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International Journal of Differential Equations
VolumeΒ 2012Β (2012), Article IDΒ 154085, 13 pages
http://dx.doi.org/10.1155/2012/154085
Research Article

Axisymmetric Solutions to Time-Fractional Heat Conduction Equation in a Half-Space under Robin Boundary Conditions

1Institute of Mathematics and Computer Science, Jan Dlugosz University, 42200 Czestochowa, Poland
2Department of Computer Science, European University of Informatics and Economics (EWSIE), 03741 Warsaw, Poland

Received 23 May 2012; Revised 18 July 2012; Accepted 23 July 2012

Academic Editor: NikolaiΒ Leonenko

Copyright Β© 2012 Y. Z. Povstenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The time-fractional heat conduction equation with the Caputo derivative of the order 0<𝛼<2 is considered in a half-space in axisymmetric case under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of the values of temperature and the values of its normal derivative and the physical condition with the prescribed linear combination of the values of temperature and the values of the heat flux at the boundary.

1. Introduction

The generalized Fourier law, the time-nonlocal dependence between the heat flux vector πͺ and the temperature gradient grad𝑇 with the β€œlong-tail memory” power kernel [1, 2] (see also [3]) as π‘˜πͺ(𝑑)=βˆ’πœ•Ξ“(𝛼)ξ€œπœ•π‘‘π‘‘0(π‘‘βˆ’πœ)π›Όβˆ’1π‘˜grad𝑇(𝜏)π‘‘πœ,0<𝛼≀1;πͺ(𝑑)=βˆ’ξ€œΞ“(π›Όβˆ’1)𝑑0(π‘‘βˆ’πœ)π›Όβˆ’2grad𝑇(𝜏)π‘‘πœ,1<𝛼≀2,(1.1) where Ξ“(𝛼) is the gamma function, can be interpreted in terms of the fractional calculus: πͺ(𝑑)=βˆ’π‘˜π·1βˆ’π›Όπ‘…πΏgrad𝑇(𝑑),0<𝛼≀1,πͺ(𝑑)=βˆ’π‘˜πΌπ›Όβˆ’1grad𝑇(𝑑),1<𝛼≀2,(1.2) and in combination with the law of conservation of energy as πœŒπ‘šπΆπ‘‘π‘‡π‘‘π‘‘=βˆ’divπͺ(1.3) results in the time-fractional heat conduction equation with the Caputo fractional derivative πœ•π›Όπ‘‡πœ•π‘‘π›Ό=π‘ŽΞ”π‘‡,0<𝛼≀2.(1.4) Here, πœŒπ‘š is the mass density, 𝐢 denotes the specific heat capacity, and π‘Ž=π‘˜/(πœŒπ‘šπΆ) is the thermal diffusivity coefficient.

Recall that the Riemann-Liouville fractional integral 𝐼𝛼𝑓(𝑑) and derivative 𝐷𝛼𝑅𝐿𝑓(𝑑) are defined as follows (see [4–6]): 𝐼𝛼1𝑓(𝑑)=ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’πœ)π›Όβˆ’1𝐷𝑓(𝜏)π‘‘πœ,𝛼>0,(1.5)𝛼𝑅𝐿𝑑𝑓(𝑑)=π‘šπ‘‘π‘‘π‘šξ‚Έ1ξ€œΞ“(π‘šβˆ’π›Ό)𝑑0(π‘‘βˆ’πœ)π‘šβˆ’π›Όβˆ’1𝑓(𝜏)π‘‘πœ,π‘šβˆ’1<𝛼<π‘š,(1.6) whereas the Caputo fractional derivative has the following form [5–7]: 𝑑𝛼𝑓(𝑑)𝑑𝑑𝛼=1ξ€œΞ“(π‘šβˆ’π›Ό)𝑑0(π‘‘βˆ’πœ)π‘šβˆ’π›Όβˆ’1π‘‘π‘šπ‘“(𝜏)π‘‘πœπ‘šπ‘‘πœ,π‘šβˆ’1<𝛼<π‘š.(1.7)

A detailed explanation of derivation of time-fractional heat conduction equation (1.4) from the constitutive equations (1.2) and the law of conservation of energy (1.3) can be found in [8]. Here we briefly present the main idea. In the case 0<𝛼≀1, as a consequence of (1.2), (1.3), and (1.6), we have πœ•π‘‡πœ•πœ•π‘‘=π‘Žξ‚Έ1πœ•π‘‘ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’πœ)π›Όβˆ’1ξ‚Ή,Δ𝑇(𝜏)π‘‘πœ(1.8) or after integration with respect to time as 𝑇(𝑑)βˆ’π‘‡(0)=π‘ŽπΌπ›ΌΞ”π‘‡.(1.9) Applying to both sides of (1.9) the Caputo derivative πœ•π›Ό/πœ•π‘‘π›Ό, and taking into account that for 𝛼>0 [6], πœ•π›Όπœ•π‘‘π›ΌπΌπ›Όπ‘‡(𝑑)=𝑇(𝑑),(1.10) we obtain the time-fractional heat conduction equation (1.4) for 0<𝛼≀1.

Similarly, for 1<𝛼≀2, we get πœ•π‘‡πœ•π‘‘=π‘ŽπΌπ›Όβˆ’1Δ𝑇.(1.11) Applying πœ•π›Όβˆ’1/πœ•π‘‘π›Όβˆ’1 to both sides of (1.11) gives (1.4) for 1<𝛼≀2 as the following equality fulfills [5] πœ•π›Όβˆ’1πœ•π‘‘π›Όβˆ’1πœ•π‘‡=πœ•πœ•π‘‘π›Όπ‘‡πœ•π‘‘π›Ό.(1.12)

If the heat conduction equation is investigated in a bounded domain, the boundary conditions should be prescribed. The mathematical Robin boundary condition is a specification of a linear combination of the values of temperature and the values of its normal derivative at the boundary of the considered domain 𝑐1𝑇+𝑐2πœ•π‘‡ξ‚|||πœ•π‘›π‘†=𝐹0𝐱𝑆,,𝑑(1.13) with some nonzero constants 𝑐1 and 𝑐2, while the physical Robin boundary condition specifies a linear combination of the values of temperature and the values of the heat flux at the boundary of the domain. For example, the condition of convective heat exchange between a body and the environment with the temperature 𝑇𝑒||πͺ⋅𝐧𝑆𝑇||=β„Žπ‘†βˆ’π‘‡π‘’ξ€Έ,(1.14) where β„Ž is the convective heat transfer coefficient, leads to ξ‚€β„Žπ‘‡+π‘˜π·1βˆ’π›Όπ‘…πΏπœ•π‘‡ξ‚|||πœ•π‘›π‘†=β„Žπ‘‡π‘’ξ€·π±π‘†ξ€Έξ‚€,𝑑,0<𝛼≀1,β„Žπ‘‡+π‘˜πΌπ›Όβˆ’1πœ•π‘‡ξ‚|||πœ•π‘›π‘†=β„Žπ‘‡π‘’ξ€·π±π‘†ξ€Έ,𝑑,1<𝛼≀2.(1.15)

The literature on mathematical aspects concerning correctness of initial-boundary-value problems for time-fractional diffusion equation and form and properties of its solutions is quite extensive (see, e.g., [9–16], among others). Geometrical explanation of fractional calculus is given in [17–19].

There are only a few papers [20, 21] in which the fractional diffusion is investigated under the mathematical Robin boundary condition. In previous publications, problems for a cylinder [22] and a sphere [23] under mathematical and physical Neumann boundary conditions were considered. In the present paper, for the first time, the solutions to time-fractional heat conduction equation in a half-space are studied under both the mathematical and physical Robin boundary conditions. The Laplace integral transform with respect to time 𝑑, the Hankel transform with respect to the spatial coordinate π‘Ÿ, and the sin-cos-Fourier transforms with respect to spatial coordinate 𝑧 are used. The solutions under the mathematical and physical Neumann boundary conditions are obtained as particular cases.

2. Mathematical Preliminaries

2.1. Laplace Transform

The Laplace transform is defined as β„’{𝑓(𝑑)}=π‘“βˆ—ξ€œ(𝑠)=∞0𝑓(𝑑)π‘’βˆ’π‘ π‘‘π‘‘π‘‘,(2.1) where 𝑠 is the transform variable.

The inverse Laplace transfrom is carried out according to the Fourier-Mellin formula: β„’βˆ’1ξ€½π‘“βˆ—ξ€Ύ1(𝑠)=𝑓(𝑑)=ξ€œ2πœ‹π‘–π‘+π‘–βˆžπ‘βˆ’π‘–βˆžπ‘“βˆ—(𝑠)𝑒𝑠𝑑𝑑𝑠,𝑑>0,(2.2) where 𝑐 is a positive fixed number.

The Laplace transform of the Riemann-Liouville fractional integral of the order 𝛼 is carried out according to the formula similar to the Laplace transform of π‘š-fold primitive of a function as β„’{𝐼𝛼1𝑓(𝑑)}=π‘ π›Όπ‘“βˆ—(𝑠).(2.3)

The Caputo derivative for its Laplace transform requires the knowledge of the initial values of the function 𝑓(0+) and its integer derivatives 𝑓(π‘˜)(0+) of the order π‘˜=1,2,…,π‘šβˆ’1ℒ𝑑𝛼𝑓(𝑑)𝑑𝑑𝛼=π‘ π›Όπ‘“βˆ—(𝑠)βˆ’π‘šβˆ’1ξ“π‘˜=0𝑓(π‘˜)ξ€·0+ξ€Έπ‘ π›Όβˆ’1βˆ’π‘˜,π‘šβˆ’1<𝛼<π‘š,(2.4) whereas the Riemann-Liouville derivative for its Laplace transform rule requires the knowledge of the initial values of the fractional integral πΌπ‘šβˆ’π›Όπ‘“(0+) and its derivatives of the order π‘˜=1,2,…,π‘šβˆ’1ℒ𝐷𝛼𝑅𝐿𝑓(𝑑)=π‘ π›Όπ‘“βˆ—(𝑠)βˆ’π‘šβˆ’1ξ“π‘˜=0π·π‘˜πΌπ‘šβˆ’π›Όπ‘“ξ€·0+ξ€Έπ‘ π‘šβˆ’1βˆ’π‘˜,π‘šβˆ’1<𝛼<π‘š.(2.5) The reader interested in applications of integral transforms in fractional calculus is referred to [4–7, 24].

2.2. Hankel Transform

The Hankel transform is used to solve problem in cylindrical coordinates in the domain 0β‰€π‘Ÿ<∞ and is defined as β„‹{𝑓(π‘Ÿ)}=π‘“βˆ—ξ€œ(πœ‚)=∞0𝑓(π‘Ÿ)π½πœˆβ„‹(πœ‚π‘Ÿ)π‘Ÿπ‘‘π‘Ÿ,βˆ’1ξ€½π‘“βˆ—ξ€Ύξ€œ(πœ‚)=𝑓(π‘Ÿ)=∞0π‘“βˆ—(πœ‚)𝐽𝜈(πœ‚π‘Ÿ)πœ‚π‘‘πœ‚,(2.6) where 𝐽𝜈(π‘Ÿ) is the Bessel function of the order 𝜈.

The following formula is fulfilled: ℋ𝑑2𝑓(π‘Ÿ)π‘‘π‘Ÿ2+1π‘Ÿπ‘‘π‘“(π‘Ÿ)βˆ’πœˆπ‘‘π‘Ÿ2π‘Ÿ2𝑓(π‘Ÿ)=βˆ’πœ‚2π‘“βˆ—(πœ‚).(2.7)

2.3. Sin-Cos-Fourier Transform

In the case of boundary condition of the third kind with the prescribed boundary value of linear combination of a function and its normal derivative 𝑧=00π‘₯00899βˆΆβˆ’π‘‘π‘“π‘‘π‘§+𝐻𝑓=πœ‘0,(2.8) the following sin-cos-Fourier transform [25] is employed: β„±{𝑓(𝑧)}=π‘“βˆ—ξ€œ(πœ‰)=∞0ℱ𝐾(𝑧,πœ‰)𝑓(𝑧)𝑑𝑧,βˆ’1ξ€½π‘“βˆ—ξ€Ύ2(πœ‰)=𝑓(𝑧)=πœ‹ξ€œβˆž0𝐾(𝑧,πœ‰)π‘“βˆ—(πœ‰)π‘‘πœ‰,(2.9) with the kernel 𝐾(𝑧,πœ‰)=πœ‰cos(π‘§πœ‰)+𝐻sin(π‘§πœ‰)βˆšπœ‰2+𝐻2.(2.10)

Application of the sin-cos-Fourier transform to the second derivative of a function gives ℱ𝑑2𝑓(𝑧)𝑑𝑧2ξ‚Ό=βˆ’πœ‰2π‘“βˆ—πœ‰(πœ‰)βˆ’βˆšπœ‰2+𝐻2𝑑𝑓(𝑧)ξ‚Ή|||||π‘‘π‘§βˆ’π»π‘“(𝑧)𝑧=0.(2.11)

3. Solution to the Problem under Mathematical Robin Boundary Condition

Consider the axisymmetric time-fractional heat conduction equation in cylindrical coordinates πœ•π›Όπ‘‡πœ•π‘‘π›Όξ‚΅πœ•=π‘Ž2π‘‡πœ•π‘Ÿ2+1π‘Ÿπœ•π‘‡+πœ•πœ•π‘Ÿ2π‘‡πœ•π‘§2ξ‚Ά,0β‰€π‘Ÿ<∞,0<𝑧<∞,0<𝑑<∞,0<𝛼≀2,(3.1) with zero initial conditions 𝑑=00π‘₯00899βˆΆπ‘‡=0,0<𝛼≀2,𝑑=00π‘₯00899βˆΆπœ•π‘‡πœ•π‘‘=0,1<𝛼≀2,(3.2) and the mathematical Robin boundary condition 𝑧=00π‘₯00899βˆΆπ»π‘‡βˆ’πœ•π‘‡πœ•π‘§=𝑓(π‘Ÿ,𝑑).(3.3)

The zero conditions at infinity are also assumed limπ‘Ÿβ†’βˆžπ‘‡(π‘Ÿ,𝑧,𝑑)=0,limπ‘§β†’βˆžπ‘‡(π‘Ÿ,𝑧,𝑑)=0.(3.4)

The solution to the initial-boundary-value problem (3.1)–(3.4) can be written as ξ€œπ‘‡=𝑑0ξ€œβˆž0𝑓(𝜌,𝜏)π’’π‘š(π‘Ÿ,𝑧,𝜌,π‘‘βˆ’πœ)πœŒπ‘‘πœŒπ‘‘πœ,(3.5) where π’’π‘š(π‘Ÿ,𝑧,𝜌,𝑑) is the fundamental solution being the solution of the following problem: πœ•π›Όπ’’π‘šπœ•π‘‘π›Όξ‚΅πœ•=π‘Ž2π’’π‘šπœ•π‘Ÿ2+1π‘Ÿπœ•π’’π‘š+πœ•πœ•π‘Ÿ2π’’π‘šπœ•π‘§2ξ‚Ά,0β‰€π‘Ÿ<∞,0<𝑧<∞,0<𝑑<∞,0<𝛼≀2,𝑑=00π‘₯00899βˆΆπ’’π‘š=0,0<𝛼≀2,𝑑=00π‘₯00899βˆΆπœ•π’’π‘šπœ•π‘‘=0,1<𝛼≀2,𝑧=00π‘₯00899βˆΆπ»π’’π‘šβˆ’πœ•π’’π‘š=1πœ•π‘§π‘Ÿπ›Ώ(π‘Ÿβˆ’πœŒ)𝛿(𝑑).(3.6) Here, 𝛿(π‘Ÿ) is the Dirac delta function.

The Laplace transform with respect to time 𝑑, the Hankel transform with respect to the spatial coordinate π‘Ÿ, and the sin-cos-Fourier transform with respect to the spatial coordinate 𝑧 result in π’’π‘šβˆ—βˆ—βˆ—=π‘Žπœ‰π½0(πœŒπœ‚)βˆšπœ‰2+𝐻21π‘ π›Όξ€·πœ‰+π‘Ž2+πœ‚2ξ€Έ,(3.7) where each of the integral transforms is denoted by the asterisk.

Invertion of the integral transforms leads to π’’π‘š(π‘Ÿ,𝑧,𝜌,𝑑)=2π‘Žπ‘‘π›Όβˆ’1πœ‹ξ€βˆž0πœ‰2cos(π‘§πœ‰)+πœ‰π»sin(π‘§πœ‰)πœ‰2+𝐻2×𝐸𝛼,π›Όξ€Ίξ€·πœ‰βˆ’π‘Ž2+πœ‚2𝑑𝛼𝐽0(π‘Ÿπœ‚)𝐽0(πœŒπœ‚)πœ‚π‘‘πœ‰π‘‘πœ‚.(3.8) Here, 𝐸𝛼,𝛽 is the Mittag-Leffler function in two parameters 𝛼 and 𝛽 defined by the series representation: 𝐸𝛼,𝛽(𝑧)=βˆžξ“π‘š=0π‘§π‘šΞ“(π›Όπ‘š+𝛽),𝛼>0,𝛽>0,π‘§βˆˆπΆ.(3.9) The essential role of the Mittag-Leffler function in fractional calculus results from the following formula for the inverse Laplace transform β„’βˆ’1ξ‚»π‘ π›Όβˆ’π›½π‘ π›Όξ‚Ό+𝑏=π‘‘π›½βˆ’1𝐸𝛼,𝛽(βˆ’π‘π‘‘π›Ό).(3.10) Consider several particular cases of solution (3.8).

