# Parallel Implementation of Modeling of Fractional-Order State-Space Systems Using the Fixed-Step Euler Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Parallel Algorithms

#### 3.1. Fractional-Order Difference

Algorithm 1: Parallel algorithm for one-step computation of fractional-order difference. |

**Remark**

**1.**

**Remark**

**2.**

#### 3.2. Fractional-Order System

Algorithm 2: Parallel algorithm for one-step calculation of fractional-order system. |

Algorithm 3: Hierarchical parallel algorithm for one-step calculation of fractional-order system. |

**Remark**

**3.**

## 4. Simulation Examples

#### 4.1. CPU-Based Hardware

**Example**

**4.**

**Example**

**5.**

#### 4.2. GPU-Based Hardware

**Example**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bauer, W.; Rydel, M. Application of Reduced Models of Non-Integer Order Integrator to the Realization of PI
^{λ}D Controller. In Proceedings of the 39th International Conference on Telecommunications and Signal Processing, Vienna, Austria, 27–29 June 2016. [Google Scholar] - Ferreira, R.A.C.; Tenreiro Machado, J. An Entropy Formulation Based on the Generalized Liouville Fractional Derivative. Entropy
**2019**, 21, 638. [Google Scholar] [CrossRef] - Kaczorek, T. Practical stability of positive fractional discrete-time linear systems. Bull. Pol. Acad. Sci. Tech. Sci.
**2008**, 56, 313–317. [Google Scholar] - Kaczorek, T. Singular fractional linear systems and electrical circuits. Int. J. Appl. Math. Comput. Sci.
**2011**, 21, 379–384. [Google Scholar] [CrossRef] - Karci, A. Fractional order entropy: New perspectives. Optik
**2016**, 127, 9172–9177. [Google Scholar] [CrossRef] - Latawiec, K.J.; Stanisławski, R.; ukaniszyn, M.; Czuczwara, W.; Rydel, M. Fractional-order modeling of electric circuits: Modern empiricism vs. classical science. In Proceedings of the Progress in Applied Electrical Engineering, Kościelisko, Poland, 25–30 June 2017. [Google Scholar]
- Monje, C.; Chen, Y.; Vinagre, B.; Xue, D.; Feliu, V. Fractional-order Systems and Controls; Springer-Verlag: London, UK, 2010. [Google Scholar]
- Muhammad Altaf, K.; Atangana, A. Dynamics of Ebola Disease in the Framework of Different Fractional Derivatives. Entropy
**2019**, 21, 303. [Google Scholar] [CrossRef] - Rydel, M.; Stanislawski, W. Selection of reduction parameters for complex plant MIMO LTI models using the evolutionary algorithm. Math. Comput. Simul.
**2017**, 140, 94–106. [Google Scholar] [CrossRef] - Stanisławski, R.; Rydel, M.; Latawiec, K.J. Modeling of discrete-time fractional-order state space systems using the balanced truncation method. J. Frankl. Inst.
**2017**, 354, 3008–3020. [Google Scholar] [CrossRef] - Ubriaco, M.R. Entropies based on fractional calculus. Phys. Lett. A
**2009**, 373, 2516–2519. [Google Scholar] [CrossRef][Green Version] - Wang, Y.; Zhang, Y.F.; Liu, J.G.; Iqbal, M. A short review on analytical methods for fractional equations with He’s fractional derivative. Therm. Sci.
**2017**, 21, 1567–1574. [Google Scholar] [CrossRef] - Zúñiga-Aguilar, C.J.; Romero-Ugalde, H.M.; Gómez-Aguilar, J.F.; Escobar-Jiménez, R.F.; Valtierra-Rodríguez, M. Solving fractional differential equations of variable-order involving operators with Mittag-Leffer kernel using artificial neural networks. Chaos Solitons Fractals
**2017**, 103, 382–403. [Google Scholar] [CrossRef] - Oprzędkiewicz, K.; Mitkowski, W. A Memory-Efficient Noninteger-Order Discrete-Time State-Space Model of a Heat Transfer Process. Int. J. Appl. Math. Comput. Sci.
**2018**, 28, 649–659. [Google Scholar] [CrossRef] - Wang, K.L.; Liu, S.Y. He’s fractional derivative and its application for fractional Fornberg-Whitham equation. Therm. Sci.
**2017**, 21, 2049–2055. [Google Scholar] [CrossRef] - Lichae, B.H.; Biazar, J.; Ayati, Z. The Fractional Differential Model of HIV-1 Infection of CD
^{+}T-Cells with Description of the Effect of Antiviral Drug Treatment. Comput. Math. Methods Med.**2019**, 2019, 4059549. [Google Scholar] [CrossRef] [PubMed] - Solís-Pérez, J.E.; Gómez-Aguilar, J.F.; Torres, L.; Escobar-Jiménez, R.F.; Reyes-Reyes, J. Fitting of experimental data using a fractional Kalman-like observer. ISA Trans.
