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.

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.

Eosinophilic vasculitis: an inhabitual and resistant manifestation of a vasculitis

VASA ◽  
2010 ◽  
Vol 39 (4) ◽  
pp. 344-348 ◽  
Author(s):  
Jandus ◽  
Bianda ◽  
Alerci ◽  
Gallino ◽  
Marone
Keyword(s):  
Skin Biopsy ◽  
Intravenous Iron ◽  
Small Vessel Vasculitis ◽  
High Dose ◽  
Raynaud Phenomenon ◽  
The Third ◽  
Left Elbow ◽  
Right Hand ◽  
The Right ◽  
Trunk Skin

A 55-year-old woman was referred because of diffuse pruritic erythematous lesions and an ischemic process of the third finger of her right hand. She was known to have anaemia secondary to hypermenorrhea. She presented six months before admission with a cutaneous infiltration on the left cubital cavity after a paravenous leakage of intravenous iron substitution. She then reported a progressive pruritic erythematous swelling of her left arm and lower extremities and trunk. Skin biopsy of a lesion on the right leg revealed a fibrillar, small-vessel vasculitis containing many eosinophils.Two months later she reported Raynaud symptoms in both hands, with a persistent violaceous coloration of the skin and cold sensation of her third digit of the right hand. A round 1.5 cm well-delimited swelling on the medial site of the left elbow was noted. The third digit of her right hand was cold and of violet colour. Eosinophilia (19 % of total leucocytes) was present. Doppler-duplex arterial examination of the upper extremities showed an occlusion of the cubital artery down to the palmar arcade on the right arm. Selective angiography of the right subclavian and brachial arteries showed diffuse alteration of the blood flow in the cubital artery and hand, with fine collateral circulation in the carpal region. Neither secondary causes of hypereosinophilia nor a myeloproliferative process was found. Considering the skin biopsy results and having excluded other causes of eosinophilia, we assumed the diagnosis of an eosinophilic vasculitis. Treatment with tacrolimus and high dose steroids was started, the latter tapered within 12 months and then stopped, but a dramatic flare-up of the vasculitis with Raynaud phenomenon occurred. A new immunosupressive approach with steroids and methotrexate was then introduced. This case of aggressive eosinophilic vasculitis is difficult to classify into the usual forms of vasculitis and constitutes a therapeutic challenge given the resistance to current immunosuppressive regimens.

scholarly journals Computation of Energy Release Rates for a Nearly Circular Crack

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Nik Mohd Asri Nik Long ◽  
Lee Feng Koo ◽  
Zainidin K. Eshkuvatov
Keyword(s):  
Integral Equation ◽  
Energy Release Rate ◽  
Energy Release ◽  
Release Rate ◽  
Linear Equations ◽  
Circular Crack ◽  
Plane Elasticity ◽  
Hypersingular Integral Equation ◽  
Hypersingular Integral ◽  
Energy Release Rates

This paper deals with a nearly circular crack, in the plane elasticity. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over a considered domain, and it is then transformed into a similar equation over a circular region, , using conformal mapping. Appropriate collocation points are chosen on the region to reduce the hypersingular integral equation into a system of linear equations with unknown coefficients, which will later be used in the determination of energy release rate. Numerical results for energy release rate are compared with the existing asymptotic solution and are displayed graphically.

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