Thermoelasticity of thin shells based on the time-fractional heat conduction equation

The time-nonlocal generalizations of Fourier’s law are analyzed and the equations of the generalized thermoelasticity based on the time-fractional heat conduction equation with the Caputo fractional derivative of order 0 < α ≤ 2 are presented. The equations of thermoelasticity of thin shells are obtained under the assumption of linear dependence of temperature on the coordinate normal to the median surface of a shell. The conditions of Newton’s convective heat exchange between a shell and the environment have been assumed. In the particular case of classical heat conduction (α = 1) the obtained equations coincide with those known in the literature.     

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations 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. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.     

Critical properties of the antiferromagnetic layered Ising model on a cubic lattice with competing interactions

The critical properties of the antiferromagnetic layered Ising model on a cubic lattice with regard to the nearest-neighbor and next-nearest-neighbor interactions are investigated by the Monte Carlo method using the replica algorithm. The investigations are carried out for the ratios of exchange nearest-neighbor and next-nearest-neighbor interactions r = J ₂/J ₁ in the range of 0 ≤ r ≤ 1.0. Using the finite-size scaling theory, the static critical indices of specific heat α, order parameter β, susceptibility γ, correlation radius ν, and Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is retained in the range of 0 ≤ r ≤ 0.4. It is established that the change in the next-nearest-neighbor interaction value in this model in the range of r > 0.8 leads to the same universality class as the three-dimensional fully frustrated Ising model on the cubic lattice.

     

Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods

A nonlinear conformable time-fractional parabolic equation with exponential nonlinearity is explored, in this article. First, under the specific transformations, the time-fractional parabolic equation is changed into a nonlinear ODE of integer order, and then, the reduced equation is solved using two lately established techniques called the \({ \exp }\left( { - \varphi \left( \varepsilon \right)} \right)\)-expansion and modified Kudryashov methods. Several exact solutions in various wave forms for the nonlinear conformable time-fractional parabolic equation with exponential nonlinearity are formally constructed.     

Exact solutions of triple-order time-fractional differential equations for anomalous relaxation and diffusion I: The accelerating case

In recent years the interest around the study of anomalous relaxation and diffusion processes is increased due to their importance in several natural phenomena. Moreover, a further generalization has been developed by introducing time-fractional differentiation of distributed order which ranges between 0 and 1. We refer to accelerating processes when the driving power law has a changing-in-time exponent whose modulus tends from less than 1 to 1, and to decelerating processes when such an exponent modulus decreases in time moving away from the linear behaviour. Accelerating processes are modelled by a time-fractional derivative in the Riemann-Liouville sense, while decelerating processes by a time-fractional derivative in the Caputo sense. Here the focus is on the accelerating case while the decelerating one is considered in the companion paper. After a short reminder about the derivation of the fundamental solution for a general distribution of time-derivative orders, we consider in detail the triple-order case for both accelerating relaxation and accelerating diffusion processes and the exact results are derived in terms of an infinite series of H-functions. The method adopted is new and it makes use of certain properties of the generalized Mittag-Leffler function and the H-function, moreover it provides an elegant generalization of the method introduced by Langlands (2006) [T.A.M. Langlands, Physica A 367 (2006) 136] to study the double-order case of accelerating diffusion processes.     

Magnetohydrodynamic Natural Convection in a Rotating Enclosure

Magnetohydrodynamic natural convection heat transfer in a rotating, differentially heated enclosure is studied numerically in this article. The governing equations are in velocity, pressure and temperature formulation and solved using the staggered grid arrangement together with MAC method. The governing parameters considered are the Hartmann number, 0≤$Ha$≤70, the inclination angle of the magnetic field, 0$^◦$≤$θ$≤90$^◦$, the Taylor number, 8.9×10$^4$≤$Ta$≤1.1×10$^6$ and the centrifugal force is smaller than the Coriolis force and the both forces were kept below the buoyancy force. It is found that a sufficiently large Lorentz force neutralizes the effect of buoyancy, inertial and Coriolis forces. Horizontal or vertical direction of the magnetic field is the most effective in reducing the global heat transfer.     

Surface radiation influence on the regimes of conjugate natural convection in an enclosure with local energy source

A numerical analysis of the unsteady regimes of the natural convection and thermal radiation in a square enclosure with heat-conducting walls has been carried out in the presence of a heat source of finite sizes located in the base zone under the conditions of convective exchange with the ambient medium. The mathematical model formulated in the dimensionless variables “stream function — vorticity — temperature” has been implemented numerically by a finite difference method. The influence of the reduced emissivity of internal surfaces of bounding walls on the local characteristics (the streamlines and temperature fields) and on the integral complex (the mean Nusselt number on typical boundaries) has been analyzed in detail for 0 ≤ ? < 1, the location of the heat source 0.1 ≤ l/L ≤ 0.4, and its length 0.2 ≤ l _hs /L ≤ 0.6 for Ra = 10⁶, Pr = 0.7. The approximation relations have been derived for the mean convective and radiation Nusselt numbers depending on the reduced emissivity of the internal surfaces of bounding walls and the energy source location relative to vertical walls.     

Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer

In this work, a new model of the magneto-thermoelasticity theory has been constructed in the context of a new consideration of heat conduction with fractional derivative. A one-dimensional application for a conducting half-space of thermoelectric elastic material, which is thermally shocked in the presence of a magnetic field, has been solved using Laplace transform and state-space techniques (Ezzat, 2008 [1]). According to the numerical results and its graphs, a conclusion about the new theory of magneto-thermoelasticity has been constructed. The theories of coupled magneto-thermoelasticity and of generalized magneto-thermoelasticity with one relaxation time follow as limited cases. The result provides a motivation to investigate conducting thermoelectric materials as a new class of applicable materials.     

Estimating the occupation probabilities of single-particle orbits in nuclei

Occupation probabilities of neutron and proton single-particle orbits are estimated for a number of spherical and close to spherical nuclei with 20 ≤ Z ≤ 50 and 20 ≤ N ≤ 82 near the Fermi energy. The estimates are made according to the formula of the BCS theory with single-particle energies calculated using the mean-field model with dispersive optical potential. The closeness of the occupation probabilities to 0 and 1 is demonstrated for nuclei with traditional and new magic numbers.     

Non-commutative standard model: model building

A non-commutative version of the usual electro-weak theory is constructed. We discuss how to overcome the two major problems: (1) although we can have non-commutative U(n) (which we denote by _U* (n)) gauge theory we cannot have non-commutative SU(n) and (2) the charges in non-commutative QED are quantized to just . We show how the latter problem with charge quantization, as well as with the gauge group, can be resolved by taking the gauge group and reducing the extra U(1) factors in an appropriate way. Then we proceed with building the non-commutative version of the standard model by specifying the proper representations for the entire particle content of the theory, the gauge bosons, the fermions and Higgs. We also present the full action for the non-commutative standard model (NCSM). In addition, among several peculiar features of our model, we address the inherent CP violation and new neutrino interactions. Received: 23 January 2003, Published online: 18 June 2003     

The Caputo–Fabrizio time-fractional Sharma–Tasso–Olver–Burgers equation and its valid approximations

Studying the dynamics of solitons in nonlinear time-fractional partial differential equations has received substantial attention, in the last decades. The main aim of the current investigation is to consider the time-fractional Sharma–Tasso–Olver–Burgers (STOB) equation in the Caputo–Fabrizio (CF) context and obtain its valid approximations through adopting a mixed approach composed of the homotopy analysis method (HAM) and the Laplace transform. The existence and uniqueness of the solution of the time-fractional STOB equation in the CF context are investigated by demonstrating the Lipschitz condition for $\varphi \left(x,t;u\right)$ as the kernel and giving some theorems. To illustrate the CF operator effect on the dynamics of the obtained solitons, several two- and three-dimensional plots are formally considered. It is shown that the mixed approach is capable of producing valid approximations to the time-fractional STOB equation in the CF context.     

Dielectric investigations of solid solutions SrTiO3-KTaO3 and SrTiO3-KNbO3

The temperature dependences of the dielectric constant and dielectric hysteresis loops in ceramic samples of (1 ? x)SrTiO_3?x KNbO₃ and (1 ? x)SrTiO_3?x KTaO₃ (0 ≤ x ≤ 0.3) solid solutions prepared using different heat treatments have been investigated. Phase diagrams of the studied solid solutions have been constructed in the T-x coordinates. It has been shown that, after quenching of samples (spontaneous cooling at room temperature after long-term heating at the sintering temperature of the ceramic samples), the temperature of the induced phase transition increases because of the weakening of random electric fields associated with nonisovalent impurities due to their “frozen” nonequilibrium redistribution. For small concentrations x, strong dielectric relaxation is observed in the temperature range of 150–250 K. A model of relaxing centers, which is based on the local charge compensation of heterovalent impurities, has been proposed.     

Low temperature specific heat of (Mo0.60Ru0.40)100−xBx metallic glasses: Evidence for compositional variation of atomic-scale structure

The heat capacity of (Mo_0.60Ru_0.40)_100?xB_x metallic glasses has been measured in the temperature range 2–15 K for glasses with 14 ≤ × ≤ 22. The electronic specific heat coefficient shows a large monotonic variation as a function of composition, in agreement with predictions based on the theory of dirty superconductors. In addition, the Debye temperature is observed to vary sharply near x = 18% boron. These effects are discussed in terms of a change in the short range order of the metal atoms. These materials appear to be BCS-like superconductors with weak to intermediate electron-phonon coupling.     

