scholarly journals Solution of a dual integral equation arising in the contact problems of elasticity theory with the full Fourier series as the right-hand side

2017 ◽  
Vol 132 ◽  
pp. 03011
Author(s):  
Andrey Vasiliev ◽  
Sergei Volkov
Keyword(s):  
Integral Equation ◽  
Fourier Series ◽  
Elasticity Theory ◽  
Contact Problems ◽  
Dual Integral Equation ◽  
Right Hand ◽  
The Right

scholarly journals Research paper on a 3d-hypersingular equation of a problem for a crack

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Stefan Samko
Keyword(s):  
Integral Equation ◽  
Elasticity Theory ◽  
Research Paper ◽  
Singular Kernel ◽  
Hypersingular Integral Equation ◽  
Hypersingular Integral ◽  
The Third ◽  
Right Hand ◽  
The Right

AbstractWe show that a certain axisymmetric hypersingular integral equation arising in problems of cracks in the elasticity theory may be explicitly solved in the case where the crack occupies a plane circle. We give three different forms of the resolving formula. Two of them involve regular kernels, while the third one involves a singular kernel, but requires less regularity assumptions on the the right-hand side of the equation.

On Fourier Series in Eigenfunctions of Elliptic Boundary Value Problems

2003 ◽  
Vol 10 (3) ◽  
pp. 401-410
Author(s):  
M. S. Agranovich ◽  
B. A. Amosov
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Fourier Coefficients ◽  
Elliptic Boundary ◽  
A Priori ◽  
Adjoint Problem ◽  
Elliptic Boundary Value ◽  
Right Hand ◽  
The Difference ◽  
The Right

Abstract We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿2(Ω) corresponding to this problem has an orthonormal basis {𝑢𝑙} of eigenfunctions, which are infinitely smooth in . However, the system {𝑢𝑙} is not a basis in Sobolev spaces 𝐻𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻𝑡 (Ω) one can explicitly construct a function 𝑢0 ∈ 𝐻𝑡 (Ω) such that the Fourier series of the difference 𝑢 – 𝑢0 in the functions 𝑢𝑙 converges to this difference in 𝐻𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 – 𝑢0. The function 𝑢0 is obtained by applying some linear operator to these right-hand sides.

XLI.—On Sommerfeld's Form of Fourier's Repeated Integrals

1912 ◽  
Vol 31 ◽  
pp. 587-603
Author(s):  
W. H. Young
Keyword(s):  
Fourier Series ◽  
Applied Mathematics ◽  
Lebesgue Integral ◽  
General Hypothesis ◽  
Differential Coefficient ◽  
Right Hand ◽  
Fourier Sine ◽  
Repeated Integral ◽  
The Right

§ 1. In his treatise on Fourier Series and Integrals Carslaw quotes without proof Sommerfeld's theorem thatwhen the limit on the right-hand side exists. In applied mathematics, he remarks, it is this limit, rather than the corresponding Fourier repeated integral which occurs.In the present paper I propose to extend this result in various ways. After proving Sommerfeld's result on the general hypothesis, not considered by him, that the integral is a Lebesgue integral, I show that the limit in question is whenever the origin is a point at which f(u) is the differential coefficient of its integral, and I obtain the corresponding results for In all their generality these statements are only true when the interval (0, p) is a finite one. I then show how, under a variety of hypotheses with respect to the nature of f(x) at infinity, they can be extended so as to be still true when p = + ∞ . These hypotheses correspond precisely to those which have been proved f to be sufficient for the corresponding statements as to the Fourier sine and cosine repeated integrals in their usual forms.

scholarly journals On a dual integral equation with a trigonometric kernel

1977 ◽  
Vol 18 (2) ◽  
pp. 175-177 ◽  
Author(s):  
D. C. Stocks
Keyword(s):  
Integral Equation ◽  
Elasticity Theory ◽  
Integral Equations ◽  
Crack Problem ◽  
Dual Integral Equation ◽  
Dual Integral Equations ◽  
Cosine Transform ◽  
Fourier Cosine ◽  
Fourier Cosine Transform ◽  
Image Position

In this note we formally solve the following dual integral equations:where h is a constant and the Fourier cosine transform of u–1 φ(u) is assumed to exist. These dual equations arise in a crack problem in elasticity theory.

On Some Unbonded Contact Problems in Plane Elasticity Theory

1976 ◽  
Vol 43 (2) ◽  
pp. 263-267 ◽  
Author(s):  
G. M. L. Gladwell
Keyword(s):  
Integral Equation ◽  
Elasticity Theory ◽  
Contact Pressure ◽  
Chebyshev Polynomials ◽  
Contact Region ◽  
Contact Problems ◽  
Elastic Cylinder ◽  
Plane Elasticity ◽  
Elastic Strip ◽  
Plane Elasticity Theory

Paper considers plane, frictionless, unbonded contact problems. It is shown that the integral equation relating the unknown contact pressure to the specified displacement in the contact region may be solved approximately by using an expansion in terms of Chebyshev polynomials. Three examples are chosen, a beam resting on a half plane, a rigid cylinder pressed into an elastic strip, and an elastic cylinder pressed between rigid planes. Graphs of results are presented.

Approximate Solution of a Hypersingular Integral Equation of the First Kind Bounded at Both Ends of the Integration Segment Using Chebyshev Series

2021 ◽  
pp. 33-38
Author(s):  
Shalva S. Khubezhty
Keyword(s):  
Integral Equation ◽  
Unknown Function ◽  
Quadrature Formulas ◽  
Hypersingular Integral Equation ◽  
Calculation Error ◽  
Hypersingular Integral ◽  
Right Hand ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
The Right

A hypersingular integral equation on the interval of integration is considered. The hypersingular integral is understood in the sense of Hadamard, that is, in the finite part. The class of such equations is widely used in problems of mathematical physics, in technology, and most importantly: in recent years, they are one of the main devices for modeling problems in electrodynamics. With the use of Chebyshev polynomials of the second kind, the unknown function, the right-hand side and the kernel are replaced in the equation. The expansion coefficients of these functions are calculated using quadrature formulas of the highest algebraic degree of accuracy, i.e., Gauss quadrature formulas. Thus, the equation is discretized. The result is an infinite system of linear algebraic equations for the expansion coefficients of the unknown function. The fact that the hypersingular integral equation in the case under consideration has a unique solution in the class of sufficiently smooth functions is taken into account. The constructed computational scheme is substantiated using the general theory of functional analysis. The calculation error is estimated under certain conditions relative to the right-hand side and the kernel of the equation. The described method for solving the hypersingular integral equation is illustrated by test examples that show the high efficiency of the method.

Editorial Note

1946 ◽  
Vol 11 (1) ◽  
pp. 2-2
Keyword(s):  
Editorial Note ◽  
Speech Sounds ◽  
Left Hand ◽  
Hand Column ◽  
Right Hand ◽  
Infant Speech ◽  
The Right

In the article “Infant Speech Sounds and Intelligence” by Orvis C. Irwin and Han Piao Chen, in the December 1945 issue of the Journal, the paragraph which begins at the bottom of the left hand column on page 295 should have been placed immediately below the first paragraph at the top of the right hand column on page 296. To the authors we express our sincere apologies.

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