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1

Optimized Stochastic Approach for Integral Equations

Todorov Venelin, Fidanova Stefka, Dimov Ivan, Georgieva Rayna

Annals of Computer Science and Information Systems

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2021

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Vol. 25

239--242

EN

An optimized Monte Carlo approach (OPTIMIZED MC) for a Fredholm integral equations of the second kind is presented and discussed in the present paper. Numerical examples and results are discussed and MC algorithms with various initial and transition probabilities are compared.

2

Axisymmetric Torsion of an Elastic Layer Sandwiched between Two Elastic Half-Spaces with Two Interfaced Cracks

Madani Fateh, Kebli Belkacem

Studia Geotechnica et Mechanica

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2019

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Vol. 41, nr 2

57--66

EN

The present article examines the problem related to the axisymmetric torsion of an elastic layer by a circular rigid disc at the symmetry plane. The layer is sandwiched between two similar elastic half-spaces with two penny-shaped cracks symmetrically located at the interfaces between the two bonded dissimilar media. The mixed boundary-value problem is transformed, by means of the Hankel integral transformation, to dual integral equations, that are reduced, to a Fredholm integral equation of the second kind. The numerical methods are used to convert the resulting system to a system of infinite algebraic equations. Some physical quantities such as the stress intensity factor and the moment are calculated and presented numerically according to some relevant parameters. The numerical results show that the discontinuities around the crack and the inclusion cause a large increase in the stresses that decay with distance from the disc-loaded. Furthermore, the dependence of the stress intensity factor on the disc size, the distance between the crack and the disc, and the shear parameter is also observerd.

3

Axisymmetric torsion of an internally cracked elastic medium by two embedded rigid discs

Madani F., Kebli B.

Mechanics and Mechanical Engineering

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2017

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Vol. 21, nr 2

363--377

EN

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material at the symmetry plane, under an axisymmetric torsion by two circular rigid discs symmetrically located in the elastic medium. The two discs rotate with the same angle in the different direction about the axis passing through their centers. The general solution of this problem is obtained by using the Hankel transforms method. The corresponding doubly mixed boundary value problem associated with the rigid disc and the penny-shaped is reduced to a system of dual integral equations, which are transformed, to a Fredholm integral equations of the second kind. Using the quadrature rule, the resulting system is converted to a system of infinite algebraic equations. The variation in the displacement, stress and stress intensity factor are presented for some particular cases of the problem.

4

Równania całkowe Fredholma w przybliżonej metodzie rozwiązywania dynamicznych zagadnień w ośrodkach lepkosprężystych

Kamiński H., Sypniewska-Kamińska G.

Vibrations in Physical Systems

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2000

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Vol. 19

152--153

5

A note on interval Fredholm integral equations

Skrzypczyk J.

Zeszyty Naukowe. Budownictwo / Politechnika Śląska

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1998

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z. 85

75-83

PL

W pracy badane jest istnienie i jednoznaczność zbioru rozwiązań przedziałowego równania całkowego Fredholma II rodzaju z niejednorodnością, która jest funkcją przedziałową, całkowalną z kwadratem, natomiast jądro równania całkowego jest całkowalne z kwadratem na zbiorze [a,b]x[a,b]. W pierwszej kolejności badane jest zagadnienie zbieżności ciągu aproksymującego do dokładnego zbioru rozwiązań, a następnie problem jednoznaczności zbioru przedziałowego, zawierającego zbiór rozwiązań. W celu wyznaczenia tej aproksymacji zastosowano analizę przedziałową. Teoria zilustrowana jest prostym przykładem analitycznym.

EN

In the paper basic concepts of the analysis of the solutions to the interval Fredholm integral equations are considered, where a free term is taken to be an interval square-integrable function and non-interval kernel is square-integrable in [a,b]x[a,b]. All arithmetic operations and integration are in interval interval sense. The existence and uniqueness of the interval integral equation we apply interval calculus. At the end the theory is illustrated by a simple analytical example.