2023/01/03 by Pierre Amenoagbadji, Amenoagbadji, Pierre, Sonia Fliss +2 · 1 citation
Computer Science · Mathematics · #35J05 #42A75 #78A50 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2301.01159
openalex publication_date 2023/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
This work is devoted to the resolution of the Helmholtz equation -(μu')' - ρω2 u = f in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation.