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1
Content available Potential method in the coupled linear quasi-static theory of thermoelasticity for double porosity materials
Mikelashvili M.
Archives of Mechanics
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2023
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Vol. 75, nr 5
559--590
EN
This paper concerns the coupled linear quasi-static theory of thermoelasticity for materials with double porosity under local thermal equilibrium. The system of equations of this theory is based on the constitutive equations, Darcy’s law of the flow of a fluid through a porous medium, Fourier’s law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. By virtue of Green’s identity the uniqueness theorems for classical solutions of the internal and external quasi-static boundary value problems (BVPs) are proved. The fundamental solution of the system of steady vibration equations in the considered theory is constructed and its basic properties are established. Then, the surface and volume potentials are presented and their basic properties are given. Finally, on the basis of these results the existence theorems for classical solutions of the above mentioned BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.
2
Content available Problems of steady vibrations in the coupled linear theory of double-porosity viscoelastic materials
Svanadze M. M.
Archives of Mechanics
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2021
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Vol. 73, nr 4
365--390
EN
In the present paper the coupled linear theory of double-porosity viscoelastic materials is considered and the basic boundary value problems (BVPs) of steady vibrations are investigated. Indeed, in the beginning, the systems of equations of motion and steady vibrations are presented. Then, Green’s identities are established and the uniqueness theorems for classical solutions of the BVPs of steady vibrations are proved. The fundamental solution of the system of steady vibration equations is constructed and the basic properties of the potentials (surface and volume) are given. Finally, the existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (the boundary integral equations method) and the theory of singular integral equations.
3
Content available remote Potential method in the theory of thermoelasticity for materials with triple voids
Svanadze M.
Archives of Mechanics
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2019
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Vol. 71, nr 2
113--136
EN
In the present paper the linear theory of thermoelasticity for isotropic and homogeneous solids with macro-, meso- and microporosity is considered. In this theory the independent variables are the displacement vector field, the changes of the volume fractions of pore networks and the variation of temperature. The fundamental solution of the system of steady vibrations equations is constructed explicitly by means of elementary functions. The basic internal and external boundary value problems (BVPs) are formulated and the uniqueness theorems of these problems are proved. The basic properties of the surface (single-layer and double-layer) and volume potentials are established and finally, the existence theorems for regular (classical) solutions of the internal and external BVPs of steady vibrations are proved by using the potential method (boundary integral equation method) and the theory of singular integral equations.
4
Content available remote Boundary value problems of steady vibrations in the theory of thermoelasticity for materials with a double porosity structure
Svanadze M.
Archives of Mechanics
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2017
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Vol. 69, nr 4-5
347--370
EN
The purpose of the present paper is to develop the classical potential method in the linear theory of thermoelasticity for materials with a double porosity structure based on the mechanics of materials with voids. The fundamental solution of the system of equations of steady vibrations is constructed explicitly by means of elementary functions and its basic properties are established. The Sommerfeld-Kupradze type radiation conditions are established. The basic internal and external boundary value problems (BVPs) are formulated and the uniqueness theorems of these problems are proved. The basic properties of the surface (single-layer and double-layer) and volume potentials are established and finally, the existence theorems for regular (classical) solutions of the internal and external BVPs of steady vibrations are proved by using the potential method and the theory of singular integral equations.
5
Content available remote Uniqueness of differential polynomials of meromorphic functions sharing a small function without counting multiplicity
Kieu-Oanh B. T., Thuy N. T. T.
Fasciculi Mathematici
|
2016
|
Nr 57
121--135
EN
The paper concerns interesting problems related to the field of Complex Analysis, in particular Nevanlinna theory of meromorphic functions. The author have studied certain uniqueness problem on differential polynomials of meromorphic functions sharing a small function without counting multiplicity. The results of this paper are extension of some problems studied by K. Boussaf et. al. in [2] and generalization of some results of S.S. Bhoosnurmath et. al. in [4].
6
Content available remote Plane waves and problems of steady vibrations in the theory of viscoelasticity for Kelvin-Voigt materials with double porosity
Svanadze M. M.
