scholarly journals Airfoils moving in air close to a dynamic water surface

1986 ◽  
Vol 27 (3) ◽  
pp. 327-345 ◽  
Author(s):  
I. H. Grundy
Keyword(s):  
Integral Equation ◽  
Potential Flow ◽  
Water Surface ◽  
Linear Integral Equation ◽  
One Dimensional ◽  
Short Waves ◽  
The One ◽  
Linear Integral ◽  
Parameter Ranges ◽  
Steady Potential

AbstractSteady potential flow about a thin wing, flying in air above a dynamic water surface, is analysed in the asymptotic limit as the clearance-to-length ratio tends to zero. This leads to a non-linear integral equation for the one-dimensional pressure distribution beneath the wing, which is solved numerically. Results are compared with established “rigid-ground” and “hydrostatic” theories. Short waves lead to complications, including non-uniqueness, in some parameter ranges.

scholarly journals On non-autonomous differential-difference AKP, BKP and CKP equations

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200717
Author(s):  
Wei Fu ◽  
Frank W. Nijhoff
Keyword(s):  
Integral Equation ◽  
Difference Equations ◽  
Integrable Systems ◽  
Three Dimensional ◽  
Linear Integral Equation ◽  
Discrete Integrable Systems ◽  
Link Type ◽  
Direct Linearization ◽  
The One ◽  
Linear Integral

Based on the direct linearization framework of the discrete Kadomtsev–Petviashvili-type equations presented in the work of Fu & Nijhoff (Fu W, Nijhoff FW. 2017 Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations. Proc. R. Soc. A 473 , 20160915 ( doi:10.1098/rspa.2016.0915 )), six novel non-autonomous differential-difference equations are established, including three in the AKP class, two in the BKP class and one in the CKP class. In particular, one in the BKP class and the one in the CKP class are both in (2 + 2)-dimensional form. All the six models are integrable in the sense of having the same linear integral equation representations as those of their associated discrete Kadomtsev–Petviashvili-type equations, which guarantees the existence of soliton-type solutions and the multi-dimensional consistency of these new equations from the viewpoint of the direct linearization.

scholarly journals On sound waves of finite amplitude

1953 ◽  
Vol 216 (1127) ◽  
pp. 539-547 ◽  
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Arbitrary Function ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Sound Waves ◽  
One Dimensional ◽  
The One ◽  
Complex Integral ◽  
Gas Cloud

An analytical solution of Riemann’s equations for the one-dimensional propagation of sound waves of finite amplitude in a gas obeying the adiabatic law p = k ρ γ is obtained for any value of the parameter γ. The solution is in the form of a complex integral involving an arbitrary function which is found from the initial conditions by solving a generalization of Abel’s integral equation. The results are applied to the problem of the expansion of a gas cloud into a vacuum.

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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