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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

A Solution of Fuzzy Time Fractional Wave Equation via Two Modified Implicit Finite Difference Schemes

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)

Received: October 20, 2023; Revised: August 15, 2024; Accepted: September 30, 2024

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.

Maryam Almutairi is a Ph.D. student at the University of Science, Malaysia (USM). She received her M.A. from the University of Science, Malaysia (USM). Her research interests include numerical solutions and computational optimization fields in fuzzy fractional partial differential equations.

E-mail: maryam.almutairi@student.usm.my

Norazrizal Aswad bin Abdul Rahman is a senior lecturer in the School of Mathematical Sciences at the University of Science Malaysia (USM). His research interests include fuzzy set theory and fuzzy logic, statistical analysis (SPSS, Minitab), graphics technology, operational research, and mathematics education.

E-mail: aswad.rahman@usm.my

Article

Original Article

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.

A Solution of Fuzzy Time Fractional Wave Equation via Two Modified Implicit Finite Difference Schemes

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)

Received: October 20, 2023; Revised: August 15, 2024; Accepted: September 30, 2024

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.

Abstract

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

Fig 1.

Figure 1.

Analytical lower solution of Eq. (15) at t = 0.005, x = 1.8, and r = 0.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 407-415https://doi.org/10.5391/IJFIS.2024.24.4.407

Fig 2.

Figure 2.

Analytical upper solution of Eq. (15) at t = 0.005, x = 1.8, and r = 0.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 407-415https://doi.org/10.5391/IJFIS.2024.24.4.407

Fig 3.

Figure 3.

Analytical and FDM solutions to Eq. (15) at α = 1.5, x = 1.8, t = 0.005 for all r ∈ [0, 1].

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 407-415https://doi.org/10.5391/IJFIS.2024.24.4.407

Fig 4.

Figure 4.

Fuzzy analytical and fuzzy numerical solutions of Eq. (15) by (a) CTCS and (b) C-N at t = 0.005 and x = 1.8 for all r, β ∈ [0, 1].

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 407-415https://doi.org/10.5391/IJFIS.2024.24.4.407

Table 1 . Numerical results of Eq. (15) by CTCS and C-N at t = 0.005 and x = 0.9 for all r, β ∈ [0, 1].

βrCTCSC-N
ũ(1.8, 0.05; r, β)(1.8, 0.05; r, β)ũ(1.8, 0.05; r, β) (1.8, 0.05; r, β)
β = 0 Lower solution0–1.8920941.46803 × 10−3–1.8770981.35277 × 10−2
0.1–1.7028851.32123 × 10−3–1.6893891.21749 × 10−2
0.3–1.3244661.02762 × 10−3–1.3139689.46938 × 10−3
0.5–0.9460477.34015 × 10−4–0.9385496.76384 × 10−3
0.7–0.5676284.40409 × 10−4–0.5631294.0583 × 10−3
0.9–0.18920941.46803 × 10−4–0.1877091.35277 × 10−3
10000

β = 1 Upper solution01.8920941.46803 × 10−31.8770981.35277 × 10−2
0.11.7028851.32123 × 10−31.6893891.21749 × 10−2
0.31.3244661.02762 × 10−31.3139689.46938 × 10−3
0.50.9460477.34015 × 10−40.9385496.76384 × 10−3
0.70.5676284.40409 × 10−40.5631294.0583 × 10−3
0.90.18920941.46803 × 10−40.1877091.35277 × 10−3
10000

Table 2 . Numerical solutions to Eq. (15) by CTCS and C-N at t = 0.005 and x = 1.8 for all r, β ∈ [0, 1].

βrCTCSC-N
ũ(1.8, 0.05; r, β)(1.8, 0.05; r, β)ũ(1.8, 0.05; r, β) (1.8, 0.05; r, β)
β = 0.40–0.37841892.93606 × 10−4–0.3754192.70554 × 10−3
0.1–0.3405772.64245 × 10−4–0.3378782.43498 × 10−3
0.3–0.2648932.05524 × 10−4–0.2627941.89388 × 10−3
0.5–0.1892091.46803 × 10−4–0.1877101.35277 × 10−3
0.7–0.1135268.80818 × 10−5–0.1111408.11661 × 10−4
0.9–0.0378422.93606 × 10−5–0.0375422.70554 × 10−4
10000

β = 0.600.37841892.93606 × 10−40.3754192.70554 × 10−3
0.10.3405772.64245 × 10−40.3378782.43498 × 10−3
0.30.2648932.05524 × 10−40.2627941.89388 × 10−3
0.50.1892091.46803 × 10−40.1877101.35277 × 10−3
0.70.1135268.80818 × 10−50.1111408.11661 × 10−4
0.90.0378422.93606 × 10−50.0375422.70554 × 10−4
10000