Archives of Mechanics
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2016
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Vol. 68, nr 6
441--458
EN
In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with double porosity is considered. Some basic properties of plane harmonic waves are established and the boundary value problems (BVPs) of steady vibrations are investigated. Indeed on the basis of this theory three longitudinal and two transverse plane harmonic waves propagate through a Kelvin–Voigt material with double porosity and these waves are attenuated. The basic properties of the singular integral operators and potentials (surface and volume) are presented. The uniqueness and existence theorems for regular (classical) solutions of the BVPs of steady vibrations are proved by using the potential method (boundary integral equations method) and the theory of singular integral equations.
7
Content available Uniqueness, reciprocity theorems and plane wave propagation in different theories of thermoelasticy
Kumar R., Gupta V.
International Journal of Applied Mechanics and Engineering
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2013
|
Vol. 18, no. 4
1067--1086
EN
In this work, a compact form of different theories of thermoelasticity is considered. The governing equations for particle motion in a homogeneous isotropic thermoelastic medium are presented. Uniqueness and reciprocity theorems are proved. The plane wave propagation in a homogeneous isotropic thermoelastic medium is studied. For a three dimensional problem there exist four waves, namely a P-wave, two transverse waves (S1, S2) and a thermal wave (T). From the obtained results the different characteristics of waves such as the phase velocity and attenuation coefficient are computed numerically and presented graphically. Some special cases are also discussed.
8
Content available remote Non linear differential polynomials sharing one value
Banerjee A.
Demonstratio Mathematica
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2007
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Vol. 40, nr 1
65-75
EN
We prove three uniqueness theorems concerning non linear dierential polynomials which will improve and supplement some earlier results given by Yang and Hua, Lahiri.
9
Content available remote Sequentially continuous non-linear fundamental systems of solutions of affine equations in locally convex space
Shkarin S. S.
Demonstratio Mathematica
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2003
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Vol. 36, nr 3
611--626
EN
According to Mickael's selection theorem any surjective continuous linear operator from one Prechet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if E is a Frechet space and T : E -> E is a continuous linear operator such that the Cauchy problem x = T x, x(0) = X0 is solvable in [0,1] for any X06 E, then for anyf zawiera się C([0, 1],E), there exists a continues map S : [0,1] x E -> E, (t x) ->o StX such that for any X0 zawiera się w E, the function x(t) = StX0 is a solution of the Cauchy problem x(t) = Tx(t) +- f(t), x(0) = X0 (they call S a fundamental system of solutions of the equation x = Tx + f). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Frechet spaces and strong duals of Frechet-Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.
10
Content available remote Uniqueness theorem for nonlinear hyperbolic equations with order degeneration
Semerdjieva R.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2003
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Vol. 51, no 1
99-105
EN
Let k(y) > O, l(y) > O for y > O, k(0) = l(0) = 0; then the equation L(u) := k(y)u(xx) - (delta)y(l(y)uy) +a(x,y)ux = f (x,y,u) is strictly hyperbolic for y > O and its order degenerates on the line y = 0. We consider the boundary value problem Lu = f (x,y,u) in G, u\(AC) = 0, where G is a simply connected domain in R-2 with piecewise smooth boundary [delta]G = AB boolean Or AC boolean OR BC; AB = {(x, 0) : 0 less than or equal to x less than or equal to 1}, AC : x = F(y) = integral(0)(y)k(t)/l(t)(1/2)dt and x = 1 - F(y) are characteristic curves. It is proved that if f satisfies the Caratheodory condition and \f{x,y,z(1)}-f{x,y,z(2))\ less than or equal to C(\z(1)\(beta) + \z(2)\(beta))\z1-z2\ with some constants C > O and beta is greater than or equal to O then there exists at most one generalized solution.
11
Content available remote The construction of a certain quasi-conformal extension of the functions defined in the unit disc, on the closed plane
Bogowski F., Wesołowski A.
Demonstratio Mathematica
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2002
|
Vol. 35, nr 3
508-520
EN
In this paper a certain method of the construction of a quasi-conformal extension of the functions denned in the unit disc, on the closed plane C is given. The earlier known results are received in particular cases, [1], [2], [3], [9).
12
Content available remote On the existence, uniqueness of solution of a nonlinear vibrations equation
Long N., Thuyet T.
Demonstratio Mathematica
|
1999
|
Vol. 32, nr 4
749-758
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