# A Fractional Single-Phase-Lag Model of Heat Conduction for Describing Propagation of the Maximum Temperature in a Finite Medium

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## Abstract

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## 1. Introduction

**q**is the heat flux vector,

**r**is the point in the considered region, t is the time, k is the thermal conductivity of the material, $\mathbf{\nabla}$ is the gradient operator and T is the temperature. This relationship implies a nonphysical infinite speed of a thermal signal in the medium. To avoid this disagreement between the mathematical model and the observations, the single-phase-lag was introduced to the heat conduction model. Namely, the relationship (Equation (1)) is replaced by the following one [2]

**q**can be eliminated. As a result, the fractional heat conduction equation is obtained:

## 2. Formulation of the Problem

## 3. Solution to the Problem

## 4. Numerical Analysis and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Temperature distributions $\widehat{T}\left(\widehat{x},\widehat{t}\right)$ in the hollow cylinder as functions of the space variable $\widehat{x}$ for $\alpha =0.15$, $\beta =0.9$ and different dimensionless time $\widehat{t}$.

**Figure 2.**The curves of locations of maxima temperatures in the hollow cylinder for different values of parameter $\alpha $; (

**a**) $\beta =0.9$; (

**b**) $\beta =0.95$; (

**c**) $\beta =1.0$.

**Figure 3.**The curves of the propagation speed of maxima temperatures in the hollow cylinder for $\beta =1.0$ and different values of parameter $\alpha $.

**Figure 4.**Temperature distributions $\widehat{T}\left(\widehat{x},\widehat{t}\right)$ in the hollow cylinder as functions of non-dimensional space variable $\widehat{x}$ and time $\widehat{t}$ when the temperature inside the hollow cylinder changes sinusoidaly for the fractional derivative order $\beta =0.9$; (

**a**) $\alpha =0$; (

**b**) $\alpha =0.05$; (

**c**) $\alpha =0.2$; (

**d**) $\alpha =0.5$.

**Table 1.**The functions ${\phi}_{i}\left(x\right)$ and ${\psi}_{i}\left(x\right)$ for the slab ($p=0$), hollow cylinder ($p=1$) and hollow sphere ($p=2$).

p | φ_{i} (x) | ψ_{i} (x) |
---|---|---|

0 | $\mathrm{cos}\left({\lambda}_{i}x\right)$ | $\mathrm{sin}\left({\lambda}_{i}x\right)$ |

1 | ${J}_{0}\left({\lambda}_{i}x\right)$ | ${Y}_{0}\left({\lambda}_{i}x\right)$ |

2 | $\frac{\mathrm{cos}\left({\lambda}_{i}x\right)}{x}$ | $\frac{\mathrm{sin}\left({\lambda}_{i}x\right)}{x}$ |

**Table 2.**The eigenfunctions ${\Phi}_{i}\left(x\right)$, eigenvalue equations and normalization integrals ${N}_{i}$ for the slab ($p=0$), hollow cylinder ($p=1$) and hollow sphere ($p=2$).

$\mathit{p}$ | Eigenfunction | Eigenequation | Normalization Integral |
---|---|---|---|

0 | ${\Phi}_{i}\left(x\right)=\mathrm{cos}\left(b-x\right){\lambda}_{i}$ | $\mathrm{sin}\left(b-a\right)\lambda -\frac{{h}_{a}}{k\lambda}\mathrm{cos}\left(b-a\right)\lambda =0$ | ${N}_{i}=\frac{b-a}{2}\left(1+\frac{\mathrm{sin}2{\lambda}_{i}\left(b-a\right)}{2{\lambda}_{i}\left(b-a\right)}\right)$ |

1 | $\begin{array}{ll}{\Phi}_{i}\left(x\right)& ={Y}_{1}\left(b{\lambda}_{i}\right){J}_{0}\left({\lambda}_{i}x\right)\\ & -{J}_{1}\left(b{\lambda}_{i}\right){Y}_{0}\left({\lambda}_{i}x\right)\end{array}$ | $\begin{array}{l}\frac{{h}_{a}}{k\lambda}\left({J}_{1}\left(b\lambda \right){Y}_{0}\left(a\lambda \right)-{J}_{0}\left(a\lambda \right){Y}_{1}\left(b\lambda \right)\right)\\ +{J}_{1}\left(b\lambda \right){Y}_{1}\left(a\lambda \right)-{J}_{1}\left(a\lambda \right){Y}_{1}\left(b\lambda \right)=0\end{array}$ | $\begin{array}{l}\hspace{1em}{N}_{i}=\frac{2}{{\pi}^{2}{\lambda}_{i}^{2}}-\frac{{a}^{2}}{2}\left(1+{\left(\frac{{h}_{a}}{k{\lambda}_{i}}\right)}^{2}\right)\xb7\\ {\left({J}_{1}\left(b{\lambda}_{i}\right){Y}_{0}\left(a{\lambda}_{i}\right)-{J}_{0}\left(a{\lambda}_{i}\right){Y}_{1}\left(b{\lambda}_{i}\right)\right)}^{2}\end{array}$ |

