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1
Content available Solutions to Time-Fractional Diffusion-Wave Equation in Spherical Coordinates
100%
Povstenko Y.
Acta Mechanica et Automatica
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2011
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tom Vol. 5, no. 2
108-111
EN
Solutions to time-fractional diffusion-wave equation with a source term in spherical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time t, the finite Fourier transform with respect to the angular coordinate , the Legendre transform with respect to the spatial coordinate , and the Hankel transform of the order n+1/2 with respect to the radial coordinate . In the central symmetric case with one spatial coordinate the obtained results coincide with those studied earlier.
2
Content available Axisymmetric thermal stresses in a half-space in the framework of fractional thermoelasticity
100%
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2014
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tom Vol. 19
207--216
EN
A theory of thermal stresses based on the time-fractional heat conduction equation is considered. The Caputo fractional derivative is used. The fundamental solution to the axisymmetric heat conduction equation in a half-space under the Dirichlet boundary condition and the associated thermal stresses are investigated.
3
Content available Axisymmetric solutions to the Cauchy problem for time-fractional diffusion equation in a circle
100%
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2010
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tom Vol. 15
109--117
EN
The Cauchy problems for time-fractional diffusion equation with delta pulse initial value of a sought-for function is studied in a circle domain in the axisymmetric case under zero Dirichlet and Neumann boundary conditions, respectively. The Caputo fractional derivative is used. The Laplace and finite Hankel integral transforms are employed. The results are illustrated graphically.
4
Content available remote Metoda kolokacji rozwiązywania równań różniczkowych typu parabolicznego
100%
Povstenko Y.
Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie
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2011
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tom Nr 1(1)
43--50
PL
W artykule omówiono metodę spektralną rozwiązywania równań typu parabolicznego. Opisano procedurę numeryczną i przedstawiono wyniki eksperymentu numerycznego.
EN
In this paper the spectral method for solving equations of parabolic type is discussed. Numerical procedure is described, and results of numerical experiment are presented.
5
Content available Different kinds of boundary conditions for time-fractional heat conduction equation
100%
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2011
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tom Vol. 16
61--66
EN
The time-fractional heat conduction equation with the Caputo derivative of the order 0 ˂ α ˂ 2 is considered in a bounded domain. For this equation different types of boundary conditions can be given. The Dirichlet boundary condition prescribes the temperature over the surface of the body. In the case of mathematical Neumann boundary condition the boundary values of the normal derivative are set, the physical Neumann boundary condition specifies the boundary values of the heat flux. In the case of the classical heat conduction equation (α = 1), these two types of boundary conditions are identical, but for fractional heat conduction they are essentially different. The mathematical Robin boundary condition is a specification of a linear combination of the values of temperature and the values of its normal derivative at the boundary of the domain, while the physical Robin boundary condition prescribes a linear combination of the values of temperature and the values of the heat flux at the surface of a body.
6
Content available Solutions to fractional diffusion-wave equation in a circular sector
100%
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2013
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tom Vol. 18
41--54
EN
The time-fractional diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a domain 0 ≤ r < R, 0 < ϕ < ϕ0 under different boundary conditions. The Laplace integral transform with respect to time, the finite Fourier transforms with respect to the angular coordinate, and the finite Hankel transforms with respect to the radial coordinate are used. The fundamental solutions are expressed in terms of the Mittag-Leffler function. The particular cases of the obtained solutions corresponding to the diffusion equation (α = 1) and the wave equation (α = 2) coincide with those known in the literature.
7
Content available Analysis of fundamental solutions to fractional diffusion-wave equation in polar coordinates
100%
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2009
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tom Vol. 14
97-104
EN
The diffusion-wave equation is a mathematical model of a wide range of important physical phenomena. The first and second Cauchy problems and the source problem for the diffusion-wave equation are considered in cylindrical coordinates. The Caputo fractional derivative is used. The Laplace and Hankel transforms are employed. The results are illustrated graphically.
