Quadrature formulas and the Poincare-Bertrand formula for singular integrals

1981 ◽  
Vol 21 (6) ◽  
pp. 787-797
Author(s):  
I. K. Lifanov
Keyword(s):  
Singular Integrals ◽  
Quadrature Formulas

scholarly journals ON SOLUTION OF AMPLITUDE-PHASE PROBLEM

2018 ◽  
pp. 64-68 ◽  
Author(s):  
I. V. Boykov ◽  
Ya V. Zelina
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Singular Integrals ◽  
Singular Integral Equations ◽  
Quadrature Formulas ◽  
Two Dimensional ◽  
Phase Problem ◽  
The Fourier Transform ◽  
The Hilbert Transform ◽  
Unconventional Method

The paper describes an unconventional method of solving the amplitude-phase problem. The main properties of the Hilbert transform in the discrete and continual cases for one-dimensional and two-dimensional mappings are considered. These transformations are widely used to solve amplitude-phase problem. A numerical method for solving of two-dimensional amplitudephase problem is proposed. Preliminary information about the zeros of the Fourier transform of the initial signal is not required for this method. The method is based on the apparatus of nonlinear singular integral equations. Computational schemes for solving the corresponding nonlinear singular integral equations are developed. An algorithm for finding initial values for realization of iterative methods is proposed. Quadrature formulas of the calculation of singular integrals are proposed.

scholarly journals The stress state in an elastic body with a rigid inclusion of the shape of three segments broken line under the action of the harmonic oscillation of the longitudinal shift

2019 ◽  
pp. 158-161
Author(s):  
V. G. Popov ◽  
O. V. Lytvyn
Keyword(s):  
Helmholtz Equation ◽  
Rigid Inclusion ◽  
Singular Integrals ◽  
Displacement Vector ◽  
Harmonic Oscillation ◽  
Quadrature Formulas ◽  
Antiplane Strain ◽  
Discontinuous Solutions ◽  
The Matrix ◽  
Harmonic Shear

There is a thin absolutely rigid inclusion that in a cross-section represents three segments broken line in an infinite elastic medium (matrix) that is in the conditions of antiplane strain. The inclusion is under the action of harmonic shear force Pe^{iwt} along the axis Oz. Under the conditions of the antiplane strain the only one different from 0 z-component of displacement vector W (x; y) satisfies the Helmholtz equation. The inclusion is fully couple with the matrix. The tangential stresses are discontinuous on the inclusion with unknown jumps. The method of the solution is based on the representation of displacement W (x; y) by discontinuous solutions of the Helmholtz equation. After the satisfaction of the conditions on the inclusion the system of integral equations relatively unknown jumps is obtained. One of the main results is a numerical method for solving the obtained system, which takes into account the singularity of the solution and is based on the use of the special quadrature formulas for singular integrals.

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