International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 407-415
Published online December 25, 2024
https://doi.org/10.5391/IJFIS.2024.24.4.407
© The Korean Institute of Intelligent Systems
Maryam Almutairi and Norazrizal Aswad bin Abdul Rahman
School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia
Correspondence to :
Abdul Rahman (aswad.rahman@usm.my)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fuzzy fractional partial differential equations have become a powerful approach for handling uncertainty or imprecision in real-world modeling problems. In this study, two finite difference schemes, the Crank-Nicolson and centered-time centered-space methods, were developed and used to obtain a numerical solution for double-parametric fuzzy time fractional wave equations. Fuzzy set theory principles were employed to perform fuzzy analysis and formulate the proposed numerical schemes. The Caputo formula was used to define the time-fractional derivative. To illustrate the practicality of the numerical method, a specific numerical instance was analyzed. The results are presented in tables and figures, revealing the efficacy of the schemes in terms of accuracy and ability to reduce computational expenses. A novel fuzzy computational approach known as the double-parametric form enabled these achievements.
Keywords: Fuzzy Caputo formula, Fuzzy time fractional wave equation, Finite difference method, Double parametric form
No potential conflict of interest relevant to this article was reported.
E-mail: maryam.almutairi@student.usm.my
E-mail: aswad.rahman@usm.my
International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 407-415
Published online December 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.4.407
Copyright © The Korean Institute of Intelligent Systems.
Maryam Almutairi and Norazrizal Aswad bin Abdul Rahman
School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia
Correspondence to:Abdul Rahman (aswad.rahman@usm.my)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fuzzy fractional partial differential equations have become a powerful approach for handling uncertainty or imprecision in real-world modeling problems. In this study, two finite difference schemes, the Crank-Nicolson and centered-time centered-space methods, were developed and used to obtain a numerical solution for double-parametric fuzzy time fractional wave equations. Fuzzy set theory principles were employed to perform fuzzy analysis and formulate the proposed numerical schemes. The Caputo formula was used to define the time-fractional derivative. To illustrate the practicality of the numerical method, a specific numerical instance was analyzed. The results are presented in tables and figures, revealing the efficacy of the schemes in terms of accuracy and ability to reduce computational expenses. A novel fuzzy computational approach known as the double-parametric form enabled these achievements.
Keywords: Fuzzy Caputo formula, Fuzzy time fractional wave equation, Finite difference method, Double parametric form
Analytical lower solution of
Analytical upper solution of
Analytical and FDM solutions to
Fuzzy analytical and fuzzy numerical solutions of
Table 1 . Numerical results of Eq. (15) by CTCS and C-N at
CTCS | C-N | ||||
---|---|---|---|---|---|
0 | –1.892094 | 1.46803 × 10−3 | –1.877098 | 1.35277 × 10−2 | |
0.1 | –1.702885 | 1.32123 × 10−3 | –1.689389 | 1.21749 × 10−2 | |
0.3 | –1.324466 | 1.02762 × 10−3 | –1.313968 | 9.46938 × 10−3 | |
0.5 | –0.946047 | 7.34015 × 10−4 | –0.938549 | 6.76384 × 10−3 | |
0.7 | –0.567628 | 4.40409 × 10−4 | –0.563129 | 4.0583 × 10−3 | |
0.9 | –0.1892094 | 1.46803 × 10−4 | –0.187709 | 1.35277 × 10−3 | |
1 | 0 | 0 | 0 | 0 | |
0 | 1.892094 | 1.46803 × 10−3 | 1.877098 | 1.35277 × 10−2 | |
0.1 | 1.702885 | 1.32123 × 10−3 | 1.689389 | 1.21749 × 10−2 | |
0.3 | 1.324466 | 1.02762 × 10−3 | 1.313968 | 9.46938 × 10−3 | |
0.5 | 0.946047 | 7.34015 × 10−4 | 0.938549 | 6.76384 × 10−3 | |
0.7 | 0.567628 | 4.40409 × 10−4 | 0.563129 | 4.0583 × 10−3 | |
0.9 | 0.1892094 | 1.46803 × 10−4 | 0.187709 | 1.35277 × 10−3 | |
1 | 0 | 0 | 0 | 0 |
Table 2 . Numerical solutions to Eq. (15) by CTCS and C-N at
CTCS | C-N | ||||
---|---|---|---|---|---|
0 | –0.3784189 | 2.93606 × 10−4 | –0.375419 | 2.70554 × 10−3 | |
0.1 | –0.340577 | 2.64245 × 10−4 | –0.337878 | 2.43498 × 10−3 | |
0.3 | –0.264893 | 2.05524 × 10−4 | –0.262794 | 1.89388 × 10−3 | |
0.5 | –0.189209 | 1.46803 × 10−4 | –0.187710 | 1.35277 × 10−3 | |
0.7 | –0.113526 | 8.80818 × 10−5 | –0.111140 | 8.11661 × 10−4 | |
0.9 | –0.037842 | 2.93606 × 10−5 | –0.037542 | 2.70554 × 10−4 | |
1 | 0 | 0 | 0 | 0 | |
0 | 0.3784189 | 2.93606 × 10−4 | 0.375419 | 2.70554 × 10−3 | |
0.1 | 0.340577 | 2.64245 × 10−4 | 0.337878 | 2.43498 × 10−3 | |
0.3 | 0.264893 | 2.05524 × 10−4 | 0.262794 | 1.89388 × 10−3 | |
0.5 | 0.189209 | 1.46803 × 10−4 | 0.187710 | 1.35277 × 10−3 | |
0.7 | 0.113526 | 8.80818 × 10−5 | 0.111140 | 8.11661 × 10−4 | |
0.9 | 0.037842 | 2.93606 × 10−5 | 0.037542 | 2.70554 × 10−4 | |
1 | 0 | 0 | 0 | 0 |
Analytical lower solution of
Analytical upper solution of
Analytical and FDM solutions to
Fuzzy analytical and fuzzy numerical solutions of