Exact solution of the equation for the problem of diffuse reflection and transmission with a scattering function ?0(1 +x cos?) by the method of laplace transform and theory of linear singular integral equation

1978 ◽  
Vol 58 (2) ◽  
pp. 409-417 ◽  
Author(s):  
Rabindra Nath Das
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Singular Integral Equation ◽  
Laplace Transform ◽  
Singular Integral ◽  
Diffuse Reflection ◽  
Scattering Function ◽  
Reflection And Transmission

SH-Waves in a Medium Containing a Disordered Periodic Array of Cracks

1995 ◽  
Vol 62 (2) ◽  
pp. 312-319 ◽  
Author(s):  
Y. Mikata
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Numerical Results ◽  
Crack Spacing ◽  
Periodic Array ◽  
Wave Number ◽  
Sh Wave ◽  
Reflection And Transmission ◽  
Dispersion And Attenuation

Reflection and transmission of an SH-wave by a disordered periodic array of coplanar cracks is investigated, and subsequently its application to the dispersion and attenuation of an SH-wave in a disorderedly cracked medium is also treated. This is a stochastic boundary value problem. The formulation largely follows Mikata and Achenbach (1988b). The problem is formulated for an averaged scattered field, and the governing singular integral equation is derived for a conditionally averaged crack-opening displacement using a quasi-crystalline-like approximation. Unlike our previous study (Mikata and Achenbach, 1988b) where a point scatterer approximation was used for the regular part of the integral kernel, however, no further approximation is introduced. The singular integral equation is solved by an eigenfunction expansion involving Chebyschev polynomials. Numerical results are presented for the averaged reflection and transmission coefficients of zeroth order as a function of the wave number for normal incidence, a completely disordered crack spacing, and various values of the ratio of crack length and average crack spacing. Numerical results are also presented for the dispersion and attenuation of an SH-wave in a disorderedly cracked medium.

scholarly journals Application of Jacobi Polynomials to the Approximate Solution of a Singular Integral Equation with Cauchy Kernel on the Real Half-line

2006 ◽  
Vol 6 (3) ◽  
pp. 326-335
Author(s):  
D. Pylak
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Jacobi Polynomials ◽  
Approximate Solutions ◽  
Half Line ◽  
Cauchy Kernel ◽  
The Real

AbstractIn this paper, exact solution of the characteristic equation with Cauchy kernel on the real half-line is presented. Next, Jacobi polynomials are used to derive approximate solutions of this equation. Moreover, estimations of errors of the approximated solutions are presented and proved.

scholarly journals A Galerkin-Petrov method for singular integral equations

1983 ◽  
Vol 25 (2) ◽  
pp. 261-275 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Convergence Theorem ◽  
Singular Integral Equations ◽  
Approximate Solutions ◽  
Principal Result ◽  
Cauchy Kernel

AbstractA Galerkin-Petrov method for the approximate solution of the complete singular integral equation with Cauchy kernel, based upon the use of two sets of orthogonal polynomials, is considered. The principal result of this paper proves convergence of the approximate solutions to the exact solution making use of a convergence theorem previously given by the author. In conclusion, some related topics such as a first iterate of the approximate solution and a discretized Galerkin-Petrov method are considered. The paper extends to a much more general equation many results obtained by other authors in particular cases.

Exact Two-Dimensional Contact Analysis of Piezomagnetic Materials Indented by a Rigid Sliding Punch

2012 ◽  
Vol 79 (4) ◽  
Author(s):  
Yue Ting Zhou ◽  
Kang Yong Lee
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Mixed Boundary ◽  
Piezoelectric Materials ◽  
Fundamental Solutions ◽  
Two Dimensional ◽  
Moving Contact ◽  
Cylindrical Punch

The aim of the present paper is to investigate the two-dimensional moving contact behavior of piezomagnetic materials under the action of a sliding rigid punch. Introduction of the Galilean transformation makes the constitutive equations containing the inertial terms tractable. Eigenvalues analyses of the piezomagnetic governing equations are detailed, which are more complex than those of the commercially available piezoelectric materials. Four eigenvalue distribution cases occur in the practical computation. For each case, real fundamental solutions are derived. The original mixed boundary value problem with either a flat or a cylindrical punch foundation is reduced to a singular integral equation. Exact solution to the singular integral equation is obtained. Especially, explicit form of the stresses and magnetic inductions are given. Figures are plotted both to show the correctness of the derivation of the exact solution and to reveal the effects of various parameters on the stress and magnetic induction.

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

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