A Class of Linear Integral Equation in Engineering

2014 ◽  
Vol 587-589 ◽  
pp. 2303-2306 ◽  
Author(s):  
Li Mian Zhao ◽  
Ji Ting Huang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Continuous Function ◽  
Explicit Solution ◽  
Piecewise Continuous Function ◽  
Linear Integral Equation ◽  
Initial Condition ◽  
Variation Formula ◽  
Piecewise Continuous ◽  
Linear Integral

In this paper, we discuss a class of linear integral equation with piecewise continuous function. Firstly, we change the integral equation to a differential equation with the initial condition. Secondly, the differential equation is solved by the constant variation formula and integration by parts. Explicit solution of the integral equation is given clearly.

A Generalization of Picard's Method of Successive Approximation

1936 ◽  
Vol 32 (1) ◽  
pp. 86-95 ◽  
Author(s):  
H. O. Hirschfeld
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Successive Approximation ◽  
Linear Integral Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Linear Integral ◽  
Image Position

It is well known that the boundary value problem for the non-linear differential equationcan be reduced with help of a Green's function K (x, ξ) to a non-linear integral equation of the type

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 ).

On a non-linear integral equation occurring in diffraction theory

1966 ◽  
Vol 62 (2) ◽  
pp. 249-261 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Radiative Transfer ◽  
Plane Wave ◽  
Admissible Solution ◽  
Diffraction Theory ◽  
Linear Integral Equation ◽  
Non Linear Equation ◽  
Non Linear ◽  
Linear Integral

AbstractThe problem of diffraction of a plane wave by a semi-infinite grating of iso-tropic scatterers leads to the consideration of a non-linear integral equation. This bears a resemblance to Chandrasekhar's integral equation which arises in the study of radiative transfer through a semi-infinite atmosphere. It is shown that methods which have been used with success to solve Chandrasekhar's equation are equally useful here. The solution to the non-linear equation satisfies a more simple functional equation which may be solved by factoring (in the Wiener-Hopf sense) a given function. Subject to certain additional conditions which are dictated by physical considerations, a solution is obtained which is the unique admissible solution of the non-linear integral equation. The factors and solution are found explicitly for the case which corresponds to closely spaced scatterers.

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