Article
Mathematics, Applied
Patrick Joly, Maryna Kachanovska
Summary: This work focuses on the construction and analysis of high-order transparent boundary conditions for the weighted wave equation on a fractal tree, modeling sound propagation in human lungs. The method proposed in this work, based on truncating the meromorphic series representing the symbol of the Dirichlet-to-Neumann operator, shows stability and convergence, with numerical results confirming theoretical findings.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Physics, Multidisciplinary
Filipa R. Prudencio, Mario G. Silveirinha
Summary: There has been a recent interest in topological materials and their properties. Topological band theory, initially developed for condensed matter systems, can also be applied to other wave platforms. However, the particle-hole symmetry and the dispersive nature of nonreciprocal photonic materials may lead to a breakdown of the usual topological methods.
PHYSICAL REVIEW LETTERS
(2022)
Article
Physics, Mathematical
Wenqiang Xiao, Bo Gong, Junshan Lin, Jiguang Sun
Summary: In this paper, a finite element method is proposed to compute the band structures of dispersive photonic crystals in 3D. The nonlinear Maxwell's eigenvalue problem is formulated as the eigenvalue problem of a holomorphic operator function. The Ne ' de ' lec edge elements are employed for discretization and the convergence of the eigenvalues is proven using the abstract approximation theory.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
S. Bandyopadhyay, B. Dacorogna, V Matveev, M. Troyanov
Summary: This article discusses the problem of finding a map u : (Omega) over bar -> R-n solving the equation u* (H) = G in a bounded open set Omega with certain boundary conditions.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics, Applied
Soura Sana, Bankim C. Mandal
Summary: This paper investigates the convergence behavior of the Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for time-fractional sub-diffusion and diffusion-wave equations. The paper analyzes the impact of the generalized diffusion coefficient on the convergence of the algorithms and the relationship between the convergence rate and the fractional order of the time derivative.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Correction
Mathematics
Tristan Milne, Abdol-Reza Mansouri
Summary: The proof in the article contained a flaw, which the author corrected by introducing a new approach.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Applied
Jiaqing Yang, Meng Ding, Keji Liu
Summary: This paper considers inverse problems related to reduced wave equations with unknown obstacles in a bounded domain. It is shown that both coefficients and obstacles can be simultaneously recovered when the leading and zero-order coefficients are known to be piecewise constant. The method relies on the local Dirichlet-to-Neumann map and a coupled PDE system constructed for the reduced wave equations.
Article
Mathematics
Hongfen Yuan, Valery Karachik
Summary: Dunkl operators are a family of commuting differential-difference operators associated with a finite reflection group. These operators play a key role in the area of harmonic analysis and theory of spherical functions. We study the solution of the inhomogeneous Dunkl polyharmonic equation based on the solutions of Dunkl-Possion equations. Furthermore, we construct the solutions of Dirichlet and Neumann boundary value problems for Dunkl polyharmonic equations without invoking the Green's function.
Article
Computer Science, Software Engineering
Martin J. Gander, Felix Kwok, Bankim C. Mandal
Summary: This paper introduces a new variant of the waveform relaxation algorithm for the wave equation and parabolic problems, utilizing domain decomposition and subdomain solves for information exchange. Convergence analysis and numerical experiments are conducted to compare the algorithm's performance in various scenarios.
BIT NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Gobinda Garai, Bankim C. Mandal
Summary: In this paper, non-overlapping substructuring algorithms for the Cahn-Hilliard (CH) equation are proposed and studied. The Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods are formulated and their convergence behavior is investigated for the CH equation. Different variations of these methods, including domain-decomposition based and nonlinear versions, are explored. Numerical results are provided to verify the findings.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
P. A. T. R. I. C. K. JOLY, M. A. R. Y. N. A. KACHANOVSKA
Summary: This work presents a refined error analysis of high-order transparent boundary conditions for the weighted wave equation on a fractal tree. By computing asymptotics of eigenvalues and bounds for Neumann traces of eigenfunctions, the error induced by truncation of the meromorphic series representing the symbol of the Dirichlet-to-Neumann operator is quantified. The sharpness of the obtained bounds is proven for a class of self-similar trees.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Engineering, Multidisciplinary
Most Shewly Aktar, M. Ali Akbar, Kottakkaran Sooppy Nisar, Haifa Alrebdi, A. Abdel-Aty
Summary: This article investigates dispersive wave propagations described by nonlinear models and computes solitary wave solutions and different classes of localized coherent structures with their interaction properties. The impact of steeping and wave spread on waveforms is also studied.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Physics, Multidisciplinary
Abdolrasoul Gharaati, Najme Abbasi Tolgari
Summary: The proposed method calculates the Faraday rotation and transmittance spectrum of two-dimensional magneto-optic photonic crystals by taking into account all three components of the electromagnetic fields and considering the permittivity tensor with off-diagonal imaginary components representing the Faraday effect. The accuracy and convergence of the method are verified through numerical results.
Article
Mathematics, Applied
Lauri Oksanen, Tianyu Yang, Yang Yang
Summary: In this paper, a linearized boundary control method is developed for the inverse boundary value problem of the acoustic wave equation. The reconstruction formula and the stability analysis are provided for both zero potential and nonzero potential cases. Numerical experiments are conducted to validate the proposed method.
Article
Mathematics, Applied
Hao Li, Zhen-Hu Ning, Fengyan Yang
Summary: This study establishes a framework to investigate the stability of the critical defocusing semilinear wave equation with distributed locally damping and Dirichlet-Neumann boundary condition on a bounded domain. Unique continuation properties and observability inequalities were proved using Morawetz estimates in Euclidean spaces, followed by the application of compactness-uniqueness arguments to prove the main stabilization result.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)