Noether Theorem of a Kind of Singular Integral Equation with Hilbert Kernel on Closed Contours

2013 ◽  
Vol 347-350 ◽  
pp. 2596-2599
Author(s):  
Li Xia Cao
Keyword(s):  
Integral Equation ◽  
General Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Noether Theorem ◽  
Hilbert Kernel ◽  
Plemelj Formula ◽  
Complex Functions

We considered a kind of singular integral equation with Hilbert kernel on closed contours. By using the method of complex functions, we obtain the extended Plemelj Formula with Hilbert kernel, and based on this, we obtained the related conditions of solvability and the general solution for the characteristic singular integral equation with Hilbert kernel on closed contours.

Noether Theorem of a Kind of Singular Integral Equation with Hilbert Kernel on Open Arcs

2013 ◽  
Vol 765-767 ◽  
pp. 695-698
Author(s):  
Li Xia Cao
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Noether Theorem ◽  
Hilbert Kernel ◽  
General Solutions ◽  
Plemelj Formula ◽  
Complex Functions

We discussed a kind of singular integral equation with Hilbert kernel on open arcs lying in a period strip. By using the method of complex functions, we obtained the extended Plemelj Formula with Hilbert kernel, and based on this, we obtained the general solutions and the solvable conditions for this kind of characteristic singular integral equation with Hilbert kernel on open arcs.

Regularization Method for Complete Singular Integral Equation with Hilbert Kernel on Open Arcs

2013 ◽  
Vol 765-767 ◽  
pp. 643-646
Author(s):  
Li Xia Cao
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Regularization Method ◽  
Singular Integral Equations ◽  
Noether Theorem ◽  
Hilbert Kernel

We considered the regularization method for a kind of complete singular integral equation with Hilbert kernel on open arcs lying in a period strip. And based on this, we obtained the solvable Noether theorem for this kind of complete singular integral equations.

scholarly journals Investigation of the semi-strip’s stress state in the case of steady-state oscillations

2019 ◽  
pp. 54-57
Author(s):  
N. D. Vaysfeld ◽  
Z. Yu. Zhuravlova ◽  
O. P. Moyseenok ◽  
V. V. Reut
Keyword(s):  
Integral Equation ◽  
Steady State ◽  
General Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Partial Solution ◽  
Initial Problem ◽  
Dimensional Problem ◽  
One Dimensional ◽  
The One

The elastic semi-strip under the dynamic load concentrated at the centre of the semi-strip’s short edge is considered. The lateral sides of the semi-strip are fixed. The case of steady-state oscillations is considered. The initial problem is reduced to the one-dimensional problem with the help of the semi-infinite sin-, cos-Fourier’s transform. The one-dimensional problem is formulated in the vector form. Its solution is constructed as a superposition of the general solution for the homogeneous equation and the partial solution for the inhomogeneous equation. The general solution for the homogeneous vector equation is found with the help of the matrix differential calculations. The partial solution is expressed through Green’s matrixfunction, which is constructed as the bilinear expansion. The inverse Fourier’s transform is applied to the derived expressions for the displacements. The solving of the initial problem is reduced to the solving of the singular integral equation. Its solution is searched as the series of the orthogonal Chebyshev polynomials of the second kind. The orthogonalization method is used for the solving of the singular integral equation. The stress-deformable state of the semi-strip is investigated regarding both the frequency of the applied load, and the load segment’s length.

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