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1

Solvability of a quadratic integral equation of Fredholm type via a modified argument

Dal İlyas, Temizer Ӧmer Faruk

Journal of Mathematics and Applications

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2020

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Vol. 43

47--66

EN

This article concerns with the existence of solutions of thea quadratic integral equation of Fredholm type with a modified argument, [wzór], where p, k are functions and F is an operator satisfying the given conditions. Using the properties of the Hölder spaces and the classical Schauder fixed point theorem, we obtain the existence of solutions of the equation under certain assumptions. Also, we present two concrete examples in which our result can be applied.

2

A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions

Repin O. A., Kumykova S. K.

Journal of Applied Analysis

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2016

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Vol. 22, nr 1

27--36

EN

The paper is devoted to the study of a boundary-value problem for an equation of mixed type with generalized operators of fractional differentiation in boundary conditions. We prove uniqueness of solutions under some restrictions on the known functions and on the different orders of the operators of generalized fractional differentiation appearing in the boundary conditions. Existence of solutions is proved by reduction to a Fredholm equation of the second kind, for which solvability follows from the uniqueness of the solution of our original problem.

3

Dimensional contact problem for the prestressed cylindrical punch and layer lying without friction on a rigid foundation

Iaretska N.

Problemy Nauk Stosowanych

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2014

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T. 2

101--108

EN

Introduction and aim: The article deals with the problem of the elastic cylindrical die pressure on the layer with initial stresses within the framework of linearized elasticity theory. In general, the research was carried out for the theory of great initial deformations and two variants of the theory of small initial deformations with arbitrary structure of elastic potential. Material and methods: The mode of deformation in the elastic layer with initial stresses is defined with the help of harmonic functions by way of Henkel integrals. It reduces the task to Fredholm equations and the method of consecutive approximations. Results: We obtained a correlation between the components of potential vector and tensor of deformations in the case of equal roots. The solutions are defined by way of lines with the help of infinitive system of constants, derivated from the regular and linear algebraic system. Conclusion: The research investigates the influence of initial stresses on the law of distribution of contact stresses in the layer and punch with initial stresses.

PL

Wstęp i cel: W ramach liniowej teorii sprężystości, należy rozważyć zadania o ciśnieniu sprężystego cylindrycznego stempla na warstwę z początkowym napięciem. Badanie wykonane w sposób ogólny do teorii znacznych odkształceń i dwóch wariantów teorii początkowych małych odkształceń przy dowolnej strukturze sprężystego potencjału. Materiał i metody: Naprężenie-odkształcenie stan w elastyczne warstwie z początkowym napięciem określamy poprzez harmoniczne funkcje w postaci całki Henkelego, które pozwalają sprowadzić zadanie do całkowania równań typu Fredgolma i metody kolejnych przybliżeń. Wyniki: Otrzymano korelację pomiędzy współrzędnymi wektora przemieszczeń i tensora naprężeń w przypadku równych stopni. Rozwiązanie przedstawione w postaci szeregu przez nieskończony system stałych. Te stałe są zdefiniowane z układem regularnych równań liniowych. Wniosek: Zbadana kwestię wpływu naprężeń początkowych na prawo dystrybucji kontaktowych w warstwie i stempli z naprężeniami początkowymi.

4

Metody badań niejednorodności energetycznej powierzchni katalizatorów i adsorbentów

Woszczyński P.

Wiadomości Chemiczne

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2013

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[Z] 67, 7-8

635--664

EN

The knowledge of the properties and a surface structure of catalysts and adsorbents is of great importance in the selection of these materials to the relevant objectives. Interesting structural information can be obtained in many ways, for example: with the use of spectroscopic or microscopic techniques or in direct examination of the adsorption isotherms. This article focuses on these last-mentioned methods, which can be a source of information on energy heterogeneity of the catalyst or adsorbent surface. Heterogeneity is usually determined by measuring adsorption isotherms of a selected adsorbate on the examined adsorbent, which is dependent of adsorbate coverage on the adsorbent relative to the equilibrium pressure under isothermal conditions. Among the many mathematical models describing this relationship particularly interesting is the adsorption isotherm model described by generalized integral Fredholm equation. The solution of this equation is density function with the assumed local isotherm model. There are different ways to solve the Fredholm equation, depending on measurement methods of obtained adsorption isotherms. For example, an application of static techniques (gravimetric or volumetric) needs to use advanced, sophisticated numerical methods for directly solving integral equations, other techniques (e.g. such as calorimetric or chromatographic) provide specific values that simplify these calculations. The resulting energy density function allows to observe active centers as peaks or inflections of the curve on the energy spectrum graph.

5

Exact Solution of a Dielectric Crack of Mode III in Magneto-Electro-Elastic Half-Space

Rogowski B.

Acta Mechanica et Automatica

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2011

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Vol. 5, no. 4

86-94

EN

This paper investigated the fracture behaviour of a piezo-electro-magneto-elastic material subjected to electro-magneto-mechanical loads. The PEMO-elastic medium contains a straight-line crack which is parallel to its poling direction and loaded surface of the half-space. Fourier transform technique is used to reduce the problem to the solution of one Fredholm integral equation. This equation is solved exactly. The semi-permeable crack-face magneto-electric boundary conditions are utilized. Field intensity factors of stress, electric displacement, magnetic induction, crack displacement, electric and magnetic potentials, and the energy release rate are determined. The electric displacement and magnetic induction of crack interior are discussed. Strong coupling between stress and electric and magnetic field near the crack tips has been found.

6

Direct and inverse acoustic scattering problem over a two-part impedance ground in a moving fluid by using Wiener-Hopf technique

Zaman F. D., Masood K.

Journal of Technical Physics

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2005

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Vol. 46, no 1

23-35

EN

The aim of this paper is to solve the direct and inverse problem in a moving fluid. We consider the direct and inverse scattering problem of acoustic line source by a two-part boundary of a half-space, having a small variation in propagation speed in the presence of a moving fluid. The problem reduces to the solution of two integral equations by using the Fourier transform and Green's function. One of these equations is solved exactly by the Wiener-Hopf technique while the other is reduced to a Fredholm equation of the first kind whose kernel involves the solution to the first equation. The procedure can be applied to recover the variation in the wave speed over a nonhomogeneous impedance ground.