Integral Equations, Origin, and Basic Tools Modeling of Problems as Integral Equations Volterra Integral Equations The Green's Function Fredholm Integral Equations Existence of the Solutions: Basic Fixed Point Theorems Higher Quadrature Rules for the Numerical Solutions Appendices Answers to Exercises References Index.

Introduction to integral equations with applications, by Abdul J. Jerri. Pp 254. $47·50. 1985. ISBN 0-8247-7293-8 (Marcel Dekker)

In this paper, we considered a mixed integral equation (MIE) of the second kind in the space 
 
 
 
 L
 
 
 2
 
 
 
 
 −
 b
 ,
 b
 
 
 ×
 C
 
 
 0
 ,
 T
 
 
 ,
 T
 <
 1.
 
 The kernel of position has a singularity and takes some different famous forms, while the kernels of time are positive and continuous. Using an asymptotic method of separating the variables, we have a Fredholm integral equation (FIE) in position with variable parameters in time. Then, using the Toeplitz matrix method (TMM), we obtain a linear algebraic system (LAS) that can be solved numerically. Some applications with the aid of the maple 18 program are discussed when the kernel takes Coleman function, Cauchy kernel, Hilbert kernel, and a generalized logarithmic function. Also the error estimate, in each case, is computed.

https://downloads.hindawi.com/journals/jmath/2022/8063971.pdf

An Asymptotic Model for Solving Mixed Integral Equation in Position and Time

Numerical solution of two-dimensional Hammerstein integral equations via quadratic spline collocation

This paper presents a collocation method for the approximate solution of two-dimensional mixed Volterra–Fredholm integral equations of the Hammerstein type. For a reformulation of the equation, we consider the domain of integration as a planar triangle and use a special type of linear interpolation on triangles. The resulting quadrature formula has a higher degree of precision than expected, leading to a collocation method that is superconvergent at the collocation nodes. The convergence of the method is established, as well as the rate of convergence. Numerical examples are considered, showing the applicability of the proposed scheme and the agreement with the theoretical results.

/pdf/a-collocation-method-for-mixed-volterra-fredholm-integral-2vyq64bq.pdf

A Collocation Method for Mixed Volterra–Fredholm Integral Equations of the Hammerstein Type

In many problems of mathematical physics, theory of elasticity, visco-dynamics fluid and mixed problems of mechanics of continuous media reduces to the integral and intro-differential equation (Volterra or Fredholm type) first or second kind. In the early XXth (around 1910) century, to solve some problems in the field of mechanics Volterra had to solve integro-differential equations (IDEs) with variable boundaries. One of the popular methods for solving Volterra IDEs is the method of quadratures when problem of consideration is a linear problem. When IDEs are nonlinear or integral kernel are complicated (singular or nonlinear) then quadrature rule is not most suitable therefore other type of methods are needed to develop. In this note, we have used the standard and new-development of homotopy analysis method (ND-HAM) for the general second order linear integral-differential equations (IDEs) of the second kind. Both methods reduce IDEs into iterative sequence of algebraic integral equations with known integrant function. Gauss-Legendre quadrature formulas are applied for kernel integrations. Obtained system of algebraic equations are solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and reliable. 

New development of HAM for approximating linear integro-differential equation of order two

This study is focused on the numerical solutions of the nonlinear Volterra-Fredholm integral equations (NV-FIEs) of the second kind, which have several applications in physical mathematics and contact problems. Herein, we develop a new technique that combines the modified Adomian decomposition method and the quadrature (trapezoidal and Weddle) rules that used when the definite integral could be extremely difficult, for approximating the solutions of the NV-FIEs of second kind with a phase lag. Foremost, Picard's method and Banach's fixed point theorem are implemented to discuss the existence and uniqueness of the solution. Furthermore, numerical examples are presented to highlight the proposed method's effectiveness, wherein the results are displayed in group of tables and figures to illustrate the applicability of the theoretical results.

Ahmed S. Rahby

Papers

Correction: Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag