Bipolar Fuzzy Solution of the Bipolar Fuzzy Wave Equation Using Bipolar Fuzzy Fourier Sine Transform Under GH-Differentiability

The bipolar fuzzy wave equation is crucial for modeling wave phenomena in systems characterized by dual uncertainty involving both positive and negative aspects of imprecise information. This study presents an innovative analytical framework for solving bipolar fuzzy wave equations using bipolar fuzzy Fourier sine transform under generalized Hukuhara differentiability (GH-differentiability). A novel bipolar fuzzy Fourier sine transform is introduced, and its fundamental properties are rigorously established. Using this transform, an analytical solution methodology for solving bipolar fuzzy wave equations is formulated. The proposed methodology extends classical transform procedures to the bipolar fuzzy setting, enabling the treatment of uncertainty in both positive and negative aspects. To assess the validity and effectiveness of the proposed methodology, several illustrative examples are presented. Graphical analysis is conducted to examine the impact of various forms of GH-differentiability on solution behavior and to assess parameter-dependent dynamics. Furthermore, a real-world application involving the modeling of neural oscillations is presented, illustrating the capability of the proposed framework to capture brain wave dynamics under bipolar fuzzy uncertainty. The results demonstrate that the bipolar fuzzy Fourier sine transform provides an effective and systematic tool for examining wave equations under bipolar fuzzy uncertainty.