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1

Solving linear Fredholm integro-differential equation by Nyström method

Tair Boutheina, Guebbai Hamza, Segni Sami, Ghiat Mourad

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 3

53--64

EN

The study of the solution’s existence and uniqueness for the linear integro-differential Fredholm equation and the application of the Nyström method to approximate the solution is what we will present in this paper. We use the Neumann theorem to construct a sufficient condition that ensures the solution’s existence and uniqueness of our problem in the Banach space C1 [a,b]. We have applied the Nyström method based on the trapezoidal rule to avoid adding other conditions in order to the approximation method’s convergence. The Nyström method discretizes the integro-differential equation into solving a linear system. Only with the existence and uniqueness condition, we show the solution’s existence and uniqueness of the linear system and the convergence of the numerical solution to the exact solution in infinite norm sense. We present two theorems to give a good estimate of the error. Also, to show the efficiency and accuracy of the Nyström method, some numerical examples will be provided at the end of this work.

2

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.

3

On approximate conformal mapping of a disk and an annulus with radial and circular slits onto multiply connected domains

Ivanshin Pyotr N., Shirokova Elena A.

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 2

73--84

EN

The method of boundary curve reparametrization is generalized to the case of multiply connected domains. We construct the approximate analytical conformal mapping of the unit disk with m circular slits and n-m radial slits and an annulus with (m-1) circular slits and n-m radial slits onto an arbitrary given (n+1) multiply connected finite domain with a smooth boundary. The method is based on extension of the Lichtenstein-Gershgorin equation to a multiply connected domain. The proposed method is reduced to the solution of a linear system with unknown Fourier coefficients. The approximate mapping function has the form of a Cauchy integral. Numerical examples demonstrate that the proposed method is effective in computations.

4

Frictionless contact between a rigid indentor and a transversely isotropic functionally graded layer

Patra R., Barik S. P., Chaudhuri P. K.

International Journal of Applied Mechanics and Engineering

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2018

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Vol. 23, no. 3

655--671

EN

This article is concerned with the study of frictionless contact between a rigid punch and a transversely isotropic functionally graded layer. The rigid punch is assumed to be axially symmetric and is supposed to be pressing the layer by an applied concentrated load. The layer is resting on a rigid base and is assumed to be sufficiently thick in comparison with the amount of indentation by the rigid punch. The graded layer is modeled as a non-homogeneous medium. The relationship between the applied load P and the contact area is obtained by solving the mathematically formulated problem through using the Hankel transform of different order. Numerical results have been presented to assess the effects of functional grading of the medium and the applied load on the stress distribution in the layer as well as on the relationship between the applied load and the area of contact.

5

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.

6

Transformation of the second order boundary value problem into integral form : different approaches and a numerical solution

Siedlecki J., Ciesielski M., Błaszczyk T.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 3

103--108

EN

In this paper we present different approaches to the transformation of the second order ordinary differential equation, with respect to adequate boundary conditions, into integral equations. The obtained equations are Fredholm integral equations of the second kind. Next, a numerical method based on quadrature methods has been proposed to get an approximate solution of these equations.

7

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

8

Analysis methods for interval Fredholm integral equations with small or degenerate kernels

Skrzypczyk J.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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1999

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z. 10

267-272

EN

In the paper basic concepts of the structure of the solutions to the interval Fredholm integral equations are considered, where a free terra is taken to be an interval square-integrable function and non-interval kernel square-integrable in [a,b]x[a,b] and degenerate or small in some sense. At first, the existence of the exact set-valued solution is investigated. In addition the hull of the solution set is obtained. For including a set of solutions of the interval integral equation we apply interval calculus. At the end the theory is illustrated by a simple analytical example.

PL

W pracy badane jest istnienie i struktura zbioru rozwiązań przedziałowego równania całkowego Fredholma II rodzaju z niejednorodnością, która jest funkcją przedziałową, całkowalną z kwadratem, natomiast jądro równania całkowego jest jądrem całkowalnym z kwadratem na zbiorze [a,b]x[a,b]. W pierwszej kolejności badane jest zagadnienie dokładnego zbioru rozwiązań, a następnie problem wyznaczenia najmniejszego zbioru przedziałowego zawierającego dokładny zbiór rozwiązań. Dla wyznaczenia tej aproksymacji zastosowano analizę przedziałową. Teoria zilustrowana jest prostym przykładem analitycznym.

9

On the bi-scalar approach to boundary-integral handling of the electromagnetic field problems

Kucharska M., Pawluk K.

Journal of Technical Physics

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1998

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Vol. 39, no 3-4

557-568

EN

Bi-scalar boundary-integral approach to the static and monoharmonic electromagnetic field is given. Scalar and vector virtual boundary quanties used in the field models are defined. The analysis of the magnetostatic field of a permanent magnet is presented.