Lie symmetry analysis,conservation laws and explicit solutions for the time fractional Rosenau-Haynam equation

Under investigation in this paper is the invariance properties of the time-fractional Rosenau–Haynam equation, which can be used to describe the formation of patterns in liquid drops. Using the Lie group analysis method, the vector fields and symmetry reductions of the equation are derived, respectively. Moreover, based on the power series theory, a kind of explicit power series solutions for the equation is well constructed with a detailed derivation. The wave propagation pattern of these solutions are presented along the x axis with different t. Finally, using the new conservation theorem, two kinds of conservation laws of the equation are well constructed with a detailed derivation.     

A new global potential energy surface of the ground state of SiH2+ (X2A1) system and dynamics calculations of the Si+ + H2 (v0 = 2, j0 = 0) → SiH+ + H reaction

A global potential energy surface (PES) of the ground state of SiH$_{2}^{+}$ system is built by using neural network method based on 18223 ab initio points. The topographic properties of PES are presented and compared with previous theoretical and experimental studies. The results indicate that the spectroscopic parameters obtained from the new PES are in good agreement with the experimental data. In order to further verify the validity of the new PES, a test dynamics calculation of the Si$^{+} +$ H$_{2}$ ($v_0 = 2, j_{0} = 0$) $\to $ H $+$ SiH$^{+}$ reaction has been carried out by using the time-dependent wave packet method. The integral cross sections and rate constants are computed for the title reaction. The reasonable dynamical behavior indicates that the newly constructed PES is suitable for relevant dynamics investigations.     

Dark energy and graviton mass in the nearby universe

The problem of the local Hubble flow on scales of several Mpc induced by the dark energy realized by a scalar quintessence field is considered within the framework of relativistic gravity theory (RGT). The observational Hubble Space Telescope data are shown to be well described in RGT by a model similar to the Chernin–Baryshev–Teerikorpi model constructed in general relativity, with the local Hubble constant being smaller than the cosmological Hubble constant. A stringent constraint has been placed on the quintessence parameter, 0 ≤ ν ≤ 0.05.     

Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition

The axisymmetric time-fractional diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a cylinder under the prescribed linear combination of the values of the sought function and the values of its normal derivative at the boundary. The fundamental solutions to the Cauchy, source, and boundary problems are investigated. The Laplace transform with respect to time and finite Hankel transform with respect to the radial coordinate are used. The solutions are obtained in terms of Mittag-Leffler functions. The numerical results are illustrated graphically.     

Natural convection with damped thermal flux in a vertical circular cylinder

Natural convection flows of an incompressible Newtonian fluid inside a circular cylinder are studied. The heat transfer process is described by a generalized fractional constitutive equation for the thermal flux-temperature gradient. Caputo time-fractional derivative operator, which provides the damping of thermal flux, is considered into the studied model.Analytical solutions to the fluid temperature, thermal flux, fluid velocity and volume flow rate are obtained with the integral transforms method (Laplace transform and finite Hankel transform).Temperature behaviors for small and large values of the time t, as well as the post-transient and transient velocity components are determined. The influence of the memory parameter (the order of the time-fractional derivative) on the temperature, thermal flux, velocity and the volume flow rate is numerically and graphically studied.     

Phase transitions and spin excitations in new multiferroics with modulated magnetic structure

Coexistence of an antiferromagnetic (modulated) structure and electric polarization has been revealed in single crystals of Eu_{1 ? x} Y_xMnO₃ (0.2 ≤ x ≤ 0.5) and Gd_{1 ? x} Y_xMnO₃ (0 ≤ x ≤ 0.2) manganites. Hence, these compounds can be considered as a new family of multiferroics. Various phase transitions, both spontaneous and induced by magnetic fields up to 250 kOe, accompanied by anomalies in magnetization, magnetostriction, permittivity, and electric polarization, have been found, and phase T-x diagrams have been constructed. In the submillimeter range (8–40 cm^?1), new spin excitations—electromagnons—have been revealed; they are excited by an electric field. It is established that suppression of the modulated structure by a magnetic field leads to the disappearance of electromagnons; this process is accompanied by significant changes in the permittivity in a wide frequency range.     

Mathematical simulation of melting inside a square cavity with a local heat source

Numerical simulation of transient melting regimes inside an enclosure in the presence of a local heat source has been carried out. Mathematical model formulated in terms of dimensionless variables such as stream function, vorticity, and temperature has been numerically solved by finite difference method. Effects of the Rayleigh number 4·10⁵ ≤ Ra ≤ 5·10⁷, Stefan number 2.21 ≤ Ste ≤ 5.53, and dimensionless time on velocity and temperature fields as well as on the local Nusselt number along the heat source surface have been analyzed in detail. The transient effects of the considered process at high values of the Rayleigh number have been identified.