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Fuzzy analysis of 2-D wave equation through Hukuhara differentiability coupled with AOS technique
Muhammad Usman1, Hidayat Ullah Khan1, Kamal Shah1,2
1Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan.
This study introduces a novel mechanism for finding fuzzy solutions to the two-dimensional fuzzy fractional wave equation using a fuzzy conformable fractional derivative and a fuzzy traveling wave method with additive operating splitting.
Area of Science:
- Mathematics
- Applied Mathematics
- Numerical Analysis
Background:
- Fuzzy fractional differential equations (FFDEs) are essential for modeling complex systems with uncertainty.
- Existing methods for solving fuzzy wave equations often lack comprehensive analytical solutions.
- The Hukuhara conformable fractional derivative (HCFD) offers a robust framework for fractional calculus in fuzzy environments.
Purpose of the Study:
- To develop and present a new mechanism for obtaining triangular analytical fuzzy solutions (TAFS).
- To investigate the two-dimensional fuzzy fractional order wave equation (2-D FFWE).
- To utilize the Hukuhara conformable fractional derivative (HCFD) and fuzzy differentiability concepts.
Main Methods:
- Extension of HCFD to the fuzzy conformable fractional derivative (FCFD).
- Detailed discussion of FCFD properties: -differentiability, switching point, fuzzy chain rule.
- Coupling of the fuzzy traveling wave method with the additive operating splitting (AOS) procedure.
Main Results:
- The proposed mechanism successfully yields triangular analytical fuzzy solutions for the 2-D FFWE.
- Demonstration of the effectiveness of the fuzzy traveling wave method combined with AOS.
- Validation of the theoretical framework through illustrative examples.
Conclusions:
- The developed mechanism provides an effective approach for solving 2-D fuzzy fractional wave equations.
- The study enhances the understanding of fuzzy fractional calculus applications in wave phenomena.
- This research offers a valuable tool for analyzing fuzzy mathematical models in science and engineering.
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