A Finite Difference Approach to the Diffusion Equation in Electrical Conductivity Tomography

in the study of electrical conductivity tomography using low frequency electromagnetic waves,the equation that describes the EM field is a diffusion equation. In this paper,a stable finite difference method called Peaceman-Rachford scheme is used to solve the diffusion equation. The solution is derived in detail and a computer simulation is given. The calculated EM data are shown in waveforms. The simulation result shows that this method is simple and accurate.The EM data in time domain can be transformed to a wave field in time-like domain by an integral transformation and the traveltime data can be calculated. Compared with the theoretical value, the first travel time data obtained through transformation have a normalized mean square error or 5 percent, which proves the method to be accurate.

An Efficient Approach for Solving One-Dimensional Fractional Heat Conduction Equation

Several researchers have dealt with the one-dimensional fractional heat conduction equation in the last decades,but as far as we know,no one has investigated such a problem from the perspective of developing suitable fractional-order methods.This has actually motivated us to address this problem by the way of establishing a proper fractional approach that involves employing a combination of a novel fractional difference formula to approximate the Caputo differentiator of orderαcoupled with the modified three-point fractional formula to approximate the Caputo differentiator of order 2α,where 0<α≤1.As a result,the fractional heat conduction equation is then reexpressed numerically using the aforementioned formulas,and by dividing the considered mesh into multiple nodes,a system is generated and algebraically solved with the aid of MATLAB.This would allow us to obtain the desired approximate solution for the problem at hand.

Variational Approach to 2D and 3D Heat Conduction Modeling

The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.

Operator Fredholm Integration Equation in Intermediate Coordinate-Momentum Representation and Its Analogue to Thermo Conduction Equation

Using the technique of integration within an ordered product （IWOP） of operators we construct intermediate coordinate-momentum representation, with which we build a type of operator Fredholm integration equation that is an operator generalization of the solution of thermo conduction equation. Then we seach for the solution of operator Fredholm integration equations, which provides us with a new approach for deriving some operator identities.

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.

A CLASS OF TWO-LEVEL EXPLICIT DIFFERENCE SCHEMES FOR SOLVING THREE DIMENSIONAL HEAT CONDUCTION EQUATION

A class of two-level explicit difference schemes are presented for solving three-dimensional heat conduction equation. When the order of truncation error is 0(Deltat + (Deltax)(2)), the stability condition is mesh ratio r = Deltat/(Deltax)(2) = Deltat/(Deltay)(2) = Deltat/(Deltaz)(2) less than or equal to 1/2, which is better than that of all the other explicit difference schemes. And when the order of truncation error is 0((Deltat)(2) + (Deltax)(4)), the stability condition is r less than or equal to 1/6, which contains the known results.

Prediction of the Conductance of Strong Electrolytes and the Calculation of the Ionization Constant of Weak Electrolytes in a Dilute Solution by a New Equation

In order to predict the conductance for dilute 1 1 valent electrolyte solutions, a new conductance equation was proposed based on the Onsager and Onsagar Fuoss Chen conductance equation. It has only one parameter A , which can be obtained directly from the data of ionic limiting molar conductivity Λ ∞ m, and its expression is very simple. The new equation has been verified by the experimental molar conductivities of some single strong electrolyte and mixed electrolyte solutions at 298.15 K reported in literatures. The results are in good agreement with the experimental data. Meanwhile the ionization constants of some weak electrolyte solutions were calculated by a modified equation of this new equation, and it was also found that the calculation results are in good agreement with the data in the literature.

3D Radiative Transfer Equation Coupled with Heat Conduction Equation with Realistic Boundary Conditions Applied on Complex Geometries

This paper presents the solution of coupled radiative transfer equation with heat conduction equation in complex three-dimensional geometries. Due to very different time scales for both physics, the radiative problem is considered steady-state but solved at each time iteration of the transient conduction problem. The discrete ordinate method along with the decentered streamline-upwind Petrov-Galerkin method is developed. Since specular reflection is considered on borders, a very accurate algorithm has been developed for calculation of partition ratio coefficients of incident solid angles to the several reflected solid angles. The developed algorithms are tested on a paraboloid-shaped geometry used for example on concentrated solar power technologies.

BLOW－UP AND DIE－OUT OF SOLUTIONS OF NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS OF GENERALIZED NERVE CONDUCTION TYPE

This paper considers the initial boundary value problems with three types of the boundary conditions for nonlinear pseudo-hyperbolic equations of generalized nerve conduction type, using foe eigenfunction method, the conditions for which the solutions blow-up and die-out in the finile time are got.

Hlder Stability Estimate for an Inverse Parabolic Problem

This paper deals with a parabolic system in a multi dimentional bounded domain ΩR n with the smooth boundary Ω. We discuss an inverse parabolic problem of determining the indirectly measurable internal heat distribution at any intermediate moment from the heat distribution measurements in arbitrary accessible subdomain ωΩ at some time interval. Our main result is the Hlder stability estimate in the inverse problem and the proof is completed with a Carleman estimate and a eigenfunction expansion for parabolic equations.

