THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre(generalized Poisson equation).As a first step,the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence,and,as a second step,makes use of the semigroup or the reproducing kernel property of each of the expanding entries.Experiments show the effectiveness and efficiency of the proposed series solutions.

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.

Performance enhancement of a DC-operated micropump with electroosmosis in a hybrid nanofluid:fractional Cattaneo heat flux problem

The purpose of this investigation is to theoretically shed some light on the effect of the unsteady electroosmotic flow(EOF)of an incompressible fractional secondgrade fluid with low-dense mixtures of two spherical nanoparticles,copper,and titanium.The flow of the hybrid nanofluid takes place through a vertical micro-channel.A fractional Cattaneo model with heat conduction is considered.For the DC-operated micropump,the Lorentz force is responsible for the pressure difference through the microchannel.The Debye-H¨ukel approximation is utilized to linearize the charge density.The semianalytical solutions for the velocity and heat equations are obtained with the Laplace and finite Fourier sine transforms and their numerical inverses.In addition to the analytical procedures,a numerical algorithm based on the finite difference method is introduced for the given domain.A comparison between the two solutions is presented.The variations of the velocity heat transfer against the enhancements in the pertinent parameters are thoroughly investigated graphically.It is noticed that the fractional-order parameter provides a crucial memory effect on the fluid and temperature fields.The present work has theoretical implications for biofluid-based microfluidic transport systems.

Fractional Cattaneo heat equation in a semi-infinite medium

To better describe the phenomenon of non-Fourier heat conduction, the fractional Cattaneo heat equation is introduced from the generalized Cattaneo model with two fractional derivatives of different orders. The anomalous heat conduction under the Neumann boundary condition in a semi-infinity medium is investigated. Exact solutions are obtained in series form of the H-function by using the Laplace transform method. Finally, numerical examples are presented graphically when different kinds of surface temperature gradient are given. The effects of fractional parameters are also discussed.

Fractional Order Generalized Thermoelastic Infinite Medium with Cylindrical Cavity Subjected to Harmonically Varying Heat

In this work, a mathematical model of an elastic material with cylindrical cavity will be constructed. The governing equations will be taken into the context of the fractional order generalized thermoelasticity theory (Youssef 2010). Laplace transform and direct approach will be used to obtain the solution when the boundary of the cavity is exposed to harmonically heat with constant angular frequency of thermal vibration. The inverse of Laplace transforms will be computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to present the effect of the fractional order parameter and the angular frequency of thermal vibration on all the studied felids.

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.

Daftardar-Jafari Method for Fractional Heat-Like and Wave-Like Equations with Variable Coefficients

In this paper, Daftardar-Gejji and Jafari method is applied to solve fractional heat-like and wave-like models with variable coefficients. The method is proved for a variety of problems in one, two and three dimensional spaces where analytical approximate solutions are obtained. The examples are presented to show the efficiency and simplicity of this method.

Continued Fraction Method for Approximation of Heat Conduction Dynamics in a Semi-Infinite Slab

Heat conduction dynamics are described by partial differential equations. Their approximations with a set of finite number of ordinary differential equations are often required for simpler computations and analyses. Rational approximations of the Laplace solutions such as the Pade approximation can be used for this purpose. For some heat conduction problems appearing in a semi-infinite slab, however, such rational approximations are not easy to obtain because the Laplace solutions are not analytic at the origin. In this article, a continued fraction method has been proposed to obtain rational approximations of such heat conduction dynamics in a semi-infinite slab.

Application of the Homotopy Perturbation Method to Nonlinear Heat Conduction and Fractional Van der Pol Damped Nonlinear Oscillator

In this paper, a powerful analytical method, called He’s homotopy perturbation method is applied to obtaining the approximate periodic solutions for some nonlinear differential equations in mathematical physics via Van der Pol damped non-linear oscillators and heat transfer. Illustrative examples reveal that this method is very effective and convenient for solving nonlinear differential equations. Comparison of the obtained results with those of the exact solution, reveals that homotopy perturbation method leads to accurate solutions.

Natural Convection of Water-Based Nanofluids in a Square Enclosure with Non-Uniform Heating of the Bottom Wall

In this paper, a numerical study of natural convection in a square enclosure with non-uniform temperature distribution maintained at the bottom wall and filled with nanofluids is carried out using different types of nanoparticles. The remaining walls of the enclosure are kept at a lower temperature. Calculations are performed for Rayleigh numbers in the range 5 × 103 ≤ Ra ≤ 106 and different solid volume fraction of nanoparticles 0 ≤ χ ≤ 0.2. An enhancement in heat transfer rate is observed with the increase of nanoparticles volume fraction for the whole range of Rayleigh numbers. It is also observed that the heat transfer enhancement strongly depends on the type of nanofluids. For Ra = 106, the pure water flow becomes unsteady. It is observed that the increase of the volume fraction of nanoparticles makes the flow return to steady state.

