# An Analysis of the One-Phase Stefan Problem with Variable Thermal Coefficients of Order p

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Approximate Solutions

**Lemma 1**.

**Remark 1**.

**Proof.**

#### 2.1. The First Approximation

#### 2.2. The Second Approximation

#### 2.3. A Special Case ${\mathrm{\Psi}}_{1}\left(x\right)={\mathrm{\Psi}}_{2}\left(x\right)$

## 3. The Values of $\mathit{\lambda}$ in Terms of the Stefan Number $\mathit{Ste}$

#### 3.1. Case 1: ${\mathrm{\Psi}}_{1}\left(x\right)={\mathrm{\Psi}}_{2}\left(x\right)$

**Lemma 2**.

#### 3.2. Case 2: ${\mathrm{\Psi}}_{1}\left(x\right)\ne {\mathrm{\Psi}}_{2}\left(x\right)$

**Lemma 3**.

**Lemma 4**.

## 4. Remarks on the General Case: ${\mathbf{\delta}}_{\mathbf{1}}>-\mathbf{1}$ and ${\mathbf{\delta}}_{\mathbf{2}}\ge \mathbf{0}$

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The variation of $\lambda $, in terms of the Stefan constant $Ste$, from Equation (28). For ${\delta}_{1}={\delta}_{2}=1$ (

**left panel**), ${\delta}_{1}={\delta}_{2}=5$ (

**right panel**), different values of $p=0.5,1,1.5,2.5$ from the highest line to the lowest, respectively.

**Figure 2.**The variation of $\lambda $, in terms of the Stefan constant $Ste$, from Equations (31) and (34). (

**Left panel**) from Equation (31), (

**right panel**) from Equation (34), for ${\delta}_{1}=1,{\delta}_{2}=0.1$, and different values of $p=0.5,1,5,10$ from the highest line to the lowest, respectively.

**Figure 3.**The numerical solution for the boundary $\lambda =1$ for different values of p. (

**Left panel**) ${\delta}_{1}={\delta}_{2}=1$, (

**right panel**) ${\delta}_{1}={\delta}_{2}=5$. Solid black line: $p=1$, dashed blue line: $p=5$, and dashed–dotted red line: $p=10$.

**Figure 4.**The analytical solution in implicit form given by Equation (23) for the boundary $\lambda =1$ for different values of p. (

**Left panel**) ${\delta}_{1}={\delta}_{2}=1$, (

**right panel**) ${\delta}_{1}={\delta}_{2}=5$. Solid black line: $p=1$, dashed blue line: $p=5$, and dashed–dotted red line: $p=10$.

**Figure 5.**Comparison between the numerical solutions of BVP (1) and the first approximation in Equation (16) for the boundary $\lambda =1$, for different values of p. The (

**left panel**) represents the numerical solutions, while the (

**right panel**) represents the first approximation as in Equation (16). In both cases ${\delta}_{1}=1,{\delta}_{2}=0.1$, solid black line: $p=1$, dashed blue line: $p=5$, and dashed–dotted red line: $p=10$. For the function, $\beta \left(x\right)=\frac{1}{2}{e}^{-(1+{\delta}_{2}){x}^{2}}$.

**Table 1.**Comparison between the numerical value of the first derivative ${y}^{\prime}\left(\lambda \right)$ and analytic expression $(-2\lambda /Ste)$ from Equation (28) for the case ${\delta}_{1}={\delta}_{2}=1$.

p | 0.1 | 0.5 | 1 | 1.5 | 2 | 5 | 10 |
---|---|---|---|---|---|---|---|

$-2\lambda /Ste$ | −0.7758 | −0.6773 | −0.6096 | −0.5689 | −0.5418 | −0.4741 | −0.4433 |

${y}^{\prime}\left(\lambda \right)$ | −0.4972 | −0.6441 | −0.6102 | −0.5698 | −0.5427 | −0.4760 | −0.4433 |

**Table 2.**Comparison between the numerical solution ${y}_{num}$ of BVP (1) for the boundary $\lambda =1$ and the analytical approximation ${y}_{app}$ given by Equation (23) for ${\delta}_{1}={\delta}_{2}=5$ and for different values of $p=1,5,10$. For the function, $\beta \left(x\right)$, see the text.

$\mathit{p}=1$ | $\mathit{p}=5$ | $\mathit{p}=10$ | ||||
---|---|---|---|---|---|---|

x | ${\mathit{y}}_{\mathit{app}}$ | ${\mathit{y}}_{\mathit{num}}$ | ${\mathit{y}}_{\mathit{app}}$ | ${\mathit{y}}_{\mathit{num}}$ | ${\mathit{y}}_{\mathit{app}}$ | ${\mathit{y}}_{\mathit{num}}$ |

0 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 |

0.1 | 0.909960 | 0.909904 | 0.949665 | 0.949618 | 0.957271 | 0.957223 |

0.2 | 0.816690 | 0.816590 | 0.889224 | 0.889194 | 0.896205 | 0.896159 |

0.3 | 0.721090 | 0.720860 | 0.814792 | 0.814763 | 0.800557 | 0.800429 |

0.4 | 0.623984 | 0.623694 | 0.720319 | 0.720219 | 0.659203 | 0.659085 |

0.5 | 0.526044 | 0.525706 | 0.599310 | 0.599076 | 0.505868 | 0.505774 |

0.6 | 0.427654 | 0.427191 | 0.456012 | 0.455911 | 0.367732 | 0.367636 |

0.7 | 0.328695 | 0.328249 | 0.312872 | 0.312859 | 0.248849 | 0.248768 |

0.8 | 0.228078 | 0.227540 | 0.187551 | 0.187508 | 0.148830 | 0.148778 |

0.9 | 0.122416 | 0.122437 | 0.0837442 | 0.0837148 | 0.0664419 | 0.0664179 |

1.0 | $4\times {10}^{-6}$ | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

**Table 3.**Comparison between the numerical value of the first derivative ${y}^{\prime}\left(\lambda \right)$ and analytic expression $(-2\lambda /Ste)$ from Equation (31) for ${\delta}_{1}=1,{\delta}_{2}=0.1$.

p | 0.1 | 0.5 | 1 | 1.5 | 2 | 5 | 10 |
---|---|---|---|---|---|---|---|

$-2\lambda /Ste$ | −0.7310 | −0.6382 | −0.5740 | −0.5360 | −0.5102 | −0.4468 | −0.4178 |

${y}^{\prime}\left(\lambda \right)$ | −0.6851 | −0.8026 | −0.7092 | −0.6315 | −0.5855 | −0.4897 | −0.4539 |

**Table 4.**Comparison between the numerical solution ${y}_{num}$ of BVP (1) for the boundary $\lambda =1$ and the first approximation ${y}_{1}$ Equation (16) for ${\delta}_{1}=1,{\delta}_{2}=0.1$ and for different values of $p=1,5,10$. For the function, $\beta \left(x\right)=\frac{1}{2}{e}^{-(1+{\delta}_{2}){x}^{2}}$.

$\mathit{p}=1$ | $\mathit{p}=5$ | $\mathit{p}=10$ | ||||
---|---|---|---|---|---|---|

x | ${\mathit{y}}_{\mathbf{1}}$ | ${\mathit{y}}_{\mathit{num}}$ | ${\mathit{y}}_{\mathbf{1}}$ | ${\mathit{y}}_{\mathit{num}}$ | ${\mathit{y}}_{\mathbf{1}}$ | ${\mathit{y}}_{\mathit{num}}$ |

