# 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

- Schlichting, H.; Gersten, K. Boundary Layer Theory, 8th ed.; Springer: Berlin, Germany, 2000. [Google Scholar]
- Lentini, M.; Keller, H.B. Boundary value problems on semi-infinite intervals and their numerical solutions. SIAM J. Numer. Anal.
**1980**, 17, 577–604. [Google Scholar] [CrossRef] - Fazio, R. A free boundary approach and Keller’s box scheme for BVPs on infinite intervals. Int. J. Comput. Math.
**2003**, 80, 1549–1560. [Google Scholar] [CrossRef] - Weyl, H. On the differential equation of the simplest boundary-layer problems. Ann. Math.
**1942**, 43, 381–407. [Google Scholar] [CrossRef] - Blasius, H. Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math. Phys.
**1908**, 56, 1–37. [Google Scholar] - Falkner, V.M.; Skan, S.W. Some approximate solutions of the boundary layer equations. Philos. Mag.
**1931**, 12, 865–896. [Google Scholar] [CrossRef] - Fazio, R. The iterative transformation method and length estimation for tubular flow reactors. Appl. Math. Comput.
**1991**, 42, 105–110. [Google Scholar] [CrossRef] - Keller, H.B. Numerical Methods for Two-Point Boundary Value Problems, 2nd ed.; Dover Publications: New York, NY, USA, 1992. [Google Scholar]
- Fazio, R.; Evans, D.J. Similarity and numerical analysis for free boundary value problems. Int. J. Comput. Math.
**1990**, 31, 215–220. [Google Scholar] [CrossRef] - Fazio, R. A moving boundary hyperbolic problem for a stress impact in a bar of rate-type material. Wave Motion
**1992**, 16, 299–305. [Google Scholar] [CrossRef] - Fazio, R. Similarity analysis for moving boundary parabolic problems. Comput. Appl. Math. Differ. Equ.
**1992**, 2, 153–162. [Google Scholar] - Fazio, R. The iterative transformation method. Int. J. Nonlin. Mech.
**2019**, 116, 181–194. [Google Scholar] [CrossRef] [Green Version] - Fazio, R. Numerical transformation methods: A constructive approach. J. Comput. Appl. Math.
**1994**, 50, 299–303. [Google Scholar] [CrossRef] [Green Version] - Fox, V.G.; Erickson, L.E.; Fan, L.I. The laminar boundary layer on a moving continuous flat sheet in a non-newtonian fluid. AIChE J.
**1969**, 15, 327–333. [Google Scholar] [CrossRef] - Meyer, G.H. Initial Value Methods for Boundary Value Problems; Theory and Application of Invariant Imbedding; Academic Press: New York, NY, USA, 1973. [Google Scholar]
- Na, T.Y. Computational Methods in Engineering Boundary Value Problems; Academic Press: New York, NY, USA, 1979. [Google Scholar]
- Sachdev, P.L. Nonlinear Ordinary Differential Equations and Their Applications; Marcel Dekker: New York, NY, USA, 1991. [Google Scholar]
- Fazio, R. The non-iterative transformation method. Int. J. Nonlin. Mech.
**2019**, 114, 41–48. [Google Scholar] [CrossRef] [Green Version] - Fazio, R. A numerical test for the existence and uniqueness of solution of free boundary problems. Appl. Anal.
**1997**, 66, 89–100. [Google Scholar] [CrossRef] - Sakiadis, B.C. Boundary-layer behaviour on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE J.
**1961**, 7, 26–28. [Google Scholar] [CrossRef] - Sakiadis, B.C. Boundary-layer behaviour on continuous solid surfaces: II. The boundary layer on a continuous flat surface. AIChE J.
**1961**, 7, 221–225. [Google Scholar] [CrossRef] - Hartree, D.R. On the equation occurring in Falkner–Skan approximate treatment of the equations of the boundary layer. Proc. Camb. Philos. Soc.
**1937**, 33, 223–239. [Google Scholar] [CrossRef] - Stewartson, K. Further solutions of the Falkner–Skan equation. Proc. Camb. Philos. Soc.
