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An approximate approach for fractional relaxation-oscillation equation based on Taylor expansion Cover

An approximate approach for fractional relaxation-oscillation equation based on Taylor expansion

International Journal of Mathematics and Computer in Engineering
AHEAD OF PRINT
By: Mohsen Didgar and  Farzan Ekhlasi  
Open Access
|Dec 2025

Figures & Tables

Absolute errors of application 1 for α=12 \alpha = {1 \over 2} _

xMethod in [15]Suggested methodSuggested method
|ψExact – −ψEulerm = 1m = 2
0.11.96 × 10−49.47835 × 10−40
0.22.12 × 10−43.92452 × 10−30
0.31.93 × 10−49.04525 × 10−30
0.41.58 × 10−41.63858 × 10−20
0.51.07 × 10−42.60039 × 10−20
0.64.70 × 10−53.79465 × 10−20
0.71.90 × 10−55.22532 × 10−20
0.89.10 × 10−56.89581 × 10−20
0.91.68 × 10−48.80911 × 10−20
1.02.49 × 10−41.09679 × 10−10

Absolute errors of application 4 for α = 0_5_

xm = 1m = 2m = 3
0.11.83884 × 10−71.67374 × 10−80
0.28.57564 × 10−72.45555 × 10−70
0.31.52024 × 10−55.48380 × 10−70
0.44.06132 × 10−52.33971 × 10−60
0.54.92393 × 10−52.08813 × 10−50
0.67.41184 × 10−48.73476 × 10−50
0.73.30409 × 10−32.70423 × 10−40
0.81.04898 × 10−26.99303 × 10−40
0.92.76795 × 10−21.59673 × 10−30
1.06.47880 × 10−23.32226 × 10−30

Absolute errors of application 3 for α = 1_5 in [18]_

xN = 3N = 5N = 7N = 9
0.12.2655 × 10−58.68960 × 10−101.67093 × 10−103.65474 × 10−14
0.33.4118 × 10−41.60361 × 10−61.67093 × 10−104.21589 × 10−14
0.56.3397 × 10−49.63817 × 10−71.06212 × 10−99.14555 × 10−13
0.71.6705 × 10−41.71558 × 10−77.03957 × 10−108.21475 × 10−14
0.95.5043 × 10−48.98678 × 10−71.92731 × 10−101.32512 × 10−13

Absolute errors of application 5 for α=12 \alpha = {1 \over 2} _

xMethod in [19]Suggested method m = 1Suggested method m = 2
0.1252.78 × 10−31.39 × 10−50
0.2508.95 × 10−31.81 × 10−40
0.3752.06 × 10−37.90 × 10−40
0.5003.81 × 10−32.21 × 10−30
0.6253.79 × 10−34.72 × 10−30
0.7508.62 × 10−38.50 × 10−30
0.8757.19 × 10−31.34 × 10−20
1.0002.90 × 10−31.92 × 10−20

Absolute errors of application 3 for α = 1_5_

xm = 1m = 2m = 3
0.11.61568 × 10−42.16231 × 10−50
0.21.29155 × 10−31.72617 × 10−40
0.34.37732 × 10−35.80789 × 10−40
0.41.04941 × 10−21.37156 × 10−30
0.52.08974 × 10−22.66859 × 10−30
0.63.71081 × 10−24.59691 × 10−30
0.76.09707 × 10−27.28893 × 10−30
0.89.46758 × 10−21.08933 × 10−20
0.91.40749 × 10−11.55858 × 10−20
1.02.02012 × 10−12.15812 × 10−20

Absolute errors of application 4 for α = 0_5 in [18]_

xN = 3N = 5N = 7
0.11.8769 × 10−42.98543 × 10−78.029610 × 10−10
0.39.7022 × 10−43.84222 × 10−63.057123 × 10−9
0.52.6184 × 10−37.27991 × 10−66.018712 × 10−9
0.73.4247 × 10−31.24040 × 10−69.156311 × 10−9
0.96.9465 × 10−31.75728 × 10−59.754413 × 10−9

The maximum of the absolute errors in [16, 17]_

Method in [16]Method in [17]
0.44 × 10−77.22 × 10−12

The maximum of the absolute errors for application 8_

mMaximal Errors
52.27804 × 10−3
61.51433 × 10−3
71.06370 × 10−3
87.79340 × 10−4
95.90323 × 10−4
104.59348 × 10−4

Approximation errors of Legendre-collocation scheme proposed in [22] for application 8_

NMaximal Errors
50.0221
100.0067
150.0033
200.0019

Absolute errors of application 2 for α=32 \alpha = {3 \over 2} _

xm = 1m = 2
0.17.96634 × 10−50
0.24.32073 × 10−40
0.31.13436 × 10−30
0.42.21276 × 10−30
0.53.67007 × 10−30
0.65.49867 × 10−30
0.77.68955 × 10−30
0.81.02393 × 10−20
0.91.31560 × 10−20
1.01.64640 × 10−20

Absolute errors of application 7 for α=12 \alpha = {1 \over 2} _

x(Method in [20]) |ψExact −ψFFDM|Suggested method m = 1Suggested method m = 2
0.11.16 × 10−49.47835 × 10−40
0.21.56 × 10−43.92452 × 10−30
0.31.81 × 10−49.04525 × 10−30
0.42.00 × 10−41.63858 × 10−20
0.52.15 × 10−42.60039 × 10−20
0.62.27 × 10−43.79465 × 10−20
0.72.37 × 10−45.22532 × 10−20
0.82.46 × 10−46.89581 × 10−20
0.92.54 × 10−48.80911 × 10−20
1.02.61 × 10−41.09679 × 10−10

Absolute errors of application 6 for α=12 \alpha = {1 \over 2} _

x(Method in [20]) |ψExact −ψFFDM|Suggested method m = 1Suggested method m = 2
0.11.49 × 10−48.57143 × 10−40
0.22.19 × 10−43.42857 × 10−30
0.32.72 × 10−47.71429 × 10−30
0.43.17 × 10−41.37143 × 10−20
0.53.57 × 10−42.14286 × 10−20
0.63.93 × 10−43.08571 × 10−20
0.74.26 × 10−44.20000 × 10−20
0.84.56 × 10−45.48571 × 10−20
0.94.85 × 10−46.94286 × 10−20
1.05.12 × 10−48.57143 × 10−20
DOI: https://doi.org/10.2478/ijmce-2025-0030 | Journal eISSN: 2956-7068
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Language: English
Submitted on: Oct 16, 2024
Accepted on: Jan 1, 2025
Published on: Dec 20, 2025
Published by: Harran University
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:
Fractional differential equation,
fractional relaxation-oscillation equation,
RL fractional derivative,
RL fractional integral,
Taylor expansion,
error analysis
Related subjects:
Computer sciences,
Computer sciences, other,
Engineering,
Introductions and overviews,
Engineering, other,
Mathematics,
General mathematics,
Physics,
Physics, other

© 2025 Mohsen Didgar, Farzan Ekhlasi, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

AHEAD OF PRINT