Generalized Cattaneo Equation Research Articles

Photothermal Response for the Thermoelastic Bending Effect Considering Dissipating Effects by Means of Fractional Dual-Phase-Lag Theory

We analyze an extension of the dual-phase lag model of thermal diffusion theory to accurately predict the contribution of thermoelastic bending (TE) to the Photoacoustic (PA) signal in a transmission configuration. To achieve this, we adopt the particular case of Jeffrey’s equation, an extension of the Generalized Cattaneo Equations (GCEs). Obtaining the temperature distribution by incorporating the effects of fractional differential operators enables us to determine the TE effects in solid samples accurately. This study contributes to understanding the mechanisms that contribute to the PA signal and highlights the importance of considering fractional differential operators in the analysis of thermoelastic bending. As a result, we can determine the PA signal’s TE component. Our findings demonstrate that the fractional differential operators lead to a wide range of behaviors, including dissipative effects related to anomalous diffusion.

Fractional dual-phase-lag heat conduction with periodic heating and photo-thermal response

We apply an extension of dual-phase-lag in thermal systems to predict the photoacoustic signal for transmission configuration and characteristics of the open photoacoustic cell technique. For this, we consider a particular case from Jeffrey?s equation as an extension of the generalized Cattaneo equations. In this context, we obtain the temperature distribution under the effects of fractional differential operators, allowing the calculation of the Photoacoustic signal for the transmission set-up. The results show a rich class of behaviors related to the anomalous diffusion connected to these fractional operators.

Integral decomposition for the solutions of the generalized Cattaneo equation

We present the integral decomposition for the fundamental solution of the generalized Cattaneo equation with both time derivatives smeared through convoluting them with some memory kernels. For power-law kernels t^{-α},α∈(0,1] this equation becomes the time fractional one governed by the Caputo derivatives in which the highest order is 2. To invert the solutions from the Fourier-Laplace domain to the space-time domain we use analytic methods based on the Efross theorem and find out that solutions looked for are represented by integral decompositions which tangle the fundamental solution of the standard Cattaneo equation with nonnegative and normalizable functions being uniquely dependent on the memory kernels. Furthermore, the use of methodology arising from the theory of complete Bernstein functions allows us to assign such constructed integral decompositions the interpretation of subordination. This fact is preserved in two limit cases built into the generalized Cattaneo equations, i.e., either the diffusion or the wave equations. We point out that applying the Efross theorem enables us to go beyond the standard approach which usually leads to the integral decompositions involving the Gaussian distribution describing the Brownian motion. Our approach clarifies puzzling situation which takes place for the power-law kernels t^{-α} for which the subordination based on the Brownian motion does not work if α∈(1/2,1].

Fractional GCEs behaviors merged: Prediction to the photoacoustic signal obtained with subdiffusive and superdiffusive operators

In this work, we developed a theoretical model with anomalous thermal diffusion by considering the modified Generalized Cattaneo Equations (GCEs) of Compte and Metzler, GCEI and GCEII, merged in a single operator by a linear combination in function of the fractional derivative order. This operator was compared with the generalization of Youssef for GCEIII. The range of the superdiffusive and subdiffusive regions was determined through phase velocity. In conclusion, both operators present unexpected sub- and superdiffusive intervals. Moreover, the temperature variation due to the heating by a pulsed light beam was determined. At the end, the photoacoustic signal for opaque samples was simulated varying the relaxation time and the fractional-order derivative.

Generalized Cattaneo (telegrapher's) equations in modeling anomalous diffusion phenomena.

We study generalized Cattaneo (telegrapher's) equations involving memory effects introduced by smearing the time derivatives. Consistency conditions where the smearing functions obey restrict freedom in their choice but the proposed scheme goes beyond the approach based on using fractional derivatives. We find conditions under which solutions of the equations considered so far can be recognized as probability distributions, i.e., are normalizable and nonnegative on their domains. Nonnegativity of solutions is demonstrated by methods of positive definite and completely monotonic functions with the Bernstein theorem being the cornerstone of the ongoing proofs. Analysis of exactly solvable examples and relevant mean-squared displacements enables us to classify diffusion processes described by such got solutions and to identify them with either ordinary or anomalous diffusion which character may change over time. To complete the present research we compare our results with those obtained using the continuous-time random-walk and the continuous-time persistent random-walk approaches.

