Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems

In graded index medium, the ray goes along a curved path determined by Fermat principle, and the curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectory, the methods not based on ray-tracing technique need to be developed for the solution of radiative transfer in graded index medium. For this purpose, in this paper the streaming operator along a curved ray trajectory in original radiative transfer equation for graded index medium is transformed and expressed in spatial and angular ordinates and the radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems are derived. The conservative and the non-conservative forms of radiative transfer equation for three-dimensional graded index medium are given, which can be used as base equations to develop the numerical simulation methods, such as finite volume method, discrete ordinates method, and finite element method, for radiative transfer in graded index medium in cylindrical and spherical coordinate systems.     

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.     

The propagation properties and kurtosis parametric characteristics of Hermite-cosh-Gaussian beams passing through fractional Fourier transformation systems

Based on the Collins diffraction integral formula and irradiance moments, the propagation properties and kurtosis parametric characteristics of Hermite-cosh-Gaussian (HChG) beams passing through fractional Fourier transformation systems with spherically aberrated lenses are analyzed in detail. The closed-form expressions for the propagation equation and kurtosis parameter of HChG beams passing ideal fractional Fourier transformation systems are also obtained. By using an efficient algorithm some numerical calculations are also performed in this paper. The results show that the spherically aberrated lenses greatly influence on the intensity distribution and kurtosis parameter of HChG beams passage through fractional Fourier transformation systems. The intensity distribution and kurtosis parameter of HChG beams passage through two types of Lohmann's systems with spherically aberrated lenses, respectively, are no longer equal even in the case of the same fractional order and the same spherical aberration coefficients. The fundamental periods of the two types of Lohmann's systems with spherically aberrated lenses are different, 4 for the type I and 2 for the type II. Therefore, the two optical setups for implementing the fractional Fourier transformation are no longer equivalent in the presence of spherically aberrated lenses.     

The stationary state and the heat equation for a variant of Davies' model of heat conduction

We study a variant of Davies' model of heat conduction, consisting of a chain of (classical or quantum) harmonic oscillators, whose ends are coupled to thermal reservoirs at different temperatures, and where neighboring oscillators interact via intermediate reservoirs. In the weak coupling limit, we show that a unique stationary state exists, and that a discretized heat equation holds. We give an explicit expression of the stationary state in the case of two classical oscillators. The heat equation is obtained in the hydrodynamic limit, and it is proved that it completely describes the macroscopic behavior of the model.     

Gradient generalization to the extended thermodynamic approach and diffusive-hyperbolic heat conduction

We study a generalization of irreversible thermodynamics with nonlocal closing relation. Thus parabolic and hyperbolic models can be described within one single theory. In the 1-d case, Guyer–Krumhansl equation and classical Fourier heat conduction may be obtained, depending on the constitutive assumptions. The thermodynamical restrictions in form of the Clausius–Duhem inequality are studied taking into account an extra flux of entropy corresponding to nonlocal irreversible effects. Numerical solutions to the resulting initial-boundary value problem are calculated and compared with available experimental results.     

The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics

By introducing a new general ansätz, the improved fractional sub-equation method is proposed to construct analytical solutions of nonlinear evolution equations involving Jumarie?s modified Riemann-Liouville derivative. By means of this method, the space-time fractional Whitham-Broer-Kaup and generalized Hirota-Satsuma coupled KdV equations are successfully solved. The obtained results show that the proposed method is quite effective, promising and convenient for solving nonlinear fractional differential equations.