Reconstruct the Heat Conduction Model with Memory Dependent Derivative

The classical heat conduction equation is derived from the assumption that the temperature increases immediately after heat transfer, but the increase of temperature is a slow process, so the memory-dependent heat conduction model has been reconstructed. Numerical results show that the solution of the initial boundary value problem of the new model is similar to that of the classical heat conduction equation, but its propagation speed is slower than that of the latter. In addition, the propagation speed of the former is also affected by time delay and kernel function.

Fractal and Fractional Diffusion Equations of Price Changing of Commodity

In this paper, three types of modeling of diffusion equations for price changing of commodity are studied. In which, the partial derivatives of price of commodity respected to time on the left hand side are integer-derivative, fractal derivative, and fractional derivative respectively;while just a second order derivative respected to space is considered on the right hand side. The solutions of these diffusion equations are obtained by method of departing variables and initial boundary conditions, by translation of variables, and by translation of operators. The definitions of order of commodity x and the distance between commodity?xi and xj are defined as [1]. Examples of calculation of price of pork, beef and mutton mainly due to price raising of pork in 2007-07 to 2008-02 inChina are given with same market data as [1]. Conclusion is made.

IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

In this paper, we identify a space-dependent source for a fractional diffusion equation. This problem is ill-posed, i.e., the solution （if it exists） does not depend continuously on the data. The generalized Tikhonov regularization method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained, Numerical examples are presented to illustrate the validity and effectiveness of this method.

Heat Transfer Enhancement of Cu-H₂O Nanofluid with Internal Heat Generation Using LBM

Fluid flow and heat transfer analysis of Cu-H2O nanofluid in a square cavity using a Thermal Lattice Boltzmann Method (TLBM) have been studied in the present work. The LBM has built up on the D2Q9 model and the single relaxation time method called the Lattice-BGK (Bhatnagar-Gross-Krook) model. The effect of suspended nanoparticles on the fluid flow and heat transfer analysis have been investigated for different non dimensional parameters such as particle volume fraction (φ) and particle diameters (dp) in presence of internal heat generation (q) of nanoparticles. It is seen that flow behaviors and the average rate of heat transfer in terms of the Nusselt number (Nu) as well as the thermal conductivity of nanofluid are effectively changed with the different controlling parameters such as particle volume fraction (2% ≤ φ ≤ 10%), particle diameter (dp = 5 nm to 40 nm) with fixed Rayleigh number, Ra = 105. The present results of the analysis are compared with the previous experimental and numerical results for both pure and nanofluid and it is seen that the agreement is good indeed among the results.

Microstructural evolution of a PM TiAl alloy during heat treatment in α+γ phase field

In this study, the effect of temperatures and cooling rates of heat treatment on the microstructure of a powder metallurgy (PM) Ti-46Al-2Cr-2Nb-(B,W) (at.%) alloy was studied. Depending on the cooling rate and temperature, the different structures were obtained from the initial near-γ (NG) microstructures by heat treatment in the α+γ field. The results show that the microstructures of samples after furnace cooling (FC) consist primarily of equiaxed γ and α 2 grains, with a few grains containing lamellae. Duplex microstructures consist mainly of γ grains and lamellar colonies were obtained in the quenching into another furnace at 900°C (QFC) samples. However, further increasing of the cooling rate to air cooling (AC) induces the transformation of α→α_2 and results in a microstructure with equiaxed γ and α_2 grains, and no lamellar colonies are found.

Identification and PID Control for a Class of Delay Fractional-order Systems

In this paper, a new model identification method is developed for a class of delay fractional-order system based on the process step response. Four characteristic functions are defined to characterize the features of the normalized fractional-order model. Based on the time scaling technology, two identification schemes are proposed for parameters U+02BC estimation. The scheme one utilizes three exact points on the step response of the process to calculate model parameters directly. The other scheme employs optimal searching method to adjust the fractional order for the best model identification. The proposed two identification schemes are both applicable to any stable complex process, such as higher-order, under-damped U+002F over-damped, and minimum-phase U+002F nonminimum-phase processes. Furthermore, an optimal PID tuning method is proposed for the delay fractional-order systems. The requirements on the stability margins and the negative feedback are cast as real part constraints U+0028 RPC U+0029 and imaginary part constraints U+0028 IPC U+0029. The constraints are implemented by trigonometric inequalities on the phase variable, and the optimal PID controller is obtained by the minimization of the integral of time absolute error U+0028 ITAE U+0029 index. Identification and control of a Titanium billet heating process is given for the illustration. © 2014 Chinese Association of Automation.

