Comparing Cattaneo and Fractional Derivative Models for Heat Transfer Processes

More around Cattaneo equation to describe heat transfer processes

“…Incidentally, the continuous dependence of solutions to Cattaneo equation on was studied, eg, in Franchi and Straughan, 28 while a pointwise estimate for the difference between the solutions to Cattaneo and heat equations was provided in Vergara-Caffarelli and Virga. 29 In agreement with these results, it was found in Ferrillo et al 24 that indeed the solution to the initial-value problem (IVP) for (4) can be uniformly approximated as → 0 + by the solution to the heat equation, despite the fact that Cattaneo equation is singularly perturbed, and the difference between such solutions is of order of ()( ) over all closed domain. This can be seen by Fourier transforming in space and Laplace transforming in time (4) and was also confirmed by the numerical solution.…”

“…17 The latter paper includes Jeffreys-type (lagging) models, Tzou's lagged and dual-phase lagged models, 21,22 and the Green and Naghdi models. Some others, based on fractional partial differential equations (PDEs) were also proposed recently, see Fabrizio et al 23 and Ferrillo et al 24 Yet we do not claim to be exhaustive.…”

“…This can be seen by Fourier transforming in space and Laplace transforming in time (4) and was also confirmed by the numerical solution. 24 The same integral transform approach shows that, in general, the solution to the fourth-order PDE in (1) behaves similarly. In fact, Fourier transforming (1)…”

“…and hence, 1 − Dk 2 > 0. It may appear surprising that no initial layer is needed to represent the limiting behavior of the solutions to such equation, when associated to an IVP, as → 0 + , as it was pointed out also for the simpler case of Cattaneo equation in Ferrillo et al 24 : The solution to any IVP for (1) tends to that of the heat equation uniformly on any closed time interval like [0, T], for any fixed T > 0. Its time derivative however tends nonuniformly to some limit as → 0 + , while higher time derivatives will be initially unbounded.…”

Exact Solutions of the Two-Dimensional Cattaneo Model Using Lie Symmetry Transformations

“…Compte and Metzler [9] generalized the Cattaneo model from three aspects which are continuoustime random walks, delayed flux-force relation, and nonlocal transport theory. Ferrillo et al [10] compared the Cattaneo model and fractional Cattaneo model and investigated the asymptotic behavior of solutions to the Cattaneo equations. Su et al [11] compared the solution of a phase-lagging heat transport equation with the solution of the classical Cattaneo equation.…”