Fractional-order heat conduction models from generalized Boltzmann transport equation.

The relationship between fractional-order heat conduction models and Boltzmann transport equations (BTEs) lacks a detailed investigation. In this paper, the continuity, constitutive and governing equations of heat conduction are derived based on fractional-order phonon BTEs. The underlying microscopic regimes of the generalized Cattaneo equation are thereafter presented. The effective thermal conductivity κeff converges in the subdiffusive regime and diverges in the superdiffusive regime. A connection between the divergence and mean-square displacement 〈|Δx|2〉 ∼ tγ is established, namely, κeff ∼ tγ-1, which coincides with the linear response theory. Entropic concepts, including the entropy density, entropy flux and entropy production rate, are studied likewise. Two non-trivial behaviours are observed, including the fractional-order expression of entropy flux and initial effects on the entropy production rate. In contrast with the continuous time random walk model, the results involve the non-classical continuity equations and entropic concepts. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

On fractional Cattaneo equation with partially reflecting boundaries

In this paper we study the time-fractional Cattaneo equation in a bounded domain with semi-reflecting conditions. In particular, we are able to find the Laplace transform of the probability density function of the absorption time and therefore the mean-time to absorption. We show the crucial role of the time-fractional formulation. Indeed, in this case we have that the mean-time to absorption diverges due to the fact that the generalized Cattaneo equation is based on the application of integral operators with a long-tail memory kernel. We also consider the time-fractional diffusion and wave limits behaviour, recovering some previous results obtained in the literature. Finally, a section is devoted to the generalized Cattaneo equation in unbounded domain. In this case we are able to discuss the characterization of the mean square displacement for short times and asymptotically by using the Fourier–Laplace transform of the solution.

Fractional Boltzmann transport equation for anomalous heat transport and divergent thermal conductivity

Anomalous heat transport and divergent thermal conductivity have attracted increasing attention in recent years. The linearized Boltzmann transport equation (BTE) proposed by Goychuk is discussed in superdiffusive and ballistic heat conduction, which is characterized by super-linear growth of the mean-square displacement (MSD) Δx2, namely, Δx2∼tγ with 1<γ⩽2. We show that this fractional-order BTE predicts a fractional-order constitutive equation and divergent effective thermal conductivity κeff. In the long-time limit, the divergence obeys a power-law type κeff∼tα, while the asymptotics of Δx2 reads γ=α+1. This connection between κeff and Δx2 coincides with previous investigations such as the linear response and Lévy-walk model. The constitutive equation from Goychuk’s model is compared with a class of fractional-order models termed generalized Cattaneo equation (GCE). We show that Goychuk’s model is more appropriate than other models of the GCE class to describe superdiffusive and ballistic heat conduction.

On Entropic Framework Based on Standard and Fractional Phonon Boltzmann Transport Equations.

Generalized expressions of the entropy and related concepts in non-Fourier heat conduction have attracted increasing attention in recent years. Based on standard and fractional phonon Boltzmann transport equations (BTEs), we study entropic functionals including entropy density, entropy flux and entropy production rate. Using the relaxation time approximation and power series expansion, macroscopic approximations are derived for these entropic concepts. For the standard BTE, our results can recover the entropic frameworks of classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT) as if there exists a well-defined effective thermal conductivity. For the fractional BTEs corresponding to the generalized Cattaneo equation (GCE) class, the entropy flux and entropy production rate will deviate from the forms in CIT and EIT. In these cases, the entropy flux and entropy production rate will contain fractional-order operators, which reflect memory effects.

Transient fractional heat conduction with generalized Cattaneo model

The generalized Cattaneo equation is a new heat conduction equation which is based on the time-nonlocal generalization of Fourier law. The fractional calculus approach is used in the generalized Cattaneo model. With the help of the Laplace transform, the exact solution of an initial-boundary value problem is obtained and presented under series forms in terms of the H-functions, which is suitable for numerical computation. The solutions of some special cases can also be concluded from the result of this paper. Finally, the influences of the model parameters on the temperature distribution are spotlighted by graphical illustrations.

