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1
Potential method in the theory of thermoelasticity for materials with triple voids
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.
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.
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].
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.
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.
6
Non linear differential polynomials sharing one value
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.
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.
8
Uniqueness theorem for noniinear hyperbolic equations with order degeneration
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.
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).
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