aim of the present article is to introduce a new analytical technique, two-step Laplace decomposition method (TSLDM) for nonlinear Volterra integral equations. The newly proposed method efficiently find exact solution with less computation as compared with standard Laplace decomposition method (LDM). The proposed algorithm is used to solve nonlinear Volterra integral equations effectively and overcome the deficiencies in Laplace decomposition method (LDM) easily.

SOLUTION METHODS FOR INTEGRAL EQUATIONS -A SURVEY

SOLUTION METHODS FOR INTEGRAL EQUATIONS - A SURVEY, 2020

The theory of integral equations has been an active field of research for many years and is inextricably related with other areas of Mathematics such as complex and mathematical analysis, function theory, integral transforms and functional analysis. Integral Equations arise naturally in applications, in many areas of Mathematics, Engineering, Science and Technology and have been studied extensively both at the theoretical and practical level. It is significant to note that a MathSciNet keyword search on Integral Equations returns more than eleven thousand items. In this paper, we do a brief survey of the existing literature on methods of solving integral equations of Volterra and Fredholm type of the first, second and third kind, Cauchy type singular integral equations and integral equations over an infinite interval. The objective is to classify the selected methods and evaluate their applicability while discussing challenges faced by individual researchers in this field. We also provide a rather extensive bibliography for the reader who would be interested in learning more about various theoretical and computational aspects of Integral Equations.

Some Approximate Methods for Solving System of Nonlinear Integral Equations

Technology Reports of Kansai University, 2020

This paper mainly focuses on the recent advances in the semi-analytical approximated methods for solving a system of Volterra and Fredholm integral equations of the second kind by using Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM), Variational Iteration Method (VIM) and Homotopy Perturbation Method (HPM). To illustrate the methods, some examples are presented.