# Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Basic Concepts of Quaternion

#### 2.1.1. Quaternion

**Proposition**

**1.**

- $\Vert q\Vert =\sqrt{q{q}^{\mathrm{*}}}=\sqrt{{{q}_{0}}^{2}+{{q}_{1}}^{2}+{{q}_{2}}^{2}+{{q}_{3}}^{2}}$
- $\Vert {q}^{\mathrm{*}}\Vert =\Vert q\Vert $
- $\Vert pq\Vert =\Vert p\Vert \Vert q\Vert $

#### 2.1.2. Derivative of Quaternion Function

**Definition**

**1.**

**Proposition**

**2.**

- (a)
- ${\left({q}^{n}\right)}^{\mathrm{\prime}}=n{q}^{n-1}$, for $n=1,2,3,\dots $
- (b)
- ${\left({e}^{q}\right)}^{\mathrm{\prime}}={e}^{q}$
- (c)
- ${\left(\mathrm{sin}q\right)}^{\mathrm{\prime}}=\mathrm{cos}q$
- (d)
- ${\left(\mathrm{cos}q\right)}^{\mathrm{\prime}}=-\mathrm{sin}q$.

#### 2.1.3. Quaternionic Regular Function

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**,**it follows that

**Definition**

**5.**

#### 2.1.4. Quaternion Differential Equation

- Quaternion frenet frame on differential geometry.

- b.
- Quaternion differential equations in kinematic modeling and attitude dynamics.

- c.
- Quaternion differential equations in fluid mechanics.

- d.
- Quaternion differential equations in quantum mechanics.

