Numerical Solution of Volterra Linear Integral Equation of the Third Kind

2016 ◽  
pp. 111-119 ◽  
Author(s):  
Taalaybek Karakeev ◽  
Dinara Rustamova ◽  
Zhumgalbubu Bugubayeva
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Linear Integral Equation ◽  
The Third ◽  
Linear Integral

scholarly journals Numerical Solution of Integral Equation by using New Modified Adomian Decomposition Method and Newton Raphson Methods

2021 ◽  
Vol 10 (8) ◽  
pp. 5-11
Author(s):  
Najmuddin Ahmad ◽  
Balmukund Singh
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Expansion Method ◽  
Linear Integral Equation ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Newton Raphson ◽  
Linear Integral

In this paper, we discuss the numerical solution of Adomian decomposition method and Taylor’s expansion method in Volterra linear integral equation. And we apply modified Adomian decomposition method and Newton Raphson method in Volterra nonlinear integral equation with the help of example and estimated an error in MATLAB 13 versions.

scholarly journals On the numerical solution of linear integral equation

1922 ◽  
Vol 100 (705) ◽  
pp. 441-449 ◽  
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Infinite Series ◽  
Direct Methods ◽  
Neumann Series ◽  
Linear Integral Equation ◽  
Approximate Methods ◽  
Linear Integral ◽  
Great Complexity ◽  
Linear Integral Equations

The numerical solution of linear integral equations of the types studied by Volterra has formed the subject of a recent memoir by E. T. Whittaker. Numerical methods are needed also for the solution of the linear integral equation studied by Fredholm and Hilbert and the object of this paper is to describe a method which may sometimes be useful. The linear integral equation of the second kind, f(s) = Ф(s) - λ ∫ 1 0 k(s, t) Ф(t)dt , may be solved for the unknown function Ф(t) teither directly or indirectly. In the direct methods of solution the function Ф(t) is expressed by means of infinite series of terms involving repeated integrals. In the so-called method of Neumann only one infinite series is used, while in the more complete method of Fredholm the function Ф(t) is expressed as the ratio of two infinite series which converge for all values of the parameter λ. The method of Neumann has been employed on many occasions to obtain numerical results and is generally more convenient to use than Fredholm’s method on account of the great complexity of the expressions occurring in Fredholm’s series. There are occasions, however, when the Neumann series fails to converge, or converges only slowly, and then some other method such as Fredholm’s must be used. It seems desirable if possible to devise simple approximate methods which possess some of the advantages of Fredholm’s method, because in many cases the evaluation of repeated integrals becomes very tedious and the use of the direct methods of solution becomes impracticable except for a rough approxi­mation.

Solution of a Linear Integral Equation of Third Kind

2002 ◽  
Vol 9 (1) ◽  
pp. 179-196
Author(s):  
D. Shulaia
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Fredholm Theory ◽  
Linear Integral Equation ◽  
Hölder Functions ◽  
Linear Integral ◽  
Necessary And Sufficient

Abstract The aim of this paper is to study, in the class of Hölder functions, a nonhomogeneous linear integral equation with coefficient cos 𝑥. Necessary and sufficient conditions for the solvability of this equation are given under some assumptions on its kernel. The solution is constructed analytically, using the Fredholm theory and the theory of singular integral equations.

scholarly journals Symmetrisable functions and their expansion in terms of biorthogonal functions

1920 ◽  
Vol 97 (686) ◽  
pp. 401-413 ◽  
Keyword(s):  
Integral Equation ◽  
Symmetric Function ◽  
Biorthogonal System ◽  
Linear Integral Equation ◽  
Positive Type ◽  
Linear Integral ◽  
The Right

The purpose of this communication is to announce certain results relative to the expansion of a symmetrisable function k ( s , t ) in terms of a complete biorthogonal system of fundamental functions, which belong to k ( s , t ) regarded as the kernel of a linear integral equation. An indication of the method by which the results have been obtained is given, but no attempt is made to supply detailed proofs. Preliminary Explanations . 1. Let k ( s , t ) be a function defined in the square a ≤ s ≤ b , a ≤ t ≤ b . If a function ϒ ( s , t ) can be found which is of positive type in the square a ≤ s ≤ b , a ≤ t ≤ b and such that ∫ a b ϒ ( s , x ) k ( x , t ) dx is a symmetric function of s and t , k ( s , t ) is said to be symmetrisable on the left by ϒ ( s , t ) is the square. Similarly, if a function ϒ' ( s, t ) of positive type can be found such that ∫ a b k ( s , x ) ϒ' ( x , t ) dx is a symmetric function of s and t , k ( s , t ) is said to be symmetrisable on the right by ϒ' ( s , t ).

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