Fractional Cattaneo heat equation in a semi-infinite medium

Xu, Huanying; Qi, Haitao; Jiang, Xiaoyun

doi:10.1088/1674-1056/22/1/014401

Cited by 26 publications

(17 citation statements)

References 39 publications

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“…Possibility of using the fractional Cattaneo law (1.5), along with its different variations, within the theory of anomalous transport processes is explored in [10]. By fractionalizing the classical Cattaneo constitutive model (1.6) in different manners and by combining such obtained heat conduction laws with the energy balance equation, classical telegrapher's equation (1.1) is generalized in [13,23,24,27] and the corresponding problems on bounded, semi-bounded and unbounded domains are analyzed using analytical and numerical tools. Fractional Cattaneo law (1.5) is further generalized, either by considering its multi-term (or even distributed-order) version in [26], or by considering the spatial non-locality in [1,6,21,29], or even by considering the non-locality in Cattaneo-Christov heat conduction law in [19].…”

Section: Introduction and Model Formulationmentioning

confidence: 99%

Fractional telegrapher’s equation as a consequence of Cattaneo’s heat conduction law generalization

Zorica

Cveticanin

2018

Theor appl mech (Belgr)

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Fractional telegrapher's equation is reinterpreted in the setting of heat conduction phenomena and reobtained by considering the energy balance equation and fractional Cattaneo heat conduction law, generalized by taking into account the history of temperature gradient as well. Using the Laplace transform method, fractional telegrapher's equation is solved on semi-bounded domain for the zero initial condition and solution is obtained as a convolution of forcing temperature on the boundary and impulse response. Some features of such obtained solution are examined.

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Section: Introduction and Model Formulationmentioning

confidence: 99%

Fractional telegrapher’s equation as a consequence of Cattaneo’s heat conduction law generalization

Zorica

Cveticanin

2018

Theor appl mech (Belgr)

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“…with 0 < α < β < 2, or α ∈ (0, 1) . Similarly as in [5], where (10) 1 was treated analytically on infinite, semi-infinite and finite domains, in [48] the problem on the semi-infinite domain for (10) 1 was treated for a special choice of the boundary conditions. By the use of analytical methods, different versions of telegraph equation are treated on unbounded and bounded domains in [11,12,22], including the non-locality as in [49].…”

Section: Introductionmentioning

confidence: 99%

Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law

Želi,

Zorica

2017

Preprint

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Generalization of the heat conduction equation is obtained by considering the system of equations consisting of the energy balance equation and fractional-order constitutive heat conduction law, assumed in the form of the distributed-order Cattaneo type. The Cauchy problem for system of energy balance equation and constitutive heat conduction law is treated analytically through Fourier and Laplace integral transform methods, as well as numerically by the method of finite differences through Adams-Bashforth and Grünwald-Letnikov schemes for approximation derivatives in temporal domain and leap frog scheme for spatial derivatives. Numerical examples, showing time evolution of temperature and heat flux spatial profiles, demonstrate applicability and good agreement of both methods in cases of multi-term and power-type distributed-order heat conduction laws.

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“…Qi and Jiang in [37] derived the exact solution of the Cattaneo-Vernotte equation by joint Laplace and Fourier transforms. Other applications of FC to Cattaneo-Vernotte equation are given in [38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation

Gómez-Aguilar¹,

Córdova–Fraga²,

Tórres-Jiménez³

et al. 2016

Mathematical Problems in Engineering

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The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range(0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to1.

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Fractional Cattaneo heat equation in a semi-infinite medium

Abstract: Fractional Cattaneo heat equation in a semi-infinite medium * Xu Huan-Ying(续焕英) a) , Qi Hai-Tao(齐海涛) a)b) † , and Jiang Xiao-Yun(蒋晓芸) c)

Cited by 26 publications

References 39 publications

Fractional telegrapher’s equation as a consequence of Cattaneo’s heat conduction law generalization

Fractional telegrapher’s equation as a consequence of Cattaneo’s heat conduction law generalization

Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law

Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation

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