The case 𝐻=0 corresponds to the mathematical Neumann boundary condition with the prescribed boundary value of the normal derivative of temperature, and the solution reads π’’π‘š(π‘Ÿ,𝑧,𝜌,𝑑)=2π‘Žπ‘‘π›Όβˆ’1πœ‹ξ€βˆž0𝐸𝛼,π›Όξ€Ίξ€·πœ‰βˆ’π‘Ž2+πœ‚2𝑑𝛼×cos(π‘§πœ‰)𝐽0(π‘Ÿπœ‚)𝐽0(πœŒπœ‚)πœ‚π‘‘πœ‰π‘‘πœ‚.(3.11)

In the case of classical heat conduction (𝛼=1), the Mittag-Leffler function: 𝐸1,1ξ€·βˆ’π‘₯2ξ€Έ=π‘’βˆ’π‘₯2,(3.12) and taking into account the following integrals [26, 27]: ξ€œβˆž0π‘₯π‘’βˆ’π‘Žπ‘₯2𝐽0(𝑏π‘₯)𝐽01(𝑐π‘₯)𝑑π‘₯=ξ‚΅βˆ’π‘2π‘Žexp2+𝑐2𝐼4π‘Ž0ξ‚€π‘π‘ξ‚ξ€œ2π‘Ž,π‘Ž>0,∞0π‘’βˆ’π‘Žπ‘₯2√cos(𝑐π‘₯)𝑑π‘₯=πœ‹2βˆšπ‘Žξ‚΅βˆ’π‘exp2ξ‚Άξ€œ4π‘Ž,π‘Ž>0,∞01π‘₯2+𝑏2π‘’βˆ’π‘Žπ‘₯2=πœ‹cos(𝑐π‘₯)𝑑π‘₯𝑒4π‘π‘Žπ‘2ξƒ¬π‘’βˆ’π‘π‘ξƒ©βˆšerfcπ‘π‘Žπ‘βˆ’2βˆšπ‘Žξƒͺ+π‘’π‘π‘ξƒ©βˆšerfcπ‘π‘Žπ‘+2βˆšπ‘Žξ€œξƒͺξƒ­,π‘Ž>0,𝑏>0,∞0π‘₯π‘₯2+𝑏2π‘’βˆ’π‘Žπ‘₯2=πœ‹sin(𝑐π‘₯)𝑑π‘₯4π‘’π‘Žπ‘2ξƒ¬π‘’βˆ’π‘π‘ξƒ©βˆšerfcπ‘π‘Žπ‘βˆ’2βˆšπ‘Žξƒͺβˆ’π‘’π‘π‘ξƒ©βˆšerfcπ‘π‘Žπ‘+2βˆšπ‘Žξƒͺξƒ­,π‘Ž>0,𝑏>0,(3.13) where 𝐼0(π‘₯) is the modified Bessel function, erfcπ‘₯ is the complementary error function, we get π’’π‘š1(π‘Ÿ,𝑧,𝜌,𝑑)=ξ‚΅βˆ’π‘Ÿ2𝑑exp2+𝜌2+𝑧2𝐼4π‘Žπ‘‘0ξ‚€π‘ŸπœŒξ‚Γ—βŽ§βŽͺ⎨βŽͺ⎩12π‘Žπ‘‘βˆšβŽ‘βŽ’βŽ’βŽ£ξƒ©βˆšπœ‹π‘Žπ‘‘βˆ’π»expπ‘§π‘Žπ‘‘π»+2√ξƒͺπ‘Žπ‘‘2⎀βŽ₯βŽ₯βŽ¦ξƒ©βˆšerfcπ‘§π‘Žπ‘‘π»+2√ξƒͺ⎫βŽͺ⎬βŽͺ⎭.π‘Žπ‘‘(3.14)

For the wave equation (𝛼=2), 𝐸2,2ξ€·βˆ’π‘₯2ξ€Έ=sinπ‘₯π‘₯,(3.15) and after evaluation of the necessary integrals appearing in solution (3.8) (under assumption 𝜌=0) [26, 27] ξ€œβˆž0π‘₯√π‘₯2+𝑝2ξ‚€π‘βˆšsinπ‘₯2+𝑝2𝐽0⎧βŽͺ⎨βŽͺ⎩1(𝑐π‘₯)𝑑π‘₯=βˆšπ‘2βˆ’π‘2ξ‚€π‘βˆšcos𝑏2βˆ’π‘2ξ‚ξ€œ,0<𝑐<𝑏,0,0<𝑏<𝑐.∞0cos(𝑝π‘₯)cos(π‘žπ‘₯)π‘₯2+𝑏2⎧βŽͺ⎨βŽͺβŽ©πœ‹π‘‘π‘₯=𝑒2π‘βˆ’π‘π‘πœ‹cosh(π‘π‘ž),0<π‘ž<𝑝,𝑒2π‘βˆ’π‘π‘žξ€œcosh(𝑏𝑝),0<𝑝<π‘ž,𝑏>0,∞0π‘₯cos(𝑝π‘₯)sin(π‘žπ‘₯)π‘₯2+𝑏2⎧βŽͺ⎨βŽͺβŽ©βˆ’πœ‹π‘‘π‘₯=2π‘’βˆ’π‘π‘πœ‹sinh(π‘π‘ž),0<π‘ž<𝑝,2π‘’βˆ’π‘π‘žcosh(𝑏𝑝),0<𝑝<π‘ž,𝑏>0,(3.16) we obtain the solution π’’π‘š=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩√(π‘Ÿ,𝑧,0,𝑑)π‘Žβˆšπ‘Žπ‘‘2βˆ’π‘Ÿ2ξ‚ƒπ›Ώξ‚€βˆšπ‘Žπ‘‘2βˆ’π‘Ÿ2ξ‚βˆ’π‘§βˆ’π»π‘’βˆšβˆ’π»(π‘Žπ‘‘2βˆ’π‘Ÿ2βˆ’π‘§)ξ‚„βˆš0<π‘Ÿ<βˆšπ‘Žπ‘‘,0<𝑧<π‘Žπ‘‘2βˆ’π‘Ÿ2,0βˆšβˆšπ‘Žπ‘‘<π‘Ÿ<∞,π‘Žπ‘‘2βˆ’π‘Ÿ2<𝑧<∞(3.17)

Of particular interest is also the case 𝛼=1/2 for which 𝐸1/2,1/21(βˆ’π‘₯)=βˆšπœ‹βˆ’π‘₯𝑒π‘₯22erfcπ‘₯=βˆšπœ‹ξ€œβˆž0π‘’βˆ’π‘’2βˆ’2π‘₯𝑒𝒒𝑒𝑑𝑒,π‘₯β‰₯0,π‘š1(π‘Ÿ,𝑧,𝜌,𝑑)=2βˆšξ€œπœ‹π‘‘βˆž0expβˆ’π‘’2βˆ’π‘Ÿ2+𝜌2+𝑧2√8π‘Žξƒͺ𝐼𝑑𝑒0ξƒ©π‘ŸπœŒβˆš4π‘ŽξƒͺΓ—βŽ§βŽͺ⎨βŽͺ⎩1π‘‘π‘’βˆš2πœ‹π‘Žπ‘’π‘‘1/4βŽ‘βŽ’βŽ’βŽ£ξƒ©βˆšβˆ’π»exp2π‘Žπ‘’π»π‘‘1/4+𝑧2√2π‘Žπ‘’π‘‘1/4ξƒͺ2⎀βŽ₯βŽ₯βŽ¦ξƒ©βˆšΓ—erfc2π‘Žπ‘’π»π‘‘1/4+𝑧2√2π‘Žπ‘’π‘‘1/4ξƒͺ𝑑𝑒.(3.18)