An analytical model for estimating soil temperature profiles on the Qinghai-Tibet Plateau of China

Soil temperature is a key variable in the control of underground hydro-thermal processes. To estimate soil temperature more accurately, this study proposed a solution method of the heat conduction equation of soil temperature （improved heat conduction model） by applying boundary conditions that incorporate the annual and diurnal variations of soil surface temperature and the temporal variation of daily temperature amplitude, as well as the temperature difference between two soil layers in the Tanggula observation site of the Qinghai-Tibet Plateau of China. We employed both the improved heat conduction model and the classical heat conduction model to fit soil temperature by using the 5 cm soil layer as the upper boundary for soil depth. The results indicated that the daily soil temperature amplitude can be better described by the sinusoidal function in the improved model, which then yielded more accurate soil temperature simulating effect at the depth of 5 cm. The simulated soil temperature values generated by the improved model and classical heat conduction model were then compared to the observed soil temperature values at different soil depths. Statistical analyses of the root mean square error （RMSE）, the normalized standard error （NSEE） and the bias demonstrated that the improved model showed higher accuracy, and the average values of RMSE, bias and NSEE at the soil depth of 10-105 cm were 1.41℃, 1.15℃ and 22.40%, respectively. These results indicated that the improved heat conduction model can better estimate soil temperature profiles compared to the traditional model.

A preliminary analysis on lithospheric thickness and crustal ages in central basin of South China Sea

The models about lithospheric thickness and thermal conduction inside the lithosphere and the top layer of the asthenosphere have been proposed in this study for four type regions: the midoceanic ridge, the extinct spreading ridge, the lithospheric fault fissure and the mouth of the extinct submarine volcanoes which are in deep sea bottom. The solutions of the models are found to be the same. The formulas of temperature distribution inside the lithosphere and the top layer of the asthenosphere, the lithospheric thicknesses to the heat flow and the crustal ages to the heat flow are obtained. The crustal ages and the lithospheric thicknesses of the central basin are calculated. And they are used to draw the lithospheric thicknesses and crustal ages maps of the central basin (in this paper both the central basin and the basin are the central basin of the South China Sea). According to their characteristics, the central basin is divided into three regions. The lithospheric thicknesses, crustal ages and heat flow distribution characteristics are discussed respectively. The formation and evolution of the South China Sea are analysed and it is thought that the South China Sea has undergone three episode-seafloor spreadings.

APPLICATION OF NUMERICAL SIMULATION TO STUDY ON THERMAL CONDUCTION

In this paper, using computer simulation and mathematic experiment method to solve the simplified one dimensional thermal conduction equation and to obtain the temperature distribution in a metal bar when its one end was heated. According to principle of hot expansion, a holograph of temperature distribution in the bar by laser holo-technique was taken. The results of numerical simulation and experiments are in good agreement and a new method for study on thermal conduction by laser holo-technique was found.

Thermal stresses induced by a point heat source in a circular plate by quasi-static approach

The present paper deals with the determination of quasi-static thermal stresses due to an instantaneous point heat source of strength g_(pi) situated at certain circle along the radial direction of the circular plate and releasing its heat spontaneously at time t=τ.A circular plate is considered having arbitrary initial temperature and subjected to time dependent heat flux at the fixed circular boundary of r=b.The governing heat conduction equation is solved by using the integral transform method,and results are obtained in series form in terms of Bessel functions.The mathematical model has been constructed for copper material and the thermal stresses are discussed graphically.

EQ^(rot)_1 Nonconforming Finite Element Method for Nonlinear Dual Phase Lagging Heat Conduction Equations

EQrot nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O（h2） one order higher than its interpolation error O（h）, the superclose results of order O（h2） in broken Hi-norm are obtained. At the same time, the global superconvergence in broken Hi-norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O（h4） is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQrot element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.

Solutions of the modification heating conduction equations of a kind of laser thermal effect

This paper has solved the Chester modified heat conduction equation of the different relaxation time r value under different temperature conditions, different boundary conditions and the different initial conditions by different means of methods. These solutions can help to obtain temperature field of laser thermal effects.

Mathematical Modeling of Heat Transfer Processes of Coal Waste Combustion in a Chamber of Automated Energy Generating Complex

The automated energy generating complex allows obtaining heat energy from waste coal-water slurry fuel （WCF） that is a mixture of fine coal particles from coal enrichment wastes with water. The mixture is blown into the swirl chamber under the pressure through the special sprayers. The received heat energy is used in different ways. One of the important issues is to estimate the heat losses through the walls of this chamber. In this paper we solved the boundary problem of mathematical physics to estimate the temperature fields in the walls of the swirl chamber. The obtained solution allows us to estimate the heat losses through the waUs of the swid chamber. The task of the mathematical physics has been solved by a numerical finite-difference method. The method for solving this prob- lem can be used in the calculation of temperature fields and evaluation of heat losses in other thermal power units.

Simulations of lunar equatorial regolith temperature profile based on measurements of Diviner on Lunar Reconnaissance Orbiter

Lunar equatorial regolith temperature profiles were simulated using the half-limited solid heat conduction model. Based on the infrared data measured using the Diviner radiometer on the Lunar Reconnaissance Orbiter launched by the United Sates in June 2009, three factors influencing temperature profiles were analyzed. The infrared brightness temperature data from Diviner channel 7 were used to retrieve surface temperature. In simulating regolith temperature profiles, the retrieved temperature, rather than temperatures calculated from solar radiance at the lunar surface, were used as the input for surface temperature in solving the heat-conductive equation. The results showed that the bottom-layer temperature at depths of 6 m approached almost 246 K after 10000 iterations. The temperature was different to the temperature of 250 K at the same depth encountered in simulations using solar radiance. Simulations from both methods of surface temperatures over a lunar day gave similar variations. At lunar night, the temperature difference between the two was about 2 K; the main differences occurred when the solar elevation angle was very low when surface temperatures are largely affected by terrain topography. With no certainty in lunar temperature profiles at present, the advantage of the retrieval method using infrared sensor data as input to the boundary conditions in solving the lunar heat conduction equation is that simulations of surface temperature variations are more accurate. This is especially true in areas with large variations in terrain topography, where surface temperatures vary greatly because of shading from the sunlight.