On the Initial Boundary Value Problem for the Time-Fractional Wave Equation with Acoustic Boundary Conditions

This paper investigates the well-posedness of an initial boundary value problem for the time-fractional wave equation involving acoustic boundary conditions, a setting that, to the best of our knowledge, is studied here for the first time. The domain ⊂ Rⁿ Ω ⊂ R n ( n ≥ 2 n ≥ 2 ) is bounded, connected, and admits a boundary composed of two disjoint parts: homogeneous Dirichlet conditions are imposed on ₀ Γ 0 , while reactive acoustic-type conditions are enforced on ₁ Γ 1 . The time-fractional derivative is taken in the Caputo sense, introducing several analytical challenges due to its nonlocal nature. To address this, we develop a constructive Faedo–Galerkin approach tailored to the fractional framework and solve a general system of time-fractional ordinary differential equations that arises from the approximation process. As a key theoretical contribution, we also establish an extended version of the classical Picard–Lindelöf theorem, which is essential for handling the approximated systems. These results lay a foundational framework for analyzing more general nonlinear problems with fractional dynamics and nonstandard boundary conditions.