**1954**, 50, 454–465. [Google Scholar] [CrossRef] - Coppel, W.A. On a differential equation of boundary-layer theory. Philos. Trans. R. Soc. A
**1960**, 253, 101–136. [Google Scholar] - Craven, A.H.; Pelietier, L.A. On the uniqueness of solutions of the Falkner–Skan equation. Mathematika
**1972**, 19, 129–133. [Google Scholar] [CrossRef] - Craven, A.H.; Pelietier, L.A. Reverse flow solutions of the Falkner–Skan equation for λ>1. Mathematika
**1972**, 19, 135–138. [Google Scholar] [CrossRef] - Veldman, A.E.P.; van de Vooren, A.I. On the generalized Flalkner–Skan equation. J. Math. Anal. Appl.
**1980**, 75, 102–111. [Google Scholar] [CrossRef] [Green Version] - Stewartson, K. The Theory of Laminar Boundary Layers in Compressible Fluids; Oxford University Press: Oxford, UK, 1964. [Google Scholar]
- Cebeci, T.; Keller, T.H.B. Shooting and parallel shooting methods for solving the Falkner–Skan boundary-layer equation. J. Comput. Phys.
**1971**, 7, 289–300. [Google Scholar] [CrossRef] - Asaithambi, A. A numerical method for the solution of the Falkner–Skan equation. Appl. Math. Comput.
**1997**, 81, 259–264. [Google Scholar] [CrossRef] - Asaithambi, A. A finite-difference method for the solution of the Falkner–Skan equation. Appl. Math. Comput.
**1998**, 92, 135–141. [Google Scholar] - Asaithambi, A. A second order finite-difference method for the Falkner–Skan equation. Appl. Math. Comput.
**2004**, 156, 779–786. [Google Scholar] [CrossRef] - Sher, I.; Yakhot, A. New approach to the solution of the Falkner–Skan equation. AIAA J.
**2001**, 39, 965–967. [Google Scholar] [CrossRef] - Kuo, B.L. Application of the differential transformation method to the solutions of the Falkner–Skan wedge flow. Acta Mech.
**2003**, 164, 161–174. [Google Scholar] [CrossRef] - Asaithambi, A. Solution of the Falkner–Skan equation by recursive evaluation of Taylor coefficients. J. Comput. Appl. Math.
**2005**, 176, 203–214. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.; Chen, B. An iterative method for solving the Falkner–Skan equation. Appl. Math. Comput.
**2009**, 210, 215–222. [Google Scholar] [CrossRef] - Auteri, F.; Quartapelle, L. Galerkin–Laguerre spectral solution of self-similar boundary layer problems. Commun. Comput. Phys.
**2012**, 12, 1329–1358. [Google Scholar] [CrossRef] [Green Version] - Hartman, P. Ordinary Differential Equations, 2nd ed.; Birkhäuser: Boston, MA, USA, 1982. [Google Scholar]
- Fazio, R. The iterative transformation method for the Sakiadis problem. Comput. Fluids
**2015**, 106, 196–200. [Google Scholar] [CrossRef] [Green Version] - Fazio, R. The Falkner-Skan equation: Numerical solutions within group invariance theory. Calcolo
**1994**, 31, 115–124. [Google Scholar] [CrossRef] - Töpfer, K. Bemerkung zu dem Aufsatz von H. Blasius: Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math. Phys.
**1912**, 60, 397–398. [Google Scholar] - Fazio, R. Numerical transformation methods: Blasius problem and its variants. Appl. Math. Comput.
**2009**, 215, 1513–1521. [Google Scholar] [CrossRef] - Fazio, R. Blasius problem and Falkner–Skan model: Töpfer’s algorithm and its extension. Comput. Fluids
**2013**, 73, 202–209. [Google Scholar] [CrossRef] [Green Version] - Fazio, R. The Blasius problem formulated as a free boundary value problem. Acta Mech.
**1992**, 95, 1–7. [Google Scholar] [CrossRef]

**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