8
Content available Harmonic impact on the surface of a half-plane in the framework of Time-Fractional Heat Conduction
100%
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2016
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tom Vol. 21
85--92
EN
The time-fractional heat conduction equation with the Caputo derivative is considered in a half-plane. The boundary value of temperature varies harmonically in time. The integral transform technique is used; the solution is obtained in terms of integral with integrand being the Mittag-Leffler functions. The particular case of solution corresponding to the classical heat conduction equation is discussed in details.
9
Content available remote Numerical aspects of mathematical modeling of crystal imperfection
100%
Povstenko Y.
Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie
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2013
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tom Nr 1(5)
1--13
EN
In this paper, a survey of studies concerning mathematical modeling of crystal imperfections in solids is presented. The emphasis is placed on describing imperfections in nonlocal elastic continuum. Nonlocal theory reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wave-length limit.
PL
W artykule przedstawiono przegląd badań dotyczących modelowania matematycznego niedoskonałości sieci krystalicznej w ciele stałym. Nacisk kładzie się na opis niedoskonałości w nielokalnie odkształcalnym continuum. Teoria nielokalna w granicy długofalowej sprowadza się do klasycznej teorii sprężystości, a w granicy krótkofalowej – do teorii sieci atomowej.
10
Content available remote On two reinterpretations of Cosserat continuum: fiber bundle versus the motor calculus
63%
Povstenko Y. , Badur J.
Archives of Mechanics
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1998
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tom Vol. 50, nr 3
367-376
EN
It is the purpose of this paper to reinterprete the original Cosserat continuum from the point of view of both the fiber bundles geometry and the non-Abelian motor calculus. The main ideas of Cosserats are best explained in terms of the moving reper which is synonymous with the procedure of gauging in physics as well as with the procedure of constructing a fiber bundle in pure geometry. On the other hand, the classical (linear) von Mises motor calculus is extended to a non-Abelian case. It also appears that this non-Abelian version of the von Mises concept is fully equivalent with the fiber bundle description.
11
Content available Fractional heat conduction in an infinite rod with heat absorption proportional to temperature
63%
Povstenko Y. , Klekot J.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2017
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tom Vol. 22
61--71
EN
The one-dimensional time-fractional heat conduction equation with heat absorption (heat release) proportional to temperature is considered. The Caputo time-fractional derivative is utilyzed. The fundamental solutions to the Cauchy and source problems are obtained using the Laplace transform with respect to time and the exponential Fourier transform with respect to the spatial coordinate. The numerical results are illustrated graphically.
12
Content available The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption
63%
Povstenko Y. , Klekot J.
Journal of Applied Mathematics and Computational Mechanics
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2017
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tom Vol. 16, nr 2
101--112
EN
The time-fractional heat conduction equation with heat absorption proportional to temperature is considered in the case of central symmetry. The fundamental solutions to the Cauchy problem and to the source problem are obtained using the integral transform technique. The numerical results are presented graphically.
13
Content available The Cauchy problem for the time-fractional advection diffusion equation in a layer
63%
Povstenko Y. , Klekot J.
Technical Sciences / University of Warmia and Mazury in Olsztyn
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2016
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tom nr 19(3)
231--244
EN
The time-fractional advection-diffusion equation with the Caputo time derivative is studied in a layer. The fundamental solution to the Cauchy problem is obtained using the integral transform technique. The logarithmicsingularity term is separated from the solution. Expressions amenable for numerical treatment are obtained. The numerical results are illustrated graphically.
14
Content available The Dirichlet problem for the time-fractional advection-diffusion equation in a half-space
63%
Povstenko Y. , Klekot J.
Journal of Applied Mathematics and Computational Mechanics
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2015
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tom Vol. 14, nr 2
73--83
EN
The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered in a half-space. The fundamental solution to the Dirichlet problem and the solution of the problem with constant boundary condition are obtained using the integral transform technique. The numerical results are illustrated graphically.
15
Content available Fundamental solution to the Cauchy problem for the time-fractional advection-diffusion equation
63%
Povstenko Y. , Klekot J.
Journal of Applied Mathematics and Computational Mechanics
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2014
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tom Vol. 13, nr 1
95--102
EN
The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered. The fundamental solution to the Cauchy problem is obtained using the integral transform technique. The numerical results are illustrated graphically.
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