4. Solution to the Problem under Physical Robin Boundary Condition

Consider the following axisymmetric time-fractional heat conduction equation: πœ•π›Όπ‘‡πœ•π‘‘π›Όξ‚΅πœ•=π‘Ž2π‘‡πœ•π‘Ÿ2+1π‘Ÿπœ•π‘‡+πœ•πœ•π‘Ÿ2π‘‡πœ•π‘§2ξ‚Ά,0β‰€π‘Ÿ<∞,0<𝑧<∞,0<𝑑<∞,0<𝛼≀2,(4.1) under zero initial conditions 𝑑=00π‘₯00899βˆΆπ‘‡=0,0<𝛼≀2,𝑑=00π‘₯00899βˆΆπœ•π‘‡πœ•π‘‘=0,1<𝛼≀2,(4.2) and the physical Robin boundary condition 𝑧=00π‘₯00899βˆΆπ»π‘‡βˆ’π·1βˆ’π›Όπ‘…πΏπœ•π‘‡πœ•π‘§=𝑓(π‘Ÿ,𝑑),0<𝛼≀1,𝑧=00π‘₯00899βˆΆπ»π‘‡βˆ’πΌπ›Όβˆ’1πœ•π‘‡πœ•π‘§=𝑓(π‘Ÿ,𝑑),1<𝛼≀2,(4.3) where 𝐻=β„Ž/π‘˜.

The solution to the initial-boundary-value problem (4.1)–(4.3) has the form ξ€œπ‘‡=𝑑0ξ€œβˆž0𝑓(𝜌,𝜏)𝒒𝑝(π‘Ÿ,𝑧,𝜌,π‘‘βˆ’πœ)πœŒπ‘‘πœŒπ‘‘πœ,(4.4) where the fundamental solution 𝒒𝑝(π‘Ÿ,𝑧,𝜌,𝑑) fulfills the following equation πœ•π›Όπ’’π‘πœ•π‘‘π›Όξƒ©πœ•=π‘Ž2π’’π‘πœ•π‘Ÿ2+1π‘Ÿπœ•π’’π‘+πœ•πœ•π‘Ÿ2π’’π‘πœ•π‘§2ξƒͺ,0β‰€π‘Ÿ<∞,0<𝑧<∞,0<𝑑<∞,0<𝛼≀2,(4.5) under the following conditions: 𝑑=00π‘₯00899βˆΆπ’’π‘=0,0<𝛼≀2,𝑑=00π‘₯00899βˆΆπœ•π’’π‘πœ•π‘‘=0,1<𝛼≀2,𝑧=0βˆΆπ»π’’π‘βˆ’π·1βˆ’π›Όπ‘…πΏπœ•π’’π‘=1πœ•π‘§π‘Ÿπ›Ώ(π‘Ÿβˆ’πœŒ)𝛿(𝑑),0<𝛼≀1,𝑧=0βˆΆπ»π’’π‘βˆ’πΌπ›Όβˆ’1πœ•π’’π‘=1πœ•π‘§π‘Ÿπ›Ώ(π‘Ÿβˆ’πœŒ)𝛿(𝑑),1<𝛼≀2.(4.6)

The Laplace transform with respect to time 𝑑 leads to the following equation: π‘ π›Όπ’’βˆ—π‘ξƒ©πœ•=π‘Ž2π’’βˆ—π‘πœ•π‘Ÿ2+1π‘Ÿπœ•π’’βˆ—π‘+πœ•πœ•π‘Ÿ2π’’βˆ—π‘πœ•π‘§2ξƒͺ,0β‰€π‘Ÿ<∞,0<𝑧<∞,0<𝛼≀2,(4.7) and the following boundary condition: 𝑧=0βˆΆπ‘ π›Όβˆ’1π»π’’βˆ—π‘βˆ’πœ•π’’βˆ—π‘=1πœ•π‘§π‘Ÿπ›Ώ(π‘Ÿβˆ’πœŒ)π‘ π›Όβˆ’1.(4.8)

In this case, the kernel of the sin-cos-Fourier transform with respect to the spatial coordinate 𝑧 depends on the Laplace transform variable 𝑠𝐾(𝑧,πœ‰,𝑠)=πœ‰cos(π‘§πœ‰)+π‘ π›Όβˆ’1𝐻sin(π‘§πœ‰)ξ”πœ‰2+ξ€·π‘ π›Όβˆ’1𝐻2,(4.9) and in the transform domain we obtain π’’π‘βˆ—βˆ—βˆ—=π‘Žπœ‰π½0(πœŒπœ‚)ξ”πœ‰2+ξ€·π‘ π›Όβˆ’1𝐻2π‘ π›Όβˆ’1π‘ π›Όξ€·πœ‰+π‘Ž2+πœ‚2ξ€Έ.(4.10)

