<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>The discretization of fractional-order differential operators is the key to the digital realization of fractional-order controllers. This paper proposes an improved second-order fractional differential equation operation method based on power series expansion. The algorithm's operation speed and accuracy performance are analyzed. The research found that the algorithm proposed in this paper is suitable for the fractional operation of arbitrary signals, including discrete data sequences whose mathematical model is unknown and the solution of linear systems.</p>
</abstract>

The discretization of fractional-order differential operators is the key to the digital realization of fractional-order controllers. This paper proposes an improved second-order fractional differential equation operation method based on power series expansion. The algorithm's operation speed and accuracy performance are analyzed. The research found that the algorithm proposed in this paper is suitable for the fractional operation of arbitrary signals, including discrete data sequences whose mathematical model is unknown and the solution of linear systems.

Precision algorithms in second-order fractional differential equations

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Precision algorithms in second-order fractional differential equations"}

The discretization of fractional-order differential operators is the key to the digital realization of fractional-order controllers. This paper proposes an...

Algorithm name		Error	Accuracy index

This paper algorithm (memory length: N)	N = 20	0.0649	13.3
	N = 50	0.0301	28.6
	N = 100	0.0161	53.5
	Full memory	0.012	370.1
G-L definition discrete method (full memory)	T = 0.1s	0.862	1
	T = 0.05s	0.0557	15.4
	T = 0.01s	0.0194	44.5
	T = 0.001s	0.0052	165.8
Tustin PSE overall expansion	50	0.0683	12.6
	100	0.0397	21.7
	200	0.0123	70.1
	50	0.0575	15
Al-Alaoui PSE overall expansion	100	0.0543	15.9
Al-Alaoui PSE overall expansion	200	0.0536	16.1

Precision algorithms in second-order fractional differential equations

Publicado en línea: 30 dic 2021

Páginas: 155 - 164

Recibido: 17 jun 2021

Aceptado: 24 sept 2021

DOI: https://doi.org/10.2478/amns.2021.2.00157

Palabras clave
second-order fractional differential equation, fractional-order system, recursive algorithm, accuracy, power series expansion

© 2021 Chunguang Liu, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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Algorithm error and accuracy indicators

Precision algorithms in second-order fractional differential equations

Publicado en línea: 30 dic 2021

Páginas: 155 - 164

Recibido: 17 jun 2021

Aceptado: 24 sept 2021

DOI: https://doi.org/10.2478/amns.2021.2.00157

Palabras clavesecond-order fractional differential equation, fractional-order system, recursive algorithm, accuracy, power series expansion

© 2021 Chunguang Liu, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Algorithm error and accuracy indicators

Palabras clave
second-order fractional differential equation, fractional-order system, recursive algorithm, accuracy, power series expansion