2 | $\begin{array}{ll}{\Phi}_{i}\left(x\right)& =\frac{1}{x}(\mathrm{cos}{\lambda}_{i}\left(b-x\right)\\ & -\frac{1}{b{\lambda}_{i}}\mathrm{sin}{\lambda}_{i}\left(b-x\right))\end{array}$ | $\begin{array}{l}\left(1+\frac{k}{{h}_{a}}\left(\frac{1}{a}+\frac{1}{b}{\left(b\lambda \right)}^{2}\right)\right)\frac{\mathrm{sin}\left(b-a\right)\lambda}{b\lambda}\\ -\left(1+\frac{k}{{h}_{a}}\left(\frac{1}{a}-\frac{1}{b}\right)\right)\mathrm{cos}\left(b-a\right)\lambda =0\end{array}$ | $\begin{array}{l}{N}_{i}=\frac{{b}^{2}{\lambda}_{i}^{2}-1}{4{b}^{2}{\lambda}_{i}^{3}}\mathrm{sin}2{\lambda}_{i}\left(b-a\right)+\frac{1}{2{b}^{2}{\lambda}_{i}^{2}}\xb7\\ \left(\left(b-a\right){\left(b{\lambda}_{i}\right)}^{2}-a+b\mathrm{cos}2{\lambda}_{i}\left(b-a\right)\right)\end{array}$ |

**Table 3.**The relative errors ${E}^{\alpha ,\beta ,\gamma}\left(t,k\right)$ of the results obtained by using the Fixed-Talbot procedure and exact values of the function ${U}_{1}^{\alpha ,\beta ,\gamma}\left(t\right)$ for $\alpha =0.5$, $\beta =1.0$ and $\gamma =-0.5;0;0.5$.

$\widehat{\mathit{t}}=\mathit{\kappa}\mathit{t}/{\left(\mathit{b}-\mathit{a}\right)}^{2}$ | $\mathit{\gamma}=-0.5$ | $\mathit{\gamma}=0$ | $\mathit{\gamma}=0.5$ |
---|---|---|---|

0.5 | $3.36748\times {10}^{-7}$ | $2.22232\times {10}^{-6}$ | $2.52736\times {10}^{-5}$ |

1 | $4.85927\times {10}^{-7}$ | $2.84182\times {10}^{-6}$ | $5.47403\times {10}^{-5}$ |

1.5 | $2.51907\times {10}^{-7}$ | $1.83325\times {10}^{-5}$ | $5.94386\times {10}^{-4}$ |

2 | $5.96577\times {10}^{-7}$ | $5.66522\times {10}^{-6}$ | $1.12375\times {10}^{-3}$ |

2.5 | $6.85496\times {10}^{-7}$ | $7.71945\times {10}^{-6}$ | $1.77174\times {10}^{-4}$ |

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**MDPI and ACS Style**

Kukla, S.; Siedlecka, U. A Fractional Single-Phase-Lag Model of Heat Conduction for Describing Propagation of the Maximum Temperature in a Finite Medium. *Entropy* **2018**, *20*, 876.
https://doi.org/10.3390/e20110876

**AMA Style**

Kukla S, Siedlecka U. A Fractional Single-Phase-Lag Model of Heat Conduction for Describing Propagation of the Maximum Temperature in a Finite Medium. *Entropy*. 2018; 20(11):876.
https://doi.org/10.3390/e20110876

**Chicago/Turabian Style**

Kukla, Stanisław, and Urszula Siedlecka. 2018. "A Fractional Single-Phase-Lag Model of Heat Conduction for Describing Propagation of the Maximum Temperature in a Finite Medium" *Entropy* 20, no. 11: 876.
https://doi.org/10.3390/e20110876