Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition
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Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition Yuriy Povstenko1
· Joanna Klekot2
Received: 26 November 2017 / Revised: 13 January 2018 / Accepted: 24 January 2018 © The Author(s) 2018. This article is an open access publication
Abstract The time-fractional heat conduction equation with the Caputo derivative and with heat absorption term proportional to temperature is considered in a sphere in the case of central symmetry. The fundamental solution to the Dirichlet boundary value problem is found, and the solution to the problem under constant boundary value of temperature is studied. The integral transform technique is used. The solutions are obtained in terms of series containing the Mittag-Leffler functions being the generalization of the exponential function. The numerical results are illustrated graphically. Keywords Non-Fourier heat conduction · Heat absorption · Caputo fractional derivative · Dirichlet boundary condition · Mittag-Leffler function · Laplace transform · Finite Fourier transform Mathematics Subject Classification 26A33 · 35K05 · 45K05
1 Introduction The classical parabolic heat conduction equation with the source term proportional to temperature ∂T = aT − bT (1) ∂t
Communicated by José Tenreiro Machado.
B
Yuriy Povstenko [email protected] Joanna Klekot [email protected]
1
Institute of Mathematics and Computer Science, Faculty of Mathematics and Natural Sciences, Jan Długosz University in Cz¸estochowa, al. Armii Krajowej 13/15, 42-200 Cz¸estochowa, Poland
2
Institute of Mathematics, Faculty of Mechanical Engineering and Computer Science, Cz¸estochowa University of Technology, al. Armii Krajowej 21, 42-200 Cz¸estochowa, Poland
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Y. Povstenko, J. Klekot
was considered, e.g., in Carslaw and Jaeger (1959), Crank (1975), Nyborg (1988), Polyanin (2002). Here, T is temperature, t is time, a stands for the thermal diffusivity coefficient, denotes the Laplace operator, the coefficient b describes the heat absorption (heat release). In the last few decades, differential equations with derivatives of non-integer order attract the attention of the researchers as such equations provide a very suitable tool for description of many important phenomena in physics, geophysics, chemistry, biology, engineering and solid mechanics (see, for example, Gafiychuk et al. 2008; Herrmann 2011; Magin 2006; Mainardi 2010; Povstenko 2015a; Sabatier et al. 2007; Tarasov 2010; Tenreiro Machado 2011; Uchaikin 2013). In this paper, we consider the time-fractional equation ∂α T = aT − bT, 0 < α ≤ 1, ∂t α where
(2)
⎧ ⎪ ⎪ ⎨
t 1 dm f (τ ) (t − τ )m−α−1 dτ, m − 1 < α < m, α d f (t) Γ (m − α) 0 dτ m (3) = m ⎪ dt α ⎪ ⎩ d f (t) , α = m, dt m is the Caputo fractional derivative (Kilbas et al. 2006; Podlubny 1999). Equation (2) results from the time-nonlocal generalization of the Fourier law with the “long-tail” power kernel. Such a generalization can be interpreted in terms of derivatives and integrals of non-integer order. Equation (2) takes into consid
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