The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H-function using integral transformation methods. The fractional heat conduction equation in the case 0<α≤1 interpolates the standard heat conduction equation (α=1) and the Localized heat conduction equation (α→0). Finally, numerical results are presented graphically for various values of order of fractional derivative.     

Generalized density expansion in the kinetic theory and the resistivity of liquid metals

For the system of electrons and immovable interacting centers an exact equation for averaged electron Green's function is formulated. The expansion of self-energy part over the one-particle t-matrices and explicit Green's functions is derived. It represents a kind of a generalized density series containing the correlation functions of the centres. In the low approximation over t-matrix, the transition probability (t)²S in the kinetic equation is obtained (S = the structure factor of centers).     

On coherent scattering in stellar atmospheres

An exact solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a sum of elementary functions, is obtained in a simple form with the help of the author's new representation of H-functions of Radiative Transfer.     

Thermodynamic,elastic, elastic anisotropy and minimum thermal conductivity of β-GaN under high temperature

The thermodynamic, elastic, elastic anisotropy and minimum thermal conductivity of β-GaN are investigated at ambient pressure and high temperature by using first-principles calculations method with the ultrasoft psedopotential scheme. The elastic constants calculations reveal β-GaN is mechanically stability at ambient pressure and high temperature. The elastic modulus (Poisson's ratio, shear modulus and Young's modulus) decreases with increasing temperature. The calculations of anisotropy show that β-GaN has a larger elastic anisotropy in Poisson's ratio, shear modulus, Young's modulus and Zener anisotropy index. In addition, when the temperature increases from 0 to 1500 K, the elastic anisotropy decreases for β-GaN. The quasi-harmonic Debye model is successfully applied to determine the thermodynamic properties at different pressures and temperatures. Using the quasi-harmonic Debye model, the thermodynamic properties including the Debye temperature, Grüneisen parameter, the heat capacity, adiabatic bulk modulus, and the thermal expansion coefficients of β-GaN are predicted under high temperature and high pressure.     

Green's function theory of the Kondo effect in dilute magnetic alloys

The exact solution of thet-matrix integral equation derived from the self-consistent Nagaoka equations in the theory of dilute magnetic alloys is established. It is shown that the unique solution for thet-matrix involving all Kondo type anomalies can be found under quite general assumptions. Using the exact solution we have calculated thermodynamic properties of dilute magnetic alloys. It is found that the excessive specific heat of the system due to the anomalous scattering of conduction electrons from the magnetic impurities is of the order ofBoltzmann's constant per local moment at low temperatures. In the limit of vanishing temperature the specific heat goes to zero asymptotically as (lnT)^?4. Finally the entropy difference of the interacting system as compared to the free system is calculated.     

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.     

Parabolic sturmians approach to the three-body continuum Coulomb problem

The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger-type equation, where the Green’s function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the (e ^?, e ^?, He⁺⁺) system has been solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of the expansion coefficients of the solution has been obtained as the basis set used to describe the potential is enlarged.     

Φ3 theory in six dimensions and the renormalization group

Φ³ field theory in six space-time dimensions is used as a testing ground for the renormalization group. The consistency of the new method devised by 't Hooft is verified at the two-loop level by exhibiting certain cancellations among single pole coefficients, and also relations among double and single pole coefficients. This calculation is contrasted in efficiency with one using the Gell-Mann-Low equation in dimensionally regularized form for which a solution is obtained at the one-loop level, without neglecting mass dependence.The theory shares with asymptotically free ones the virtue of an asymptotically vanishing effective coupling constant. Indeed, the next to leading term in the beta function is also of opposite sign to the coupling constant.     

Lindemann's criterion and the melting of solids at high pressures

The generalized Lindemann's criterion is utilized to derive the variation of the melting temperature T_m with compression. The result incorporates the basic features of the lattice potential as well as the volume dependence of the frequency distribution through the first two moments of the mode Grüneisen parameter. This melting relation predicts that when T_m is plotted against the volume of the melting solid, the fusion curves of solids bonded by Van der Waals forces are concave to the temperature axis and in accordance with the predictions of the Simon equation. On the other hand, the predicted melting curves of ionic compounds, as the alkali-halides, are concave to the volume axis and exhibit a maxima at higher pressures. Since silicates generally possess an ionic contribution to the lattice energy, this melting relation should give a more accurate estimate of the high pressure melting temperature for these compounds than Simon's and theoretically similar equations.     

