scholarly journals A solution to a linear integral equation with an application to statistics of infinitely divisible moving averages

2021 ◽  
Author(s):  
Jochen Glück ◽  
Stefan Roth ◽  
Evgeny Spodarev
Keyword(s):  
Integral Equation ◽  
Linear Integral Equation ◽  
Moving Averages ◽  
Infinitely Divisible ◽  
Linear Integral

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.

Sluice-gate problems with gravity

1977 ◽  
Vol 81 (1) ◽  
pp. 157-175 ◽  
Author(s):  
P. J. Budden ◽  
J. Norbury
Keyword(s):  
Integral Equation ◽  
Existence Of Solutions ◽  
Fluid Flows ◽  
Linear Integral Equation ◽  
Constructive Proof ◽  
Sluice Gate ◽  
Plane Flow ◽  
Steady Plane ◽  
Linear Integral ◽  
A Free Boundary Problem

AbstractIn this paper, a free-boundary problem of the steady plane flow of an ideal fluid through a slot is solved. The fluid flows under the effect of gravity between rigid boundaries, and then out of a slot as a jet which becomes horizontal at infinity downstream. A constructive proof of the existence of solutions to a non-linear integral equation is given for a parameter range 0 < ε < ε* ≃ 1·02 (where ε, the inverse Froude number, measures the effect of gravity). Approximations to the solution are then found for ε → 0.

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