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

Abstract 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 $$\varOmega \subset \mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> ( $$n \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ) is bounded, connected, and admits a boundary composed of two disjoint parts: homogeneous Dirichlet conditions are imposed on $$\varGamma _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , while reactive acoustic-type conditions are enforced on $$\varGamma _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . 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.