Fuzzy fractional generalized Bagley–Torvik equation with fuzzy ψ -Caputo gH-differentiability

This paper investigates the fuzzy fractional generalized Bagley–Torvik equation (FFGB–TE), which models the motion of an immersed plate in a Newtonian fluid under uncertain initial conditions. Analytically studying this equation with two independent fractional orders presents significant challenges, making the development of efficient analytical methods essential. We introduce an analytical fuzzy solution by employing the concept of fuzzy ψ -Caputo generalized Hukuhara differentiability (gH-differentiability) and the fuzzy Laplace transform (FLT) technique. The closed-form solution is derived for both homogeneous and non-homogeneous cases, expressed in terms of the Mittag-Leffler function (MLF) involving double series. Several key results are presented and rigorously proven. To demonstrate the proposed analytical approach, illustrative examples are provided. Additionally, to highlight the novelty of this work, the FFGB–TE is applied to model the motion of an immersed plate, and graphical representations are provided to validate the theoretical findings.