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Precision algorithms in second-order fractional differential equations

Chunguang Liu

Chunguang Liu

Yantai Vocational CollegeYantai, China
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Liu, Chunguang
  
30 dic 2021
Precision algorithms in second-order fractional differential equations's Cover Image
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Pubblicato online: 30 dic 2021
Pagine: 155 - 164
Ricevuto: 17 giu 2021
Accettato: 24 set 2021
DOI: https://doi.org/10.2478/amns.2021.2.00157
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© 2021 Chunguang Liu, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Output divergence of CFE-Euler simulation model
Output divergence of CFE-Euler simulation model

Fig. 2

Maclaurin series expansion coefficient of Tustin transform
Maclaurin series expansion coefficient of Tustin transform

Fig. 3

Maclaurin series expansion coefficient of binomial power function
Maclaurin series expansion coefficient of binomial power function

Fig. 4

f(x)=((1−x)/(1+x))12 f(x) = ((1 - x)/(1 + x{))^{{1 \over 2}}} fitting curve
f(x)=((1−x)/(1+x))12 f(x) = ((1 - x)/(1 + x{))^{{1 \over 2}}} fitting curve

Fig. 5

FOC curve of function sint. FOC, fractional calculus
FOC curve of function sint. FOC, fractional calculus

Fig. 6

Comparison of algorithm curve and theoretical solution curve in this paper
Comparison of algorithm curve and theoretical solution curve in this paper

Fig. 7

The comparison curve between the full memory algorithm in this paper and the G-L definition @@LISAN method
The comparison curve between the full memory algorithm in this paper and the G-L definition @@LISAN method

Fig. 8

Comparison curve between the algorithm in this paper and the conventional PSE-Tustin method. PSE, power series expansion
Comparison curve between the algorithm in this paper and the conventional PSE-Tustin method. PSE, power series expansion

Fig. 9

Step response of CFE-Al-Alaoui discrete method. CFE, continued fraction expansion
Step response of CFE-Al-Alaoui discrete method. CFE, continued fraction expansion

Algorithm error and accuracy indicators

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
200 0.0536 16.1
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Inglese
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