# Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method

## Abstract

**:**

## 1. Introduction

## 2. The Iterative Transformation Method

Algorithm 1: The iterative algorithm. |

1. Input $\delta $, ${h}_{0}^{*}$, ${h}_{1}^{*}$, ${\eta}_{\infty}^{*}$, $\mathrm{Tol}$. 2. $j=2,3,\cdots $; repeat through step 5 until condition $\left|\mathsf{\Gamma}\right({h}^{*}\left)\right|\le \mathrm{Tol}$ is satisfied.3. Solve (9) in the starred variables on $[0,{\eta}_{\infty}^{*}]$. 4. Compute $\lambda $ by (12). 5. Use equation (14) to get ${\mathsf{\Gamma}}_{j}$. 6. Rescale according to (6). |

## 3. Existence and Uniqueness

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 4. The Transformation Function

## 5. Numerical Tests and Results

#### 5.1. Sakiadis Problem

#### 5.2. The Falkner–Skan Model

## 6. Final Remarks and Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Plot of $\mathsf{\Gamma}\left({h}^{*}\right)$ for $\frac{{d}^{2}{f}^{*}}{{d{\eta}^{*}}^{2}}}\left(0\right)=1$.

**Figure 2.**Plot of $\mathsf{\Gamma}\left({h}^{*}\right)$ for $\frac{{d}^{2}{f}^{*}}{{d{\eta}^{*}}^{2}}}\left(0\right)=-1$.

**Figure 4.**Two cases of the $\mathsf{\Gamma}\left({h}^{*}\right)$ function: top and bottom frames are related to normal and reverse flow solutions, respectively.

**Figure 5.**Normal and reverse flow solutions to Falkner–Skan model for $\beta =-1.5$. The symbols • denote values of $f\left(\eta \right)$.

**Figure 6.**Missing initial conditions to Falkner–Skan model for several values of $\beta $. Positive values determine normal flow, and instead, negative values define reverse flow solutions.

**Figure 7.**Numerical solutions to Falkner–Skan model for $\beta =-0.1988376$. We notice that $\frac{{d}^{2}f}{d{\eta}^{2}}}\left(0\right)=0$ and values of $f\left(\eta \right)$ are marked by •.

j | ${\mathit{h}}_{\mathit{j}}^{*}$ | ${\mathit{\lambda}}_{\mathit{j}}$ | $\mathbf{\Gamma}\left({\mathit{h}}_{\mathit{j}}^{*}\right)$ | $\frac{{\mathit{d}}^{2}\mathit{f}}{{\mathit{d}\mathit{\eta}}^{2}}}\left(0\right)$ |
---|---|---|---|---|

0 | 2.5 | $1.061732$ | 0.967343 | $-0.835517$ |

1 | 3.5 | $1.475487$ | $-0.261541$ | $-0.311310$ |

2 | 3.287172 | $1.417981$ | $-0.186906$ | $-0.350743$ |

3 | 2.754191 | $1.229206$ | 0.206411 | $-0.538426$ |

4 | 3.033897 | $1.339089$ | $-0.056455$ | $-0.416458$ |

5 | 2.973826 | $1.318081$ | $-0.014749$ | $-0.436690$ |

6 | 2.952581 | $1.310382$ | 0.001407 | $-0.444433$ |

7 | 2.954432 | $1.311058$ | $-3.23\mathrm{D}-05$ | $-0.443745$ |

8 | 2.954391 | $1.311043$ | $-6.93\mathrm{D}-08$ | $-0.443761$ |

9 | 2.954391 | $1.311043$ | $3.42\mathrm{D}-12$ | $-0.443761$ |

**Table 2.**Iterations for $\beta =-0.01$ with $\frac{{d}^{2}{f}^{*}}{d{\eta}^{*2}}\left(0\right)}=1$. Here and in the following, the $\mathrm{D}-k={10}^{-k}$ means a double precision arithmetic.

j | ${\mathit{h}}_{\mathit{j}}^{*}$ | $\mathbf{\Gamma}\left({\mathit{h}}_{\mathit{j}}^{*}\right)$ | $\frac{|{\mathit{h}}_{\mathit{j}}^{*}-{\mathit{h}}_{\mathit{j}-1}^{*}|}{|{\mathit{h}}_{\mathit{j}}^{*}|}$ | $\frac{{\mathit{d}}^{2}\mathit{f}}{\mathit{d}{\mathit{\eta}}^{2}}\left(0\right)$ |
---|---|---|---|---|

0 | 5. | 0.631459 | 0.431723 | |

1 | 10. | 1.791425 | 0.384034 | |

2 | 2.278111 | −0.182888 | 3.389602 | 0.454658 |

3 | 2.993420 | 0.0465208 | 0.238960 | 0.454658 |

4 | 2.848366 | 9.5$\mathrm{D}-04$ | 0.050925 | 0.456418 |

5 | 2.845340 | −5.0$\mathrm{D}-06$ | 0.001064 | 0.456455 |

6 | 2.845356 | 6.1$\mathrm{D}-08$ | 5.6$\mathrm{D}-06$ | 0.456455 |

7 | 2.845355 | 7.3$\mathrm{D}-10$ | 6.7$\mathrm{D}-08$ | 0.456455 |

**Table 3.**Iterations for $\beta =-0.01$ with $\frac{{d}^{2}{f}^{*}}{d{\eta}^{*2}}\left(0\right)}=-1$.

j | ${\mathit{h}}_{\mathit{j}}^{*}$ | $\mathbf{\Gamma}\left({\mathit{h}}_{\mathit{j}}^{*}\right)$ | $\frac{|{\mathit{h}}_{\mathit{j}}^{*}-{\mathit{h}}_{\mathit{j}-1}^{*}|}{|{\mathit{h}}_{\mathit{j}}^{*}|}$ | $\frac{{\mathit{d}}^{2}\mathit{f}}{\mathit{d}{\mathit{\eta}}^{2}}\left(0\right)$ |
---|---|---|---|---|

0 | 75. | 0.731890 | −0.059237 | |

1 | 150. | 5.263092 | −0.092368 | |

2 | 62.885833 | −0.443040 | 1.385275 | −0.028870 |

3 | 69.649620 | 0.181067 | 0.097112 | −0.046991 |

4 | 67.687299 | −0.011297 | 0.028991 | −0.042016 |

5 | 67.802542 | −2.1$\mathrm{D}-04$ | 0.001700 | −0.042315 |

6 | 67.804749 | 2.8$\mathrm{D}-07$ | 3.3$\mathrm{D}-05$ | −0.042321 |

7 | 67.804746 | 7.9$\mathrm{D}-10$ | 4.3$\mathrm{D}-08$ | −0.042321 |

**Table 4.**Comparison for the reverse flow skin-friction coefficients $\frac{{d}^{2}f}{d{\eta}^{2}}\left(0\right)$. For all cases, we used ${h}_{0}^{*}=15$ and ${h}_{1}^{*}=25$. The iterations were, from top to bottom line: 8, 7, 9, 7, and 7.

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**MDPI and ACS Style**

Fazio, R. Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method. *Math. Comput. Appl.* **2021**, *26*, 18.
https://doi.org/10.3390/mca26010018

**AMA Style**

Fazio R. Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method. *Mathematical and Computational Applications*. 2021; 26(1):18.
https://doi.org/10.3390/mca26010018

**Chicago/Turabian Style**

Fazio, Riccardo. 2021. "Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method" *Mathematical and Computational Applications* 26, no. 1: 18.
https://doi.org/10.3390/mca26010018