#### 2.2. Database and Search Strategy

#### 2.3. Data Analysis and Visualization

## 3. Results

#### 3.1. Result from Bibliometric Analysis

#### 3.2. Development of Quaternion Differential Equation

#### 3.3. Result from Systematic Literature Review

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Familton, J.C. Quaternions: A History of Complex Noncommutative Rotation Groups in Theoretical Physics. Ph.D. Thesis, Columbia University, New York, NY, USA, 2015. [Google Scholar]
- Tait, P.G. An Elementary Treatise Quaternions, 2nd ed.; Nabu Press: Charleston, SC, USA, 1878; ISBN 3663537137. [Google Scholar]
- Voight, J. Quaternion Algebras; Springer Nature: Berlin/Heidelberg, Germany, 2020. [Google Scholar]
- Kaya, O.; Onder, M. On Fibonacci and Lucas Vectors and Quaternions. Univers. J. Appl. Math.
**2018**, 13, 156–163. [Google Scholar] [CrossRef] - Catarino, P. A Note on h(x)—Fibonacci Quaternion Polynomials. Chaos Solitons Fractals
**2015**, 77, 1–5. [Google Scholar] [CrossRef] - Halici, S. On Fibonacci Quaternions. Adv. Appl. Clifford Algebr.
**2012**, 22, 321–327. [Google Scholar] [CrossRef] - Halici, S.; Karataş, A. On a Generalization for Fibonacci Quaternions. Chaos Solitons Fractals
**2017**, 98, 178–182. [Google Scholar] [CrossRef] - Horadam, A.F. Complex Fibonacci Numbers and Fibonacci Quaternions. Am. Math. Mon.
**2012**, 70, 289–291. [Google Scholar] [CrossRef] - Tan, E.; Yilmaz, S.; Sahin, M. On a New Generalization of Fibonacci Quaternions. Chaos Solitons Fractals
**2016**, 82, 1–4. [Google Scholar] [CrossRef] - Kamano, K. Analytic Continuation of the Lucas Zeta and L-Functions. Indag. Math.
**2013**, 24, 637–646. [Google Scholar] [CrossRef] - De Bie, H.; De Schepper, N.; Ell, T.A.; Rubrecht, K.; Sangwine, S.J. Connecting Spatial and Frequency Domains for the Quaternion Fourier Transform. Appl. Math. Comput.
**2015**, 271, 581–593. [Google Scholar] [CrossRef] - Bahri, M.; Hitzer, E.S.M.; Hayashi, A.; Ashino, R. An Uncertainty Principle for Quaternion Fourier Transform. Comput. Math. Appl.
**2008**, 56, 2398–2410. [Google Scholar] [CrossRef] - Hitzer, E.M.S. Quaternion Fourier Transform on Quaternion Fields and Generalizations. Adv. Appl. Clifford Algebr.
**2007**, 17, 497–517. [Google Scholar] [CrossRef] - Lian, P. The Octonionic Fourier Transform: Uncertainty Relations and Convolution. Signal Process.
**2019**, 164, 295–300. [Google Scholar] [CrossRef] - Ell, T.A.; Le Bihan, N.; Sangwine, S.J. Quaternion Fourier Transforms for Signal; Castanié, F., Ed.; ISTE Ltd.: London, UK, 2014; ISBN 9781848214781. [Google Scholar]
- Bahri, M.; Toaha, S.; Rahim, A.; Ivan, M. On One-Dimensional Quaternion Fourier Transform On One-Dimensional Quaternion Fourier Transform. J. Phys. Conf. Ser.
**2019**, 1341, 062004. [Google Scholar] [CrossRef] - Sudbery, A. Quaternionic Analysis; Cambridge University Press: Cambridge, UK, 1979. [Google Scholar] [CrossRef]
- Dzagnidze, O. On the Differentiability of Quaternion Functions. arXiv
**2012**, arXiv:1203.5619. [Google Scholar] [CrossRef] - Chudá, H. Universal Approach to Derivation of Quaternion Rotation Formulas. MATEC Web Conf.
**2019**, 292, 01060. [Google Scholar] [CrossRef] - Van Leunen, H. Quaternions and Hilbert Spaces. 2015. Available online: https://www.researchgate.net/publication/282655670_Quaternions_and_Hilbert_spaces (accessed on 19 December 2021).
- Ha, V.T.N. Helmholtz Operator in Quaternionic Analysis. Ph.D Dissertation, Freien Universitat, Berlin, Germany, 10 February 2005. [Google Scholar]
- Hashim, H.A. Special Orthogonal Group SO(3), Euler Angles, Angle-axis, Rodriguez Vector and Unit-Quaternion: Overview, Mapping and Challenges. arXiv. 2019. Available online: https://arxiv.org/abs/1909.06669 (accessed on 19 December 2021).
- Sveier, A.; Sjøberg, A.M.; Egeland, O. Applied Runge-Kutta-Munthe-Kaas Integration for the Quaternion Kinematics. J. Guid. Control. Dyn.
**2019**, 42, 2747–2754. [Google Scholar] [CrossRef] - Klitzner, H. The Culture of Quaternions The Phoenix Bird of Mathematics; New York Academy of Sciences, Lyceum Society: New York, NY, USA, 1 June 2015. [Google Scholar] [CrossRef]
- Shoemake, K. Animating Rotation with Quaternion Curves. ACM SIGGRAPH Comput. Graph.
**1985**, 19, 245–254. [Google Scholar] [CrossRef] - Dam, E.B.; Koch, M.; Lillholm, M. Quaternions, Interpolation and Animation; Institute of Computer Science University of Copenhagen: Denmark, Copenhagen, 1998. [Google Scholar]
- Waldvogel, J. Quaternions for Regularizing Celestial Mechanics—The Right Way. Celest. Mech. Dyn. Astron.
**2008**, 102, 149–162. [Google Scholar] [CrossRef] - Kwasniewski, A.K. Glimpses of the Octonions and Quaternions History and Today’ s Applications in Quantum Physics. Adv. Appl. Clifford Algebr.
**2012**, 22, 87–105. [Google Scholar] [CrossRef] - Haetinger, C.; Malheiros, M.; Dullius, E.; Kronbauer, M. A Quaternion Application to Control Rotation Movements in The Three Dimensional Space of an Articulate Mechanical Arm Type Robot Built from Low Cost Materials as a Supporting Tool for Teaching at The Undergraduate Level. In Proceedings of the Global Congress on Engineering and Technology Education, São Paulo, Brazil, 13–16 March 2005. [Google Scholar]
- Solà, J. Quaternion Kinematics for The Error-State Kalman Filter. arXiv
