ON A PROBLEM FOR AN ELASTIC INFINITE SHEET STRENGTHENED BY TWO PARALLEL STRINGERS WITH FINITE LENGTHS THROUGH ADHESIVE SHEAR LAYERS

2020 ◽  
Vol 54 (3 (253)) ◽  
pp. 153-164
Author(s):  
Aghasi V. Kerobyan
Keyword(s):  
Banach Space ◽  
Integral Equations ◽  
Shear Layers ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Fredholm Integral Equations ◽  
Shear Stresses ◽  
Characteristic Parameters ◽  
System Of Integral Equations ◽  
Adhesive Layers

The article considers the problem for an elastic infinite sheet (plate), which is strengthened on two parallel finite parts of its upper surface by two parallel finite stringers with different elastic properties. The parallel stringers are located asymmetrically with respect to the horizontal axis of the sheet and deform under the action of horizontal forces. The interaction between the infinite sheet and stringers takes place through thin elastic adhesive layers. The problem of determining unknown shear stresses acting between the infinite sheet and stringers is reduced to a system of Fredholm integral equations of second kind with respect to unknown functions, which are specified on two parallel finite intervals. It is shown that in the certain domain of the change of the characteristic parameters of the problem this system of integral equations in Banach space can be solved by the method of successive approximations. Particular cases are considered, the character and behaviour of unknown shear stresses are investigated.

TRANSFER OF LOADS FROM A FINITE NUMBER OF ELASTIC OVERLAYS WITH FINITE LENGTHS TO AN ELASTIC STRIP THROUGH ADHESIVE SHEAR LAYERS

2019 ◽  
Vol 53 (2 (249)) ◽  
pp. 109-118
Author(s):  
A.V. Kerobyan
Keyword(s):  
Finite Number ◽  
Integral Equations ◽  
Characteristic Parameter ◽  
Shear Layers ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Fredholm Integral Equations ◽  
Shear Stresses ◽  
Infinite Strip ◽  
Small Constant

This article deals with the problem of an elastic infinite strip, which is strengthened along its free boundary by a finite number of finite overlays with different elastic characteristics and small constant thicknesses. The interaction between the strip and the overlays is mediated by adhesive shear layers. The overlays are deformed under the action of horizontal forces. The problem of determination of unknown stresses acting between the strip and overlays are reduced to a system of Fredholm integral equations of the second kind for a finite number of unknown functions defined on different finite intervals. It is shown that in the certain domain of variation of the characteristic parameter of the problem this system of integral equations in Banach space may be solved by the method of successive approximations. Particular cases are discussed and the character and behaviour of unknown shear stresses are investigated.

scholarly journals METHODS FOR SOLVING FREDHOLM INTEGRAL EQUATIONS

2020 ◽  
Vol 69 (1) ◽  
pp. 174-178
Author(s):  
R.S. Ysmagul ◽  
◽  
А.Е. Nurgeldina ◽  
Keyword(s):  
Quantum Mechanics ◽  
Initial Data ◽  
Differential Equations ◽  
Integral Equations ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Fredholm Integral Equations ◽  
Future State ◽  
Continuous Sequence ◽  
The Moment

The article deals with integral equations that are widely used in various sections of physics (theory of waves on the surface of liquids, quantum mechanics, problems of spectroscopy, crystallography, acoustics, analysis and diagnostics of plasma, etc.), Geophysics (problems of gravimetry, kinematic problems of seismics), mechanics (vibrations of structures), etc. When the physics introduced aftereffect, it is not enough ordinary differential equations or partial differential equations, otherwise the initial data would determine the future state. To take into account the continuous sequence of previous States, we need to use integral and integro-differential equations, where the sign of the integral appears functions of parameters that characterize the system, which depend on time for some period preceding the moment under consideration. In this article we have considered the solution of Fredholm integral equations of the second kind by the method of successive approximations and the method of iterated nuclei.

scholarly journals Fuzzy Volterra integral equations with in-finite delay

2009 ◽  
Vol 40 (1) ◽  
pp. 19-29 ◽  
Author(s):  
P. Prakash ◽  
V. Kalaiselvi
Keyword(s):  
Integral Equations ◽  
Existence And Uniqueness ◽  
Infinite Delay ◽  
Volterra Integral Equations ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Finite Delay

In this paper, we study the existence and uniqueness of solutions for a class of fuzzy Volterra integral equations with infinite delay by using the method of successive approximations.

scholarly journals Some Bounded Linear Integral Operators and Linear Fredholm Integral Equations in the SpacesHα,δ,γ((a,b)×(a,b),X)andHα,δ((a,b),X)

2013 ◽  
Vol 2013 ◽  
pp. 1-20
Author(s):  
İsmet Özdemir ◽  
Ali M. Akhmedov ◽  
Ö. Faruk Temizer
Keyword(s):  
Banach Space ◽  
Numerical Methods ◽  
Banach Spaces ◽  
Integral Equations ◽  
Singular Integral ◽  
Integral Operators ◽  
Fredholm Integral Equations ◽  
Bounded Linear ◽  
Linear Integral ◽  
Linear Integral Operators

