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1

Trigonometric Hermite interpolation method for Fredholm linear integral equations

Ajeddar Mohamed, Lamnii Abdellah

Journal of Applied Analysis

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2023

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Vol. 29, nr 2

261--275

EN

This paper presents a new trigonometric composite Hermite interpolation method for solving Fredholm linear integral equations. This operator approximates locally both the function and its derivative, which is known on the subdivision nodes. Then we derive a class of quadrature rules with endpoint corrections based on integrating the composite Hermite interpolant.We also provide error estimation and numerical examples to illustrate that this new operator can provide highly accurate results.

2

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.

3

Construction of complex potentials for multiply connected domain

Ivanshin Pyotr N.

Journal of Applied Analysis

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2021

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Vol. 27, nr 2

209--217

EN

The method of reduction of a Fredholm integral equation to the linear system is generalized to construction of a complex potential - an analytic function in an unbounded multiply connected domain with a simple pole at infinity which maps the domain onto a plane with horizontal slits. We consider a locally sourceless, locally irrotational flow on an arbitrary given n-connected unbounded domain with impermeable boundary. The complex potential has the form of a Cauchy integral with one linear and n logarithmic summands. The method is easily computable.

4

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.

5

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.

6

P-wave interaction with a pair of rigid strips embedded in an orthotropic strip

Basu S., Mandal S. C.

Journal of Theoretical and Applied Mechanics

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2016

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Vol. 54 nr 2

579--592

EN

The present paper is concerned with the problem of scattering of the P-wave by two co- -planer finite rigid strips placed symmetrically in an infinitely long orthotropic strip. Using the Hilbert transform technique, the mixed boundary value problem has been reduced to the solution of dual integral equations which has finally been reduced to the solution of a Fredholm integral equation of the second kind. Solving this integral equation numerically, stress intensity factors have been calculated at the inner and outer edges of the rigid strips, and the vertical displacement outside the strips has been calculated and plotted graphically to show the effect of material orthotropy.

7

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.

8

The use of object-oriented software for computation of current density distribution

Nawrowski R., Pabian J., Tomczewski A.

Poznan University of Technology Academic Journals. Electrical Engineering

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2006

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No 52

131-144

EN

The ODS (Operating Deflection Shapes) method used for the estimation of the influence of weak external extortions on the functioning of chosen structural elements of the chimney is presented in this paper. The chosen method permits to build a virtual model of the investigated object and then to „reproduce'' the behavior of a real structure, using the results of executed measurements. The analysis is made for selected frequencies. Thanks to this it is possible to verify the influence of chosen vibration sources or chosen frequencies on the object.

9

Fredholm integral equations in the radiative heat transfer problems

Hącia L., Domke K.

Poznan University of Technology Academic Journals. Electrical Engineering

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2006

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No 52

115-123

EN

Basic equations of radiative heat transfer have been presented, along with typical Dirichlet and Neumann boundary conditions for established stales. Possible methods of solving integral equations describing radiative heat transfer have been identified, and these have been limited to iterative and projection methods. We restrict to method of successive approximations, discretization method, Galerkin method, collocation method and method of special kernels. Moreover, quadratures and probabilistic methods frequently used in the radiosity are presented. Presented problem is illustrated by example.

10

Solution of Fredholm integral of first kind Singular Kernel

El-Bary A.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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2000

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z. 24 [181]

27-35

EN

This paper is devoted to give and discuss the method of solving the Fredholm integral equation of the first kind with singular kernel by using the Fourier method.

11

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.