Inversion of the Laplace transform in (4.10) depends on the value of 𝛼. For 0<𝛼≀1, we have 𝒒𝑝(π‘Ÿ,𝑧,𝜌,𝑑)=2π‘Žπœ‹ξ€βˆž0ξ€œπ‘‘0πœ‚π½0(π‘Ÿπœ‚)𝐽0(πœŒπœ‚)πœπ›Όβˆ’1𝐸𝛼,π›Όξ€Ίξ€·πœ‰βˆ’π‘Ž2+πœ‚2ξ€Έπœπ›Όξ€»Γ—ξ‚»(π‘‘βˆ’πœ)βˆ’π›ΌπΈ2βˆ’2𝛼,1βˆ’π›Όξ‚Έβˆ’π»2πœ‰2(π‘‘βˆ’πœ)2βˆ’2𝛼+𝐻cos(π‘§πœ‰)πœ‰(π‘‘βˆ’πœ)1βˆ’2𝛼𝐸2βˆ’2𝛼,2βˆ’2π›Όξ‚Έβˆ’π»2πœ‰2(π‘‘βˆ’πœ)2βˆ’2𝛼sin(π‘§πœ‰)π‘‘πœπ‘‘πœ‰π‘‘πœ‚,(4.11) whereas for 1<𝛼≀2, we get 𝒒𝑝(π‘Ÿ,𝑧,𝜌,𝑑)=2π‘Žπœ‹ξ€βˆž0ξ€œπ‘‘0𝐽0(π‘Ÿπœ‚)𝐽0(πœŒπœ‚)πΈπ›Όξ€Ίξ€·πœ‰βˆ’π‘Ž2+πœ‚2ξ€Έπœπ›Όξ€»Γ—ξ‚»πœ‰2𝐻2(π‘‘βˆ’πœ)2π›Όβˆ’3𝐸2π›Όβˆ’2,2π›Όβˆ’2ξ‚Έβˆ’πœ‰2𝐻2(π‘‘βˆ’πœ)2π›Όβˆ’2ξ‚Ή+πœ‰cos(π‘§πœ‰)𝐻(π‘‘βˆ’πœ)π›Όβˆ’2𝐸2π›Όβˆ’2,π›Όβˆ’1ξ‚Έβˆ’π»2πœ‰2(π‘‘βˆ’πœ)2βˆ’2𝛼sin(π‘§πœ‰)π‘‘πœπ‘‘πœ‰π‘‘πœ‚.(4.12) The fundamental solution under physical Neumann boundary condition is obtained for𝐻=0 as 𝒒𝑝(π‘Ÿ,𝑧,𝜌,𝑑)=2π‘Žπœ‹ξ€βˆž0πΈπ›Όξ€Ίξ€·πœ‰βˆ’π‘Ž2+πœ‚2𝑑𝛼𝐽0(π‘Ÿπœ‚)𝐽0(πœŒπœ‚)πœ‚cos(π‘§πœ‰)π‘‘πœ‰π‘‘πœ‚.(4.13)

Formulae (4.11) and (4.12) simplify for 𝛼=1/2 and 𝛼=3/2, respectively, taking into account (3.12) and that 𝐸1,1/2ξ€·βˆ’π‘₯2ξ€Έ=1βˆšπœ‹[],1βˆ’2π‘₯𝐷(π‘₯)(4.14) where 𝐷(π‘₯)=π‘’βˆ’π‘₯2ξ€œπ‘₯0𝑒𝑒2𝑑𝑒(4.15) is the Dawson integral.

5. Conclusion

We have derived the analytical solutions to time-fractional heat conduction equation in a half-space under mathematical and physical Robin boundary conditions. The integral transform technique has been used. It should be emphasized that in the case of physical Robin boundary condition, the order of integral transforms is important as the kernel of the sin-cos-Fourier transform depends on the Laplace transform variable. The limiting case 𝐻=0 corresponds to solutions of problems under mathematical and physical Neumann boundary conditions with the prescribed boundary value of the normal derivative and with the prescribed boundary value of the heat flux, respectively. The difference between mathematical and physical boundary conditions (as well as the difference between the solutions) disappears in the case of standard heat conduction equation (𝛼=1).

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