scholarly journals New Model for Solving Mixed Integral Equation of the First Kind with Generalized Potential Kernel

2017 ◽  
Vol 9 (5) ◽  
pp. 18
Author(s):  
Shareefa Eisa Ali Alhazmi
Keyword(s):  
Integral Equation ◽  
Potential Function ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Contact Problems ◽  
Three Dimensions ◽  
Function Form ◽  
Special Cases ◽  
Generalized Potential ◽  
Carleman Function

New technique model is used to solve the mixed integral equation (\textbf{MIE}) of the first kind, with a position kernel contains a generalized potential function multiplying by a continuous function and continuous kernel in time, in the space $L_{2} (\Omega )\times C[0,T],\, 0\leq T<1$, $\Omega$ is the domain of integration and $T$ is the time. The integral equation arises in the treatment of various semi-symmetric contact problems, in one, two, and  three dimensions, with mixed boundary conditions in the mechanics of continuous media. The solution of the \textbf{MIE }when the kernel of position takes the potential function form, elliptic function form, Carleman function and logarithmic kernel are discussed and obtain in  this work. Moreover, many special cases are derived.

scholarly journals An asymptotic model for solving mixed integral equation in some domains

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
Mohamed Abdella Abdou ◽  
Hamed Kamal Awad
Keyword(s):  
Integral Equation ◽  
Potential Function ◽  
Theory Of Elasticity ◽  
Asymptotic Model ◽  
Logarithmic Function ◽  
Integral Term ◽  
Special Cases ◽  
Generalized Potential ◽  
Carleman Function ◽  
Axisymmetric Problems

Abstract In this paper, we discuss the solution of mixed integral equation with generalized potential function in position and the kernel of Volterra integral term in time. The solution will be discussed in the space $$L_{2} (\Omega ) \times C[0,T],$$ L 2 ( Ω ) × C [ 0 , T ] , $$0 \le t \le T < 1$$ 0 ≤ t ≤ T < 1 , where $$\Omega$$ Ω is the domain of position and $$t$$ t is the time. The mixed integral equation is established from the axisymmetric problems in the theory of elasticity. Many special cases when kernel takes the potential function, Carleman function, the elliptic function and logarithmic function will be established.

scholarly journals The dual integral equation method in hydromechanical systems

2004 ◽  
Vol 2004 (6) ◽  
pp. 447-460 ◽  
Author(s):  
N. I. Kavallaris ◽  
V. Zisis
Keyword(s):  
Integral Equation ◽  
Potential Function ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Integral Equation Method ◽  
Equation Method ◽  
Dual Integral Equation ◽  
Spherical Coordinates ◽  
Steady State Heat

Some hydromechanical systems are investigated by applying the dual integral equation method. In developing this method we suggest from elementary appropriate solutions of Laplace's equation, in the domain under consideration, the introduction of a potential function which provides useful combinations in cylindrical and spherical coordinates systems. Since the mixed boundary conditions and the form of the potential function are quite general, we obtain integral equations withmth-order Hankel kernels. We then discuss a kind of approximate practicable solutions. We note also that the method has important applications in situations which arise in the determination of the temperature distribution in steady-state heat-conduction problems.

The Multiscale Analysis of Multiple Interacting Inclusions Problem: Finite Number of Interacting Inclusions

2005 ◽  
Vol 10 (1) ◽  
pp. 25-62 ◽  
Author(s):  
V. A. Buryachenko ◽  
N. J. Pagano
Keyword(s):  
Integral Equation ◽  
Mixed Boundary ◽  
Singular Integral Equations ◽  
Mixed Boundary Conditions ◽  
Analytical Approximation ◽  
Boundary Integral ◽  
Element Analysis ◽  
Unbounded Medium ◽  
Macro Scale ◽  
Micro Level

A hybrid method based on the combination of the volume integral equation (VIE) method and the boundary integral equation (BIE) method is proposed for the micro-macro solution of elastostatic 2D and 3D multiscale problems in bounded or unbounded solids containing interacting multiple inclusions of essentially different scale. The hybrid micro-macro formulation allows decomposition of the complete problem into two associated subproblems, one residing entirely at the micro-level and the other at the macro-level at each iteration. The efficiency of the standard iterative scheme of the BIE and VIE methods for the singular integral equations involved is enhanced by the use of a modification in the spirit of a subtraction technique as well as by the advantageous choice of the initial analytical approximation for interacting inclusions (micro-level) in an unbounded medium subjected to inhomogeneous loading. The latter is evaluated by the macro-scale BIE technique capable of handling complex finite geometries and mixed boundary conditions. The iteration method proposed converges rapidly in a wide class of problems considered with high matrix-inclusion elastic contrast, with continuously varying anisotropic and nonlinear elastic properties of inclusions, as well as with sizes of interacting inclusions differing by a factor varying in the interval from 1 to 107. The accuracy and efficiency of the method are examined through comparison with results obtained from finite-element analysis and boundary element analysis as well as from analytical solution.

scholarly journals Strong convergence of discrete DG solutions of the heat equation

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Vivette Girault ◽  
Jizhou Li ◽  
Beatrice Rivière
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Strong Convergence ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Three Dimensions ◽  
Galerkin Methods

AbstractA convergence analysis to the weak solution is derived for interior penalty discontinuous Galerkin methods applied to the heat equation in two and three dimensions under general mixed boundary conditions. Strong convergence is established in the DG norm, as well as in the

New mixed boundary, true triaxial loading device for testing three-dimensional stress–strain–strength behaviour of geomaterials

2010 ◽  
Vol 47 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Jian-Hua Yin ◽  
Chun-Man Cheng ◽  
Md. Kumruzzaman ◽  
Wan-Huan Zhou
Keyword(s):  
Mixed Boundary ◽  
Three Dimensional ◽  
Mixed Boundary Conditions ◽  
Contact Problems ◽  
Large Strains ◽  
Soil Specimen ◽  
Fe Modelling ◽  
Triaxial Loading ◽  
True Triaxial ◽  
Loading Device

This paper presents a brief review of true triaxial apparatuses (TTAs) developed in the past and their advantages and limitations. Considering the limitations of previous designs, a new true triaxial loading device that provides mixed boundary conditions for a true triaxial apparatus (TTA) is introduced. This loading device consists of four sliding rigid plates and two flexible loading faces. The setup of the loading device together with the whole true triaxial system is described. Frictions between sliding plates and the soil membrane surfaces in the new loading device are examined. A three-dimensional finite element (FE) modelling study is carried out on the stress and strain distribution of a soil specimen subjected to loading from two different loading devices. It is found that stresses and strains of a soil specimen subjected to loading from the new sliding plates are far more uniform than those subjected to loading from nonsliding plates with preset gaps. Finally, the paper presents the applications of the present TTA with the new loading device for testing studies of a completely decomposed granite soil and a geofoam. Typical results are presented and discussed. It is found that the present mixed boundary loading device is very suitable for true triaxial testing on both soils and geofoam, especially under large strains or compression without corner contact problems.

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