Fuzzy analysis of 2-D wave equation through Hukuhara differentiability coupled with AOS technique

Muhammad Usman; Hidayat Ullah Khan; Kamal Shah; Bahaaeldin Abdalla; Nabil Mlaiki; Thabet Abdeljawad

doi:10.1016/j.heliyon.2024.e27719

Fuzzy analysis of 2-D wave equation through Hukuhara differentiability coupled with AOS technique

Heliyon. 2024 Mar 12;10(6):e27719. doi: 10.1016/j.heliyon.2024.e27719. eCollection 2024 Mar 30.

Authors

Muhammad Usman¹, Hidayat Ullah Khan¹, Kamal Shah^{1

2}, Bahaaeldin Abdalla², Nabil Mlaiki², Thabet Abdeljawad^{2

3

4}

Affiliations

¹ Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan.
² Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh Saudi Arabia.
³ Department of Medical Research, China Medical University Taichung 40402, Taiwan.
⁴ Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa.

Abstract

The research article in hand provides a new mechanism that deals with the investigation of the triangular analytical fuzzy solutions (TAFS) of the two-dimensional fuzzy fractional order wave equation (2-D FFWE) through the Hukuhara conformable fractional derivative (HCFD) along with the concept of $[g H]$ and $[g H - p]$ differentiability. The mechanism consists of a fuzzy traveling wave method coupled with additive operating splitting (AOS). The procedure for the aforesaid mechanism starts with the extension of the HCFD to the fuzzy conformable fractional derivative (FCFD). Furthermore, some properties of FCFD that play a vital role in this study like, $([i - g H], [i i - g H], [i - p], [i i - p])$ -differentiability, switching point, fuzzy chain rule, and traveling wave method are discussed in detail. Further, fuzzy traveling wave method coupled with the procedure of the additive operating splitting (AOS) method is adopted to investigate the triangular analytical fuzzy solution of the Two-dimensional fuzzy wave equation (2-D FWE). Finally, some examples are provided that support our arguments.

Keywords: $([i - g H], [i i - g H], [i - p], i - p])$ -differentiability; Additive operating splitting method; Fuzzy chain rule; Fuzzy wave equation; Hukuhara conformable fractional derivative; Switching point; Traveling wave method.