This study analyzes the well-posedness and global existence of nonlinear biharmonic partial differential equations (PDEs) of the form ∂ t η , α w + w tt − γ 1 ∆ 2 w − γ 2 ∆ 2 w t + γ 3 w t + γ 4 w = γ 5 ∇ w g ( w ) , involving the generalized Caputo fractional derivative of order 0 < α < 1 in time, where ( x , t ) ∈ Ω × R + . To validate our theoretical findings, we employ a finite difference approach for one-dimensional cases and use the Grünwald–Letnikov approximation. We also address the convergence of this numerical method. AMS Classification: 26A33, 34K37, 31A30, 49K40, 65N06