Time-fractional heat equations and negative absolute temperatures

The classical parabolic heat equation based on Fourier’s law implies infinite heat propagation speed. To remedy this physical flaw, the hyperbolic heat equation is used, but it may instead predict temperatures less than absolute zero. In recent years, fractional heat equations have been proposed as generalizations of heat equations of integer order. By simulating a 1D model problem of size on the order of a thermal energy carrier’s mean free path length, we have done a study of four fractional generalized Cattaneo equations from Compte and Metzler (1997) called GCE, GCE I, GCE II, and GCE III and also a fractional version of the parabolic heat equation. We have observed that when the fractional order is large enough, these equations give temperatures less than absolute zero. But if the fractional order is small enough, GCE I does not have this problem when the domain length is comparable to the mean free path length. With larger size, GCE I and GCE III also give non-oscillating solutions for both small and large values of the fractional order.

A High-Order Difference Scheme for the Generalized Cattaneo Equation

AbstractA high-order finite difference scheme for the fractional Cattaneo equation is investigated. The L1 approximation is invoked for the time fractional part, and a compact difference scheme is applied to approximate the second-order space derivative. The stability and convergence rate are discussed in the maximum norm by the energy method. Numerical examples are provided to verify the effectiveness and accuracy of the proposed difference scheme.

Hyperbolic subdiffusive impedance

We use two different hyperbolic subdiffusion equations with fractional time derivatives (the generalized Cattaneo equations) to study the transport process of electrolytes in media where subdiffusion occurs. In these models the flux is delayed in a non-zero time with respect to the concentration gradient. In particular, we obtain the formulae of electrochemical subdiffusive impedance of a spatially limited sample in the limit of large and small pulsation of the electric field. The boundary conditions at the external wall of the sample are taken in the general form as a linear combination of the subdiffusive flux and the concentration of transported particles. We also discuss the influence of the equation parameters (the subdiffusion parameter and the delay time) on the Nyquist impedance plots.

A finite element formulation for a convection–diffusion equation based on Cattaneo’s law

A number of contributions have been made during the last decades to model pure-diffusive transport problems by using the so-called hyperbolic diffusion equations [M. Zakari, D. Jou, Equations of state and transport equations in viscous cosmological models, Phys. Rev. D 48 (1993) 1597–1601; T. Ruggeri, A. Muracchini, L. Seccia, Shock waves and second sound in a rigid heat conductor: a critical temperature for NaF and Bi, Phys. Rev. Lett. 64 (1990) 2640–2643]. These equations are used for both mass and heat transport. The hyperbolic diffusion equations are obtained by substituting the classic time-independent constitutive equation (Fick’s [A. Fick, Uber diffusion, Poggendorff’s Annal. Phys. Chem. 94 (1855) 59–86] and Fourier’s [J.B. Fourier, Théorie analytique de la chaleur, Jacques Gabay, 1822] law, respectively) by a more general time-dependent equation, due to Cattaneo [M.C. Cattaneo, Sur une forme de l’equation de la chaleur liminant le paradoxe d’une propagation instantane, Comptes Rendus de L’Academie des Sciences: Series I-Mathematics 247 (1958) 431–433]. In some applications the use of a parabolic model for diffusive processes is accurate enough in spite of predicting an infinite speed of propagation (Cattaneo, 1958). However, the use of a wave-like equation that predicts a finite velocity of propagation is necessary in many other calculations [A. Compte, The generalized Cattaneo equation for the description of anomalous transport processes, J. Phys. A: Math. General 30 (1997) 7277–7289]. The studies of heat or mass transport with finite velocity of propagation have been limited, so far, to pure-diffusive situations. This paper proposes a formulation for convection–diffusion problems based on a Cattaneo-type law. The finite element solution of the proposed equations is addressed. The performance of the algorithm is verified by solving some 2D test cases. Some interesting features of the proposed model can be observed from these examples and we conclude that the proposal is a feasible alternative to the parabolic model for real engineering simulations.

The generalized Cattaneo equation for the description of anomalous transport processes

The Cattaneo equation, which describes a diffusion process with a finite velocity of propagation, is generalized to describe anomalous transport. Three possible generalizations are proposed, each one supported by a different scheme: continuous time random walks, non-local transport theory, and delayed flux-force relation. The properties of these generalizations are studied in both the long-time and the short-time regimes. In the long-time limit, we recover the mean-square displacement which is characteristic for these anomalous processes. As expected, the short-time behaviour is modified in comparison to generalized diffusion equations.