Work

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

Year: 2026

Type: article

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 $$\varOmega \subset \mathbb {R}^n$$ Ω ⊂ R n ( $$n \ge 2$$ n ≥ 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.

Source: Bulletin of the Brazilian Mathematical Society New Series

DOI: 10.1007/s00574-026-00509-3

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

Institutions: Universidade Federal de Santa Catarina, Universidade Estadual de Maringá

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funders: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Universidade Federal de Santa Catarina