On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

Review Mathematics, Applied

On local and global structures of transmission eigenfunctions and beyond

Hongyu Liu

Summary: This paper reviews the recent developments in the study of transmission eigenfunctions, discusses the underlying principles, and proposes several conjectures related to transmission eigenvalue problems and inverse scattering problems.

JOURNAL OF INVERSE AND ILL-POSED PROBLEMS (2022)

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Article Mathematics, Applied

A space-time finite element method for solving linear Riesz space fractional partial differential equations

Junjiang Lai et al.

Summary: This paper discusses numerical solutions for linear Riesz space fractional partial differential equations with a second-order time derivative, proposing a space-time finite element method for numerical solutions. The method approximates the second-order time derivative using the C-0-continuous Galerkin method in the time direction and develops the usual linear finite element method to approximate the space fractional derivative in the space direction. The stability of the discrete solution is established, optimal error estimates are investigated, and numerical tests are conducted to validate the theoretical results.

NUMERICAL ALGORITHMS (2021)

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Article Computer Science, Artificial Intelligence

Surface-Localized Transmission Eigenstates, Super-resolution Imaging, and Pseudo Surface Plasmon Modes

Yat Tin Chow et al.

Summary: The study reveals a novel global geometric structure of transmission eigenfunctions associated with the Helmholtz system, demonstrating the localization of eigenfunctions under certain conditions. A comprehensive theoretical justification is provided for the radial geometry, while extensive numerical experiments are conducted for the nonradial case. This discovery offers a new perspective on wave localization and has practical applications in super-resolution imaging and pseudo surface plasmon resonant modes.

SIAM JOURNAL ON IMAGING SCIENCES (2021)

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