The use of energy lLagrangian coordinates for a numerical solution of the one-dimensional heat conduction equation

1986 ◽  
Vol 26 (4) ◽  
pp. 195-198
Author(s):  
Yu.A. Bondarenko ◽  
V.N. Sofronov
Keyword(s):  
Heat Conduction ◽  
Numerical Solution ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
One Dimensional ◽  
The One

scholarly journals Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation

2020 ◽  
Vol 4 (3) ◽  
pp. 32
Author(s):  
Emilia Bazhlekova ◽  
Ivan Bazhlekov
Keyword(s):  
Heat Conduction ◽  
Fundamental Solution ◽  
Heat Conduction Equation ◽  
Constitutive Law ◽  
Conduction Equation ◽  
Diffusion Regime ◽  
One Dimensional ◽  
Different Types ◽  
Propagation Regime ◽  
The One

The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag–Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs.

Research on the Issue of LED’s Cooling Based on Heat Conduction Equations

2012 ◽  
Vol 507 ◽  
pp. 137-141
Author(s):  
Zhi Qin Huang ◽  
Pei Ying Quan ◽  
Yong Qing Pan
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Heat Conduction Equation ◽  
Rapid Development ◽  
Three Dimensional ◽  
Conduction Equation ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
One Dimensional ◽  
The One

With the rapid development of power type LED, the issue of the cooling of LED has been prominent. How to make the heat generated by LED chip go out quickly in order to cool the LED chip has become an urgent problem. The form of heat goes through the substrate has been widely used and has become the best way to solve the heat problem. There are three types of LED substrate. They are metal substrate, ceramic substrate and composite substrate. At first, In this paper I analyze the theoretical of three-dimensional non-steady state and steady state heat conduction equation, then the three-dimensional model is simplified as one-dimensional model and I get the results of heat conduction equation under the one-dimensional stationary and non-steady state.

scholarly journals Fractional Heat Conduction Models and Thermal Diffusivity Determination

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Monika Žecová ◽  
Ján Terpák
Keyword(s):  
Heat Conduction ◽  
Thermal Diffusivity ◽  
Numerical Methods ◽  
Fractional Order ◽  
Experimental Verification ◽  
Heat Conduction Equation ◽  
Historical Overview ◽  
One Dimensional ◽  
The One ◽  
Fractional Heat Conduction

The contribution deals with the fractional heat conduction models and their use for determining thermal diffusivity. A brief historical overview of the authors who have dealt with the heat conduction equation is described in the introduction of the paper. The one-dimensional heat conduction models with using integer- and fractional-order derivatives are listed. Analytical and numerical methods of solution of the heat conduction models with using integer- and fractional-order derivatives are described. Individual methods have been implemented in MATLAB and the examples of simulations are listed. The proposal and experimental verification of the methods for determining thermal diffusivity using half-order derivative of temperature by time are listed at the conclusion of the paper.

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