The spacesHα,δ,γ((a,b)×(a,b),ℝ)andHα,δ((a,b),ℝ)were defined in ((Hüseynov (1981)), pages 271–277). Some singular integral operators on Banach spaces were examined, (Dostanic (2012)), (Dunford (1988), pages 2419–2426 and (Plamenevskiy (1965)). The solutions of some singular Fredholm integral equations were given in (Babolian (2011), Okayama (2010), and Thomas (1981)) by numerical methods. In this paper, we define the setsHα,δ,γ((a,b)×(a,b),X)andHα,δ((a,b),X)by taking an arbitrary Banach spaceXinstead ofℝ, and we show that these sets which are different from the spaces given in (Dunford (1988)) and (Plamenevskiy (1965)) are Banach spaces with the norms∥·∥α,δ,γand∥·∥α,δ. Besides, the bounded linear integral operators on the spacesHα,δ,γ((a,b)×(a,b),X)andHα,δ((a,b),X), some of which are singular, are derived, and the solutions of the linear Fredholm integral equations of the formf(s)=ϕ(s)+λ∫abA(s,t)f(t)dt,f(s)=ϕ(s)+λ∫abA(t,s)f(t)dtandf(s,t)=ϕ(s,t)+λ∫abA(s,t)f(t,s)dtare investigated in these spaces by analytical methods.

Heat Potentials Method in the Treatment of One-dimensional Free Boundry Problems Applied in Cryomedicine

2018 ◽  
Vol 85 (1-2) ◽  
pp. 111 ◽  
Author(s):  
Fatimat K. Kudayeva ◽  
Arslan A. Kaigermazov ◽  
Elizaveta K. Edgulova ◽  
Mariya M. Tkhabisimova ◽  
Aminat R. Bechelova
Keyword(s):  
Free Boundary ◽  
Integral Equations ◽  
Free Boundary Problems ◽  
Evolutionary Equation ◽  
Fredholm Integral Equations ◽  
Integral Expression ◽  
Boundary Problems ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Heat Potential

Free boundary problems are considered to be the most difcult and the least researched in the eld of mathematical physics. The present article is concerned with the research of the following issue: treatment of one-dimensional free boundary problems. The treated problem contains a nonlinear evolutionary equation, which occurs within the context of mathematical modeling of cryosurgery problems. In the course of the research, an integral expression has been obtained. The obtained integral expression presents a general solution to the non-homogeneous evolutionary equation which contains the functions that represent simple-layer and double-layer heat potential density. In order to determine the free boundary and the density of potential a system of nonlinear, the second kind of Fredholm integral equations was obtained within the framework of the given work. The treated problem has been reduced to the system of integral equations. In order to reduce the problem to the integral equation system, a method of heat potentials has been used. In the obtained system of integral equations instead of K(ξ; x; τ - t) in case of Dirichlet or Neumann conditions the corresponding Greens functions G(ξ; x; τ - t) or N(ξ; x; τ - t) have been applied. Herewith the integral expression contains fewer densities, but the selection of arbitrary functions is reserved. The article contains a number of results in terms of building a mathematical model of cooling and freezing processes of biological tissue, as well as their effective solution development.

scholarly journals A Unified Formulation of Analytical and Numerical Methods for Solving Linear Fredholm Integral Equations

Algorithms ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 293
Author(s):  
Efthimios Providas
Keyword(s):  
Banach Space ◽  
Approximate Solution ◽  
Integral Equations ◽  
Kernel Methods ◽  
Projection Methods ◽  
Analytic Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Degenerate Kernel ◽  
Quadrature Methods

This article is concerned with the construction of approximate analytic solutions to linear Fredholm integral equations of the second kind with general continuous kernels. A unified treatment of some classes of analytical and numerical classical methods, such as the Direct Computational Method (DCM), the Degenerate Kernel Methods (DKM), the Quadrature Methods (QM) and the Projection Methods (PM), is proposed. The problem is formulated as an abstract equation in a Banach space and a solution formula is derived. Then, several approximating schemes are discussed. In all cases, the method yields an explicit, albeit approximate, solution. Several examples are solved to illustrate the performance of the technique.

On the Stresses in a Semi-Infinite Strip Subjected to a Concentrated Load

1965 ◽  
Vol 32 (2) ◽  
pp. 456-458 ◽  
Author(s):  
Chih-Bing Ling
Keyword(s):  
Exact Solution ◽  
Integral Equations ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Infinite Strip ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Method Of Images ◽  
Concentrated Load

This Note presents an exact solution for the stresses in a semi-infinite strip subjected to a symmetrically placed concentrated load. The solution is constructed by method of images. The resulting system of equations, which consists partly of integral equations and partly of algebraic equations, is solved by method of successive approximations. Convergence of the method is proved.

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