scholarly journals Generalized and Fundamental Solutions of Motion Equations of Two-Component Biot’s Medium

2020 ◽  
Author(s):  
Lyudmila Alexeyeva ◽  
Yergali Kurmanov
Keyword(s):  
Wave Propagation ◽  
Harmonic Oscillation ◽  
Delta Function ◽  
Fundamental Solutions ◽  
Generalized Functions ◽  
Integral Representations ◽  
Generalized Solutions ◽  
Motion Equations ◽  
Two Component ◽  
Water Saturated

Here processes of wave propagation in a two-component Biot’s medium are considered, which are generated by arbitrary forces actions. By using Fourier transformation of generalized functions, a fundamental solution, Green tensor, of motion equations of this medium has been constructed in a non-stationary case and in the case of stationary harmonic oscillation. These tensors describe the processes of wave propagation (in spaces of dimensions 1, 2, 3) under an action of power sources concentrated at coordinates origin, which are described by a singular delta-function. Based on them, generalized solutions of these equations are constructed under the action of various sources of periodic and non-stationary perturbations, which are described by both regular and singular generalized functions. For regular acting forces, integral representations of solutions are given that can be used to calculate the stress-strain state of a porous water-saturated medium.

Seismic stability of the underground main pipeline

2011 ◽  
Vol 8 (1) ◽  
pp. 275-286
Author(s):  
R.G. Yakupov ◽  
D.M. Zaripov
Keyword(s):  
Laplace Transformation ◽  
Seismic Waves ◽  
Generalized Functions ◽  
Seismic Stability ◽  
Main Pipeline ◽  
Motion Equations ◽  
Numerical Example ◽  
Earthquake Force ◽  
Deformed State

The stress-deformed state of the underground main pipeline under the action of seismic waves of an earthquake is considered. The generalized functions of seismic impulses are constructed. The pipeline motion equations are solved with used Laplace transformation by the time. Tensions and deformations of the pipeline have been determined. A numerical example is reviewed. Diagrams of change of the tension depending on earthquake force are provided in earthquake-points.

scholarly journals An axially symmetric forced convection problem

1975 ◽  
Vol 20 (1) ◽  
pp. 1-17
Author(s):  
J. A. Belward
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Convection Heat Transfer ◽  
Fundamental Solutions ◽  
Integral Representations ◽  
Original Equation ◽  
Transfer Model ◽  
Axially Symmetric ◽  
Simple Diffusion ◽  
Diffusion Convection

AbstractA simple diffusion-convection heat transfer model is formulated which leads to an axially symmetric partial differential equation. The equation is shown to be closely related to a second one which is adjoint to the original equation in one variable can and be interpreted as a description of another diffusion-convection model. Fundamental solutions of the original equation are constructed and interpreted with reference to both models. Some boundary value problems are solved in series form and integral representations of the solutions are also given. The boundary value problems are shown to be equivalent to an integral equation and the correspondence between the two formulations is understood in terms of the two diffusion-convection problems. A Péclet number is defined in one of the boundary value problems and the behaviour of the solutions is studied for large and small values of this parameter.

On Operators and Distributions

1968 ◽  
Vol 11 (1) ◽  
pp. 61-64 ◽  
Author(s):  
Raimond A. Struble
Keyword(s):  
Commutative Ring ◽  
Operational Calculus ◽  
Delta Function ◽  
Generalized Functions ◽  
Dirac Delta Function ◽  
Quotient Field ◽  
Dirac Delta ◽  
The Family ◽  
Continuous Complex ◽  
Complex Valued

Mikusinski [1] has extended the operational calculus by methods which are essentially algebraic. He considers the family C of continuous complex valued functions on the half-line [0,∞). Under addition and convolution C becomes a commutative ring. Titchmarsh's theorem [2] shows that the ring has no divisors of zero and, hence, that it may be imbedded in its quotient field Q whose elements are then called operators. Included in the field are the integral, differential and translational operators of analysis as well as certain generalized functions, such as the Dirac delta function. An alternate approach [3] yields a rather interesting result which we shall now describe briefly.

On the solution of certain integral equations by generalized functions

1961 ◽  
Vol 57 (4) ◽  
pp. 767-777 ◽  
Author(s):  
J. B. Miller
Keyword(s):  
Integral Equations ◽  
Generalized Functions ◽  
Generalized Solutions ◽  
Generalized Function ◽  
Usual Sense ◽  
Inverse Transformation ◽  
Extended Sense ◽  
Image Position

This note is concerned with a method by which generalized solutions can be shown to exist for certain types of integral equationswhere f(x) ∈ L2(0, ∞). The method is briefly this. An extended meaning is given to such equations by using generalized functions of a particular type. Then, if (1) denotes a transformation which has no everywhere-defined inverse in the usual sense, it may be possible to define in the extended sense an inverse transformationso that if f is given, a generalized function g = can be determined as a solution of (1).

Direct BEM for Three-Dimensional Transient Dynamic Piezoelectric Analysis

2014 ◽  
Vol 709 ◽  
pp. 113-116 ◽  
Author(s):  
Leonid Igumnov ◽  
I.P. Маrkov ◽  
A.A. Belov
Keyword(s):  
Boundary Element Method ◽  
Real Time ◽  
Three Dimensional ◽  
Fundamental Solutions ◽  
Integral Representations ◽  
Laplace Domain ◽  
Transient Dynamic ◽  
Linear Piezoelectricity ◽  
Method Formulation ◽  
Direct Boundary Element Method

Direct boundary element method formulation for transient dynamic linear piezoelectricity is presented. Integral representations of Laplace transformed dynamic piezoelectric fundamental solutions are used. Laplace domain BEM solutions inverted in real time by the stepping method. Numerical example of transient piezoelectric analysis is presented.

A Three-Dimensional BEM for Dynamic Analysis of Anisotropic Elastic Multi-Connected Bodies

2017 ◽  
Vol 743 ◽  
pp. 153-157 ◽  
Author(s):  
Leonid A. Igumnov ◽  
Ivan Markov
Keyword(s):  
Boundary Element Method ◽  
Dynamic Analysis ◽  
Three Dimensional ◽  
Fundamental Solutions ◽  
Anisotropic Elasticity ◽  
Integral Representations ◽  
Dynamic Problems ◽  
Laplace Domain ◽  
Anisotropic Elastic ◽  
Direct Boundary Element Method

In this paper, the direct boundary element method in the Laplace domain is applied for the solution of three-dimensional transient dynamic problems of anisotropic elasticity in multi-connected domains. The formulation is based upon the integral representations of anisotropic dynamic fundamental solutions. As numerical example the problem of an anisotropic elastic prismatic solid with cubic cavity is investigated.

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