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.     

Time fractional dual-phase-lag heat conduction equation

We build a fractional dual-phase-lag model and the corresponding bioheat transfer equation, which we use to interpret the experiment results for processed meat that have been explained by applying the hyperbolic conduction. Analytical solutions expressed by H-functions are obtained by using the Laplace and Fourier transforms method. The inverse fractional dual-phase-lag heat conduction problem for the simultaneous estimation of two relaxation times and orders of fractionality is solved by applying the nonlinear least-square method. The estimated model parameters are given. Finally, the measured and the calculated temperatures versus time are compared and discussed. Some numerical examples are also given and discussed.     

The physical defects of the hyperbolic heat conduction equation

In this paper the HHCE is inspected on a microscopic level from a physical point of view. Starting from the Boltzmann transport equations we study the underlying approximations. We find that the hyperbolic approach to the heat current density violates the fundamental law of energy conservation. As a consequence, the HHCE predicts physically impossible solutions with a negative local heat content. This behaviour is demonstrated in detail for a standard problem in heat conduction, the solution for a point source. Received: 29 October 1997/Accepted: 17 February 1998     

On a generalization of the Gibbs equation for heat conduction

A generalized Gibbs equation for the heat conduction problem is proposed in order to take finite wave speed into account.     

An approximate analytic solution of the heat conduction equation at nanoscale

The Adomian decomposition method (ADM) and the Adomian double decomposition method (ADDM) for solving the 3D non-Fourier heat conduction equation at nanoscale based on the dual-phase-lag framework are proposed. We show that the noise terms that appear in ADM solution can be removed, if the ADDM is employed.     

Peculiarities of solutions to the heat conduction equation in fractional derivatives

Features of solutions to the heat conduction equation in fractional derivatives taking into account diffusion and convection mechanisms of heat transfer are analyzed. One-dimensional cases of infinite straight line, semi-infinite line, and the problem with zero initial conditions are considered.     

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.     

Thermoelastic damping in thin microrings with two-dimensional heat conduction

Accurate determination of thermoelastic damping (TED) is very challenging in the design of micro-resonators. Microrings are widely used in many micro-resonators. In the past, to model the TED effect on the microrings, some analytical models have been developed. However, in the previous works, the heat conduction within the microring is modeled by using the one-dimensional approach. The governing equation for heat conduction is solved only for the one-dimensional heat conduction along the radial thickness of the microring. This paper presents a simple analytical model for TED in microrings. The two-dimensional heat conduction over the thermoelastic temperature gradients along the radial thickness and the circumferential direction are considered in the present model. A two-dimensional heat conduction equation is developed. The solution of the equation is represented by the product of an assumed sine series along the radial thickness and an assumed trigonometric series along the circumferential direction. The analytical results obtained by the present 2-D model show a good agreement with the numerical (FEM) results. The limitations of the previous 1-D model are assessed.     

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.     

The triple integral equations method for solving heat conduction equation

A new type of triple integral equation was used to determine a solution of nonstationary heat equation in axially symmetric cylindrical coordinates under mixed discontinuous boundary of the first and second kind conditions acted on the level surface of solid cylinder, with the aid of a Laplace transform, the solution of the given triple equations is introduced to a singular integral equation of the second kind.     

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.     

On the crossing of thin shells

The crossing of two dust shells is considered as a simplified model of a collapsing thick layer of dust. We use the Israel's formalism to describe the development of two shells of dust which interact only gravitationally. The formalism is developed in both Schwarzschild and Kruskal coordinates.     

New exact solutions of nonlinear conformable time-fractional Phi-4 equation

In this paper, new exact analytical solutions of time-fractional Phi-4 equation are developed using extended direct algebraic method by means of conformable fractional derivative. The obtained new results reveal that the proposed method is effective to studythe nonlinear dispersive equations in mathematical physics.     

Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation

In this paper, we studied time-fractional nonlinear partial differential equations to reach their some solutions. There are lots of explicit and analytic methods in the literature. We used Kudryashov, Exp-function, and Jacobi elliptic rational expansion methods. By using these methods, we get some solutions of time-fractional fifth-order KdV-like equation.     

Analytical solution of hyperbolic heat conduction equation in relation to laser short-pulse heating

In the present study, the hyperbolic heat conduction equation is derived from the Boltzmann transport equation and the analytical solution of the resulting equation appropriate to the laser short-pulse heating of a solid surface is presented. The time exponentially decaying pulse is incorporated as a volumetric heat source in the hyperbolic equation to account for the absorption of the incident laser energy. The Fourier transformation is used to simplify the hyperbolic equation and the analytical solution of the simplified equation is obtained using the Laplace transformation method. Temperature distribution in space and time are computed in steel for two laser pulse parameters. It is found that internal energy gain from the irradiated field, due to the presence of the volumetric heat source in the hyperbolic equation, results in rapid rise of temperature in the surface region during the early heating period. In addition, temperature decay is gradual in the surface region and as the depth below the surface increases beyond the absorption depth, temperature decay becomes sharp.     

基于热质运动概念的普适导热定律

根据爱因斯坦的质能等效关系式,热能具有的等效质量称为热质,从而在固态和气态介质中分别建立了声子气质量和热子气质量的概念.应用牛顿定律建立了含有驱动力、阻力和惯性力的热质(声子气或热子气)运动的动量守恒方程.由于热量在介质中的传递本质上就是热质(声子气和热子气)在介质中的运动,所以热质动量守恒方程就是普适的导热定律,能够统一描述各种条件下的导热规律.当热流密度不是很大从而热质惯性力可以忽略时,热质动量守恒方程就退化为傅里叶导热定律,这表明傅里叶导热定律是特殊条件下的导热定律,对于微纳尺度条件下的导热,热流密度可以极高,由速度空间变化引起的惯性力不能忽略,在稳态导热情况下也将出现非傅里叶导热,此时在计算或者实验中不能用热流密度除温度梯度求导热系数.在超快速加热条件下,必需考虑惯性力,与基于CV导热模型的波动方程相比,普适的导热定律增加了因速度空间变化引起的惯性力项,所以在介质中热波叠加时不会出现产生负温度的非物理现象,表明基于热质运动概念的普适导热定律更为合理. 关键词： 傅里叶导热定律 普适导热定律 热质运动 非傅里叶导热     

基于热传导原理电枢平均磨损率计算

在假设导轨为半无限长固体导体及电枢表面磨损主要是熔化磨损条件下,基于固体接触面导热原理,推导出电枢和导轨温度分布方程,提出了电枢平均磨损率计算方法。利用此方法分析了驱动电流、电枢和导轨材料匹配和接触面积参数对电枢平均磨损率影响。将平均磨损率计算结果与Stefani和Parker的实验结果进行对比,二者吻合较好,从而为建立电磁轨道炮磨损率与转捩之间的关系提供理论支持。     

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.     

Analysis of a heat wave induced by laser radiation absorption in an optical fiber on the basis of a 2D nonstationary heat conduction equation

A heat wave resulting from the absorption of laser radiation in the core of an optical fiber is studied using a nonstationary 2D heat conduction equation. The velocity of the wave as a function of the laser intensity is determined, and the threshold intensity generating the heat wave is calculated. At high intensities, the velocity of the wave can be qualitatively described by a well-known formula from combustion theory; i.e., the velocity is shown to be proportional to the square root of the radiation intensity. The analytical threshold laser intensities closely agree with the available experimental data.     

Symmetries of the heat equation on the lattice

Discrete versions of the heat equation on two-dimensional uniform lattices are shown to possess the same symmetry algebra as their continuum limits. Solutions with definite symmetry properties are presented.