COMPARING CATTANEO AND FRACTIONAL DERIVATIVE MODELS FOR HEAT TRANSFER PROCESSES

被引:13
作者
Ferrillo, Francesca [1 ]
Spigler, Renato [1 ]
Concezzi, Moreno [1 ]
机构
[1] Roma Tre Univ, Dept Math & Phys, 1 Largo S Leonardo Murialdo, I-00146 Rome, Italy
关键词
Cattaneo-Maxwell-Vernotte equation; heat equation; fractional differential equations; Caputo derivative; singular perturbations; 2ND SOUND; IRREVERSIBLE THERMODYNAMICS; LA CHALEUR; HELIUM II; CONDUCTION; WAVES; THERMOELASTICITY; DIFFUSION; LEQUATION; GAS;
D O I
10.1137/17M1135918
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We compare the model of heat transfer proposed by Cattaneo, Maxwell, and Vernotte with another one, formulated in terms of fractional differential equations, in one and two dimensions. These are only some of the numerous models that have been proposed in the literature over many decades to model heat transport and possibly heat waves, in place of the classical heat equation due to Fourier. These models are characterized by sound as well as by critical properties. In particular, we found that the Cattaneo model does not exhibit necessarily oscillations or negative values of the (absolute) temperature when the relaxation parameter, tau, drops below some value. On the other hand, the fractional derivative model may be affected by oscillations, depending on the specific initial profile. We also estimate the error made when the Cattaneo equation is adopted in place of the heat equation, and show that the approximation error is of order tau. Moreover, the solution of the Cattaneo equation converges uniformly to that of the heat equation as tau -> 0(+) in the full closed time interval [0,T] (for any given T > 0), while this does not occur for the time derivative, and the higher-order time derivatives blow up.
引用
收藏
页码:1450 / 1469
页数:20
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