**2019**, 88, 153–169. [Google Scholar] [CrossRef] [PubMed] - He, J.H. Fractal calculus and its geometrical explanation. Results Phys.
**2018**, 10, 272–276. [Google Scholar] [CrossRef] - Alzabut, J.; Sudsutad, W.; Kayar, Z.; Baghani, H. A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative. Mathematics
**2019**, 7, 747. [Google Scholar] [CrossRef] - Zúniga-Aguilar, C.J.; Coronel-Escamilla, A.; Gómez-Aguilar, J.F.; Alvarado-Martínez, V.M.; Romero-Ugalde, H.M. New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networks. Eur. Phys. J. Plus
**2018**, 133, 75. [Google Scholar] [CrossRef] - Tanaka, H.-A.; Nakagawa, M.; Oohama, Y. A Direct Link between Rényi-Tsallis Entropy and Hölder’s Inequality-Yet Another Proof of Rényi-Tsallis Entropy Maximization. Entropy
**2019**, 21, 549. [Google Scholar] [CrossRef] - Ibrahim, R.W.; Darus, M. Analytic Study of Complex Fractional Tsallis’ Entropy with Applications in CNNs. Entropy
**2018**, 20, 722. [Google Scholar] [CrossRef] - Herlihy, M.; Shavit, N. The Art of Multiprocessor Programming; Morgan Kaufmann: Burlington, NJ, USA, 2008. [Google Scholar]
- Pacheco, P. An Introduction to Parallel Programming; Morgan Kaufmann: Burlington, NJ, USA, 2011. [Google Scholar]
- Diethelm, K. An Efficient Parallel Algorithm for the Numerical Solution of Fractional Differential Equations. Fract. Calc. Appl. Anal.
**2011**, 14, 475–490. [Google Scholar] [CrossRef] - Bonchis, C.; Kaslik, E.; Rosu, F. HPC optimal parallel communication algorithm for the simulation of fractional-order systems. J. Supercomput.
**2019**, 75, 1014–1025. [Google Scholar] [CrossRef] - Wang, Q.; Liu, J.; Gong, C.; Tang, X.; Fu, G.; Xing, Z. An efficient parallel algorithm for Caputo fractional reaction-diffusion equation with implicit finite-difference method. Adv. Differ. Equ.
**2016**, 2016, 207. [Google Scholar] [CrossRef] - Biala, T.A.; Khaliq, A.Q.M. Parallel algorithms for nonlinear time-space fractional parabolic PDEs. J. Comput. Phys.
**2018**, 375, 135–154. [Google Scholar] [CrossRef] - OpenMP Application Programming Interface. Available online: https://www.openmp.org/wp-content/uploads/ OpenMP-API-Specification-5.0.pdf (accessed on 4 September 2018).
- Scherer, R.; Kalla, S.L.; Tang, Y.; Huang, J. The Grünwald–Letnikov method for fractional differential equations. Comput. Math. Appl.
**2011**, 62, 902–917. [Google Scholar] [CrossRef] - Stanisławski, R.; Latawiec, K.J. Normalized finite fractional differences: Computational and accuracy breakthroughs. Int. J. Appl. Math. Comput. Sci.
**2012**, 22, 907–919. [Google Scholar] [CrossRef] - Stanisławski, R.; Latawiec, K.J.; ukaniszyn, M. A Comparative Analysis of Laguerre-Based Approximators to the Grunwald Letnikov Fractional-Order Difference. Math. Probl. Eng.
**2015**, 2015, 512104. [Google Scholar] - Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1998. [Google Scholar]
- Koufaty, D.; Marr, D.T. Hyperthreading technology in the netburst microarchitecture. IEEE Micro
**2003**, 23, 56–65. [Google Scholar] [CrossRef] - Song, Y.; Kalogeropulos, S.; Tirumalai, P. Design and implementation of a compiler framework for helper threading on multi-core processors. In Proceedings of the 14th International Conference on Parallel Architectures and Compilation Techniques, St. Louis, MO, USA, 17–21 September 2005. [Google Scholar]

**Figure 7.**Execution times for fractional-order state-space system with implementation length $L={2}^{14}$.

**Figure 8.**Execution times for fractional-order state-space system with implementation length $L={2}^{15}$.

**Figure 9.**Execution times for fractional-order state-space system with implementation length $L={2}^{16}$.

**Figure 10.**Execution times for fractional-order state-space system with implementation length $L={2}^{17}$.

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**MDPI and ACS Style**

Stanisławski, R.; Kozioł, K. Parallel Implementation of Modeling of Fractional-Order State-Space Systems Using the Fixed-Step Euler Method. *Entropy* **2019**, *21*, 931.
https://doi.org/10.3390/e21100931

**AMA Style**

Stanisławski R, Kozioł K. Parallel Implementation of Modeling of Fractional-Order State-Space Systems Using the Fixed-Step Euler Method. *Entropy*. 2019; 21(10):931.
https://doi.org/10.3390/e21100931

**Chicago/Turabian Style**

Stanisławski, Rafał, and Kamil Kozioł. 2019. "Parallel Implementation of Modeling of Fractional-Order State-Space Systems Using the Fixed-Step Euler Method" *Entropy* 21, no. 10: 931.
https://doi.org/10.3390/e21100931