0 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 |

0.1 | 0.88353 | 0.894305 | 0.901384 | 0.905988 | 0.898449 | 0.902000 |

0.2 | 0.76533 | 0.785659 | 0.784270 | 0.793780 | 0.764844 | 0.772572 |

0.3 | 0.64752 | 0.676322 | 0.653688 | 0.667504 | 0.622902 | 0.633430 |

0.4 | 0.53229 | 0.567523 | 0.520863 | 0.536825 | 0.490117 | 0.501984 |

0.5 | 0.42178 | 0.460092 | 0.396590 | 0.412553 | 0.371442 | 0.383526 |

0.6 | 0.31802 | 0.355452 | 0.286588 | 0.300962 | 0.268065 | 0.279297 |

0.7 | 0.22277 | 0.255541 | 0.192562 | 0.204256 | 0.180066 | 0.189469 |

0.8 | 0.13746 | 0.161506 | 0.114270 | 0.122498 | 0.106851 | 0.113620 |

0.9 | 0.06306 | 0.0753524 | 0.050600 | 0.054836 | 0.0473141 | 0.0508684 |

1.0 | $8\times {10}^{-6}$ | 0.00000 | 0.000013 | 0.00000 | 0.00000 | 0.00000 |

**Table 5.**Comparison between the numerical solution ${y}_{num}$ of BVP (1) for the boundary $\lambda =1$ and the second approximation ${y}_{2}$ of Equation (21) for ${\delta}_{1}=0.1,{\delta}_{2}=1$ and for different values of $p=1,5,10$. For the function, $\beta \left(x\right)=\frac{1}{2}{e}^{-\frac{{x}^{2}}{1+{\delta}_{1}}}$.

$\mathit{p}=1$ | $\mathit{p}=5$ | $\mathit{p}=10$ | ||||
---|---|---|---|---|---|---|

x | ${\mathit{y}}_{\mathbf{2}}$ | ${\mathit{y}}_{\mathit{num}}$ | ${\mathit{y}}_{\mathbf{2}}$ | ${\mathit{y}}_{\mathit{num}}$ | ${\mathit{y}}_{\mathbf{2}}$ | ${\mathit{y}}_{\mathit{num}}$ |

0 | 1.0000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 |

0.1 | 0.859128 | 0.844884 | 0.860807 | 0.855819 | 0.859406 | 0.855716 |

0.2 | 0.723092 | 0.696854 | 0.723068 | 0.714310 | 0.719800 | 0.713261 |

0.3 | 0.594257 | 0.559926 | 0.591772 | 0.580742 | 0.588046 | 0.579486 |

0.4 | 0.474624 | 0.436756 | 0.470282 | 0.458281 | 0.466955 | 0.457202 |

0.5 | 0.365737 | 0.328685 | 0.360565 | 0.348677 | 0.357914 | 0.347866 |

0.6 | 0.268611 | 0.235952 | 0.263571 | 0.252739 | 0.261613 | 0.252179 |

0.7 | 0.183726 | 0.157980 | 0.179527 | 0.170570 | 0.178190 | 0.170217 |

0.8 | 0.111045 | 0.093645 | 0.108117 | 0.101710 | 0.107311 | 0.101518 |

0.9 | 0.050085 | 0.041510 | 0.0486159 | 0.0452548 | 0.0482536 | 0.0451840 |

1.0 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

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**MDPI and ACS Style**

Bougoffa, L.; Bougouffa, S.; Khanfer, A.
An Analysis of the One-Phase Stefan Problem with Variable Thermal Coefficients of Order *p*. *Axioms* **2023**, *12*, 497.
https://doi.org/10.3390/axioms12050497

**AMA Style**

Bougoffa L, Bougouffa S, Khanfer A.
An Analysis of the One-Phase Stefan Problem with Variable Thermal Coefficients of Order *p*. *Axioms*. 2023; 12(5):497.
https://doi.org/10.3390/axioms12050497

**Chicago/Turabian Style**

Bougoffa, Lazhar, Smail Bougouffa, and Ammar Khanfer.
2023. "An Analysis of the One-Phase Stefan Problem with Variable Thermal Coefficients of Order *p*" *Axioms* 12, no. 5: 497.
https://doi.org/10.3390/axioms12050497