FINITE DIFFERENCE FRACTIONAL STEP METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

Characteristic finite difference fractional step schemes are put forward. The electric potential equation is described by a seven-point finite difference scheme, and the electron and hole concentration equations are treated by a kind of characteristic finite difference fractional step methods. The temperature equation is described by a fractional step method. Thick and thin grids are made use of to form a complete set. Piecewise threefold quadratic interpolation, symmetrical extension, calculus of variations, commutativity of operator product, decomposition of high order difference operators and prior estimates are also made use of. Optimal order estimates in l2 norm are derived to determine the error of the approximate solution. The well-known problem is thorongley and completely solred.

Experimental Study of Filling Ratio Effect on the Thermal Performance in a Multi-Heat Pipe with Graphene Oxide/Water Nanofluids

This experimental study is performed to investigate heat transfer performance of a multi-heat pipe cooling device in the condition of different filling ratios (40%, 60%, 80% and 100%) and different constant heat fluxes (10 - 30 W). Here, pure water (distilled water) and graphene oxide (GO)/water nanofluids are employed respectively as working fluid. GO/water nanofluids were synthesized by the modified Hummers method with 0.05%, 0.10%, 0.15%, and 0.20% volume concentrations. Multi-heat pipe is fabricated from copper;the heating and cooling sections are the same size and both are connected by four circular parallel tubes. Temperature fields and thermal resistance are measured for different filling ratio, heat fluxes and volume concentrations. The results indicated that the thermal performance of heat pipe increased with increasing the concentration of GO nanoparticles in the base fluid, while the maximum heat transfer enhancement was observed at 0.20% volume concentration. GO/water nanofluids showed lower thermal resistance compared to pure water;the optimal thermal resistance was obtained at 100% filling charge ratio with 0.20% volume concentration. Studies were also demonstrated that heat transfer coefficient of the heat pipe significantly increases with increasing the input heat flux and GO nanoparticles concentration.

Effects of Electrode Types on the Microstructure, Tensile and Hardness Properties of 304 L Austenitic Stainless Steel Heat-Affected Zone (HAZ)

Submerged arc welding (SAW) has been well utilised for the production of weld joints in 304 L ASS for various industrial application. However, effective performance of the material in service has been hampered by improper choice of electrode. Therefore, in this study, effects of different types of electrode on the microstructure and tensile property of type 304 L austenitic stainless steel heat-affected zone (HAZ) were studied. Chemical composition of the as-received sample was determined. A number of samples were cut from the as-received sample. Afterwards, two half were joined together with 308 L, 312 L and 316 electrodes at a controlled welding speed, current and voltage of 4.6 mm/s, 160 A and 30 V to produce a constant heat input of 626.09 J/mm. An automatic SAW machine with Model Type: DX3-301, and Frequency: 50 Hz was used. And based on ASTM standard, tensile and hardness samples were prepared from the as-received and HAZs. Tensile and hardness measurements were made. Also, specimens for microscopy studies were prepared from the HAZ and as-received samples. From the results, microstructures of the HAZs revealed varied volume fraction of austenite and ferrite phases and grain sizes, and at austenite and ferrite grain boundaries, chromium carbide formation and precipitation were observed. The weld joint produced with 308 L electrode revealed optimum UTS value and YS value of 475 and 325 respectively. While weld joint produced with 316 L electrode has superior ductility of value 41%. Irrespective of the types of electrode used, the as-received sample revealed superior tensile properties over the weld joints. Also, optimum hardness value of 45.7 HRA was obtained with 308 L. Hardness value of the as-received sample was higher than that of HAZ samples.

放射形相变蓄热器的释热特性实验与数值研究

针对太阳能热利用的时空不匹配问题,设计并制备一种放射形相变蓄热器,通过实验与数值模拟研究了蓄热器释热功率及相变材料固相率在释热过程中的响应特性,探讨了冷却工质流量、进水方式、翅片型式对释热性能的影响规律。研究结果表明:蓄热器的释热过程可划分为液态显热区间、潜热区间、固态显热区间3个阶段,潜热区间相变材料的潜热作用使得释热功率下降相对缓慢;冷却工质流量越大,释热功率下降速率越快,完全凝固时间越短;不同进水方式对换热效果影响不明显,但可改善蓄热器的换热均匀性;与平板翅片相比,放射形翅片各翅片单元间相变材料的导热更均匀,凝固时间缩短31%,提高了换热速率。