Yuriy Povstenko

Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder

The time-fractional diffusion-wave equation is considered in an infinite cylinder in the case of three spatial coordinates r, ϕ and z. The Caputo fractional derivative of the order 0 < α ≤ 2 is used. Several examples of problems with Dirichlet and Neumann boundary conditions at a surface of the cylinder are solved using the integral transforms technique. Numerical results are illustrated graphically.

Theories of thermal stresses based on space-time-fractional telegraph equations

The generalized telegraph equations with time- and space-fractional derivatives are considered. The corresponding theories of thermal stresses are formulated. The proposed theories interpolate the classical thermoelasticity, the theory of Lord and Shulman, thermoelasticity without energy dissipation of Green and Naghdi, and theories of fractional thermoelasticity proposed earlier. The fundamental solution to the nonhomogeneous space-time-fractional telegraph equation as well as the corresponding thermal stresses are obtained in the axisymmetric case.

Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation

In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order @a,1<@a<2, is revisited. This equation interpolates between the diffusion and the wave equations that behave quite differently regarding their response to a localized disturbance: whereas the diffusion equation describes a process, where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. For the time-fractional diffusion-wave equation, the propagation speed of a disturbance is infinite, but its fundamental solution possesses a maximum that disperses with a finite speed. In this paper, the fundamental solution of the Cauchy problem for the time-fractional diffusion-wave equation, its maximum location, maximum value, and other important characteristics are investigated in detail. To illustrate analytical formulas, results of numerical calculations and plots are presented. Numerical algorithms and programs used to produce plots are discussed.

Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

Heat conduction in two joint half-lines is considered under the condition of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The heat conduction in one half-line is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another half-line is described by the equation with the time derivative of order β. The fundamental solutions to the first and second Cauchy problems as well as to the source problem are obtained using the Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate. The fundamental solutions are expressed in terms of the Mittag-Leffler function and the Mainardi function.

Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses

ABSTRACT Time-nonlocal generalization of the classical Fourier law with the “long-tail” power kernel can be interpreted in terms of fractional calculus (theory of integrals and derivatives of noninteger order) and leads to the time-fractional heat conduction equation with the Caputo derivative. Fractional heat conduction equation with the harmonic source term under zero initial conditions is studied. Different formulations of the problem for the standard parabolic heat conduction equation and for the hyperbolic wave equation appearing in thermoelasticity without energy dissipation are discussed. The integral transform technique is used. The corresponding thermal stresses are found using the displacement potential.

Non-central-symmetric solution to time-fractional diffusion-wave equation in a sphere under Dirichlet boundary condition

The time-fractional diffusion-wave equation is considered in a sphere in the case of three spatial coordinates r, µ, and φ. The Caputo fractional derivative of the order 0 < α ≤ 2 is used. The solution is found using the Laplace transform with respect to time t, the finite Fourier transform with respect to the angular coordinate φ, the Legendre transform with respect to the spatial coordinate µ, and the finite Hankel transform of the order n + 1/2 with respect to the radial coordinate r. In the central symmetric case with one spatial coordinate r the obtained result coincides with that studied earlier. Numerical results are illustrated graphically.

Mathematical Modeling of Phenomena Caused by Surface Stresses in Solids

Interfacial region between two bulk phases and the transition region near the line of contact of three media are considered as a two-dimensional and one-dimensional continuum, respectively. A survey of works on mathematical modeling of phenomena in such systems is presented. The equation of the linear momentum balance for an interface generalizes the classical Laplace equation and that for a contact line generalizes the Young equation of the capillarity theory. The influence of nonuniform surface tension on the stress field in an infinite cylinder is investigated. The anisotropy of wetting is discussed and explained on the basis of the generalized Young equation taking into account the tensor character of surface stresses. Several applications of the results in the theory of surface defects are also discussed.

Neumann boundary-value problems for a time-fractional diffusion-wave equation in a half-plane

The time-fractional diffusion-wave equation with the Caputo derivative of the order 0<@a<2 is considered in a half-plane. Two types of Neumann boundary condition are examined: the mathematical condition with the prescribed boundary value of the normal derivative and the physical one with the prescribed boundary value of the matter flux.

Time-fractional thermoelasticity problem for a sphere subjected to the heat flux

The theory of thermal stresses based on the heat conduction equation with the Caputo time-fractional derivative is used to study central symmetric thermal stresses in a sphere. The solution is obtained using the Laplace transform with respect to time and the finite sin-Fourier integral transform with respect to the radial coordinate. The physical Neumann problem with the prescribed boundary value of the heat flux is considered. Numerical results are illustrated graphically.

Optimal Boundary Control of Thermal Stresses in a Plate Based on Time-Fractional Heat Conduction Equation

This article presents an optimal control problem for a fractional heat conduction equation that describes a temperature field. The main purpose of the research was to find the boundary temperature that takes the thermal stress under control. The fractional derivative is defined in terms of the Caputo operator. The Laplace and finite Fourier sine transforms were applied to obtain the exact solution. Linear approximation is used to get the numerical results. The dependence of the solution on the order of fractional derivative and on the nondimensional time is analyzed.