Semiclassical quantization of the nonlinear Schrödinger equation

Using the functional integral technique of Dashen, Hasslacher, and Neveu, we perform a semiclassical quantization of the nonlinear Schrödinger equation, which reproduces McGuire's exact result for the energy levels of the theory's bound states. We show that the stability angle formalism leads to the one-loop normal ordering and self-energy renormalization expected from perturbation theory and demonstrate that taking into account center-of-mass motion gives the correct nonrelativistic energymomentum relation. We interpret the classical solution in the context of the quantum theory, relating it to the matrix element of the field operator between adjacent bound states in the limit of large quantum numbers. Finally, we quantize the NLSE as a theory of N component fermion fields and show that the semiclassical method yields the exact energy levels and correct degeneracies.     

Facteurs de structures partiels et grandeurs thermodynamiques dans les alliages liquides

From the calculation of partial structure factors in liquid alloys developed by Bhatia and Thornton, the authors establish a generalized relationship between these structure factors and the thermodynamic parameters in dilute solutions by introducing Wagner's interaction coefficients. The validity of using models describing the interactions in solution to establish such relationship is discussed.     

Localized heat perturbation in harmonic 1D crystals: Solutions for the equation of anomalous heat conduction

In this paper exact analytical solutions for the equation that describes anomalous heat propagation in a harmonic 1D lattices are obtained. Rectangular, triangular and sawtooth initial perturbations of the temperature field are considered. The solution for an initially rectangular temperature profile is investigated in detail. It is shown that the decay of the solution near the wavefront is proportional to \(1/\sqrt t \). In the center of the perturbation zone the decay is proportional to 1/t. Thus, the solution decays slower near the wavefront, leaving clearly visible peaks that can be detected experimentally.     

Superconductivity in the ternary borides CeOs3B2 and CeRu3B2: Magnetic susceptibility and specific heat measurements

The compounds CeOs₃B₂ and CeRu₃B₂ show superconductivity with T_c's of 3.5 and 1.1 K respectively. The magnetic susceptibility of both these compounds may be described as the sum of a Curie-Weiss term dominating at low temperature and a large temperature independent term. The heat capacity measurements yield the value of electronic specific heat coefficient γ of 40.1 and 15.6 mJ/mol K² for Os and Ru compounds, respectively. These compounds appear to be conventional bulk superconductors in which Ce is in a strongly mixed valent state.     

The generalized Fubini instanton

We show that (1+2) nonlinear Klein-Gordon equation with negative coupling admits an exact solution which appears to be the linear superposition of the plane wave and the nonsingular rational soliton. We show that the same approach allows to construct the solution of similar properties for the Euclidean ?⁴ model with broken symmetry. Interestingly, this regular solution will be of instanton type only in the D?5 Euclidean space. Thus one can use the generalized Fubini instantons (in quantum cosmology for example) only for the case of the single infinite extra dimension.     

A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations

With the aid of symbolic computation, a generalized auxiliary equation method is proposed to construct more general exact solutions to two types of NLPDEs. First, we present new family of solutions to a nonlinear Klein-Gordon equation, by using this auxiliary equation method including a new first-order nonlinear ODE with six-degree nonlinear term proposed by Sirendaoreji. Then, we apply an indirect F-function method very close to the F-expansion method to solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l,n,p). Taking advantage of the new first-order nonlinear ODE with six degree nonlinear term, this indirect F-function method is used to map the solutions of C(l,n,p) equations to those of that nonlinear ODE. As a result, we can successfully obtain in a unified way, many exact solutions.     

Absorption intensities for vibration-rotation bands and the dipole-moment expansion of HBr

Line intensities in the four 0–2, 0–3, 0–4, 0–5 vibration-rotation bands of HBr were measured. The electric dipole matrix elements 〈0|M|ν'〉 for vibrational transitions with ν'?5 have been calculated by using a Dunham potential and the analytical solution of the Schrödinger equation described by R.H. Tipping. A dipole-moment expansion accurate to M₅ was determined by fitting these matrix elements to the available experimental data on line intensities. The experimental results were also used to calculate the Einstein coefficients of the R₀ lines for the bands ν' → ν′' with ? ν' ? 5.