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Applied Mathematics and Nonlinear Sciences
Volume 7 (2022): Issue 1 (January 2022)
Open Access
Precision algorithms in second-order fractional differential equations
Chunguang Liu
Chunguang Liu
| Dec 30, 2021
Applied Mathematics and Nonlinear Sciences
Volume 7 (2022): Issue 1 (January 2022)
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Published Online:
Dec 30, 2021
Page range:
155 - 164
Received:
Jun 17, 2021
Accepted:
Sep 24, 2021
DOI:
https://doi.org/10.2478/amns.2021.2.00157
Keywords
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.
Fig. 1
Output divergence of CFE-Euler simulation model
Fig. 2
Maclaurin series expansion coefficient of Tustin transform
Fig. 3
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
Fig. 5
FOC curve of function sint. FOC, fractional calculus
Fig. 6
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
Fig. 8
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
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