**2017**, arXiv:1711.02508. [Google Scholar] - Xie, C.; Kumar, B.V.K.V. Quaternion Correlation Filters for Color Face Recognition. In Proceedings of the Security, Steganography, and Watermarking of Multimedia Contents VII, San Jose, CA, USA, 17–20 January 2005; Volume 5681, pp. 486–494. [Google Scholar]
- Giirlebeck, K.; SproBig, W. Quatemionic Analysis and Elliptic Boundary Value Problems; Birkhäuser Basel: Basel, Switzerland, 1990; ISBN 2013206534. [Google Scholar]
- Georgiev, S. New Aspects on Elementary Functions in the Context of Quaternionic Analysis. Cubo
**2012**, 14, 93–110. [Google Scholar] [CrossRef] - Stover, C. A Survey of Quaternionic Analysis; Florida State University: Tallahassee, FL, USA, 2014. [Google Scholar] [CrossRef]
- Dzagnidze, O. On Some New Properties of Quaternion Functions. J. Math. Sci.
**2018**, 235, 557–603. [Google Scholar] [CrossRef] - De Leo, S.; Ducati, G.C. Solving Simple Quaternionic Differential Equations. J. Math. Phys.
**2003**, 44, 2224–2233. [Google Scholar] [CrossRef] - Campos, J.; Mawhin, J. Periodic Solutions of Quaternionic-Valued Ordinary. Ann. Di Mat.
**2006**, 185, 109–127. [Google Scholar] [CrossRef] - Wilczynski, P. Quaternionic-Valued Ordinary Differential Equations. Riccati Equ.
**2009**, 247, 2163–2187. [Google Scholar] [CrossRef] - Papillon, C.; Tremblay, S. On a Three-Dimensional Riccati Differential Equation and its Symmetries. J. Math. Anal. Appl.
**2018**, 458, 611–627. [Google Scholar] [CrossRef] - Grigorian, G.A. Global Solvability Criteria for Quaternionic Riccati Equations. Arch. Math.
**2019**, 57, 83–99. [Google Scholar] [CrossRef] - Zhi, W.; Chu, J.; Li, J.; Wang, Y. A Novel Attitude Determination System Aided by Polarization Sensor. Sensors
**2018**, 10, 158. [Google Scholar] [CrossRef] - Kou, K.I.; Xia, Y.; Xia, Y.-H. Linear Quaternion Differential Equations: Basic Theory and Fundamental Results. Stud. Appl. Math.
**2018**, 141, 1–43. [Google Scholar] [CrossRef] - Kou, K.I.; Liu, W.-K.; Xia, Y.-H. Solve the Linear Quaternion-Valued Differential Equations Having Multiple Eigenvalues. J. Math. Phys.
**2019**, 60, 023510. [Google Scholar] - Cai, Z.; Kou, K.L. Laplace Transform: A New Approach in Solving Linear Quaternion Differential Equations. Math. Methods Appl. Sci.
**2017**, 41, 4033–4048. [Google Scholar] [CrossRef] - Donachali, A.K.; Jafari, H. A Decomposition Method for Solving Quaternion Differential Equations. Int. J. Appl. Comput. Math.
**2020**, 123, 1–7. [Google Scholar] [CrossRef] - Jia, Y.-B. Quaternions. Com S
**2019**, 477, 577. [Google Scholar] - Morris, D. Elementary Calculus from an Advanced Standpoint; Abane and Right: Port Mulgrave, UK, 2016. [Google Scholar]
- Morris, D. Quaternions; Abane and Right: Port Mulgrave, UK, 2015. [Google Scholar]
- Gürlebeck, K.; Sprössig, W. Quaternionic and Clifford Calculus for Physicists and Engineers; Willey: Hoboken, NJ, USA, 1998. [Google Scholar]
- Gentili, G.; Stoppato, C.; Struppa, D.C. Regular Functions of a Quaternionic Variable; Springer: Cham, Switzerland, 2022; ISBN 9783031075308. [Google Scholar] [CrossRef]
- Ellegaard, O.; Wallin, J.A. The bibliometric analysis of scholarly production: How great is the impact? Scientometrics
**2015**, 105, 1809–1831. [Google Scholar] [CrossRef] - Simmons, G.F. Differential Equations with Applications and Historical Notes, 3rd ed.; Boggess, A., Rosen, K., Eds.; CRC Press: New York, NY, USA, 2017; Volume 4, ISBN 2013206534. [Google Scholar]
- Nagy, G. Ordinary Differential Equations; Michigan State University: East Lansing, MI, USA, 2020. [Google Scholar]
- Deimling, K. Lecture Notes in Mathematics: Ordinary Differential Equations in Banach Spaces; Springer: Berlin/Heidelberg, Germany, 2013; Volume 2084, ISBN 9783319008240. [Google Scholar]
- Yang, B.; Bao, W. Complex-Valued Ordinary Differential Equation Modeling for Time Series Identification. IEEE Access
**2019**, 7, 41033–41042. [Google Scholar] [CrossRef] - Feng, Z.; Kit, C.; Kou, I. Solving Quaternion Ordinary Differential Equations with Two-Sided Coefficients. Qual. Theory Dyn. Syst.
**2017**, 17, 441–462. [Google Scholar] [CrossRef] - Hille, E. Ordinary Differential Equations in the Complex Domain by Einar Hille (z-lib.org).pdf; John Willey and Sons: San Diego, CA, USA, 1976. [Google Scholar]
- Laine, I. Complex differential equations. In Handbook of Differential Equations: Ordinary Differential Equations; Chapman and Hall/CRC: Boca Raton, FL, USA, 2008; Volume 4, pp. 269–363. ISBN 9780444530318. [Google Scholar] [CrossRef]
- Haraoka, Y. Linear Differential Equations in the Complex Domain: From Classical Theory to Forefront; Springer: Berlin/Heidelberg, Germany, 2020; Volume 2271, ISBN 9783030546625. [Google Scholar] [CrossRef]

**Figure 1.**Frenet Frame on a Curve (Modified by [42]).

**Figure 3.**(

**a**) Network visualization of co-occurrence word relation in datasets, (

**b**) Overlay visualization in datasets, (

**c**) Overlay visualization in datasets for quaternion differential equation.

Keyword | Google Scholar | Scopus | Science Direct | Dimensions |
---|---|---|---|---|

“quaternion differential equation” | 55 | 45 | 38 | 27 |

“quaternionic differential equation” | 13 | 12 | 9 | 11 |

Total | 68 | 57 | 47 | 38 |

No | Authors | Title | Year | Topic | Object |
---|---|---|---|---|---|

1 | Anthony Sudbery [17] | Quaternionic Analysis | 1978 | function derivative | $f:\mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

2 | S. Georgiev [33] | New Aspects of Elementary Functions in the Context of Quaternionic Analysis | 2012 | quaternion elementary function derivative | $f:\mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

3 | Christopher Stover [34] | A Survey of Quaternionic Analysis | 2014 | derivative/the analytic function | $f:\mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

4 | Omar Dzagnidze [18] | On the Differentiability of Quaternion Functions | 2012 | function derivative | $f:\mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

5 | Omar Dzagnidze [35] | On Some New Properties of Quaternion Function | 2018 | function derivative | $f:\mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

6 | Stefano De Leo and Gisele C. Ducati [36] | Solving simple Quaternionic differential equations | 2003 | quaternion differential equations | $f:\mathbb{\mathbb{R}}\times \mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

7 | Juan Campos and Jean Mawhin [37] | Periodic solutions of Quaternionic-valued ordinary differential equations | 2005 | quaternion differential equations | $f:\mathbb{\mathbb{R}}\times \mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

8 | Paweł Wilczynski [38] | Quaternionic-valued ordinary differential equations. The Riccati equation | 2009 | Riccati quaternion differential equation | $f:\mathbb{\mathbb{R}}\times \mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

9 | Charles Papillon, Sébastien Tremblay [39] | On a three-dimensional Riccati differential equation and its symmetries | 2018 | three-dimensional Riccati differential equation | $f:{\mathbb{\mathbb{R}}}^{3}\times \mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}\left(\mathbb{\u2102}\right)$ |

10 | G. A. Grigorian [40] | Global solvability criteria for Quaternionic Riccati equations | 2019 | Riccati quaternion differential equation | $f:\mathbb{\mathbb{R}}\times \mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

11 | Kit Ian Kou and Yong-Hui Xia [42] | Linear Quaternion Differential Equations: Basic Theory and Fundamental Results | 2018 | linear quaternion differential equations | $f:\mathbb{\mathbb{R}}\times \mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

12 | Kit Ian Kou, Wan-Kai Liu, and Yong-Hui Xia [43] | Solve the linear Quaternion-valued differential equations having multiple eigenvalues | 2019 | linear quaternion differential equations | $f:\mathbb{\mathbb{R}}\times \mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

13 | Zhen-Feng Cai dan Kit Ian Kou [44] | Laplace transform, which is a new approach for solving linear Quaternion differential equations | 2017 | linear quaternion differential equations | $f:\mathbb{\mathbb{R}}\times \mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

14 | A. Kameli Donachali dan H. Jafari [45] | A Decomposition Method for Solving Quaternion Differential Equations | 2020 | quaternion differential equations | $f:\mathbb{\mathbb{R}}\times \mathbb{\mathbb{H}}\to \mathbb{\mathbb{H}}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kartiwa, A.; Supriatna, A.K.; Rusyaman, E.; Sulaiman, J.
Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction. *Axioms* **2023**, *12*, 483.
https://doi.org/10.3390/axioms12050483

**AMA Style**

Kartiwa A, Supriatna AK, Rusyaman E, Sulaiman J.
Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction. *Axioms*. 2023; 12(5):483.
https://doi.org/10.3390/axioms12050483

**Chicago/Turabian Style**

Kartiwa, Alit, Asep K. Supriatna, Endang Rusyaman, and Jumat Sulaiman.
2023. "Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction" *Axioms* 12, no. 5: 483.
https://doi.org/10.3390/axioms12050483