<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varOmega \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) is bounded, connected, and admits a boundary composed of two disjoint parts: homogeneous Dirichlet conditions are imposed on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varGamma _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Γ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, while reactive acoustic-type conditions are enforced on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varGamma _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Γ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>. 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.</p>

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

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2026年3月31日

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Paulo M. Carvalho Neto,
Cícero L. Frota,
Pedro G. P. Torelli

摘要

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 $\varOmega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ ( $n \ge 2$ $n \geq 2$ ) is bounded, connected, and admits a boundary composed of two disjoint parts: homogeneous Dirichlet conditions are imposed on $\varGamma _0$ $Γ_{0}$ , while reactive acoustic-type conditions are enforced on $\varGamma _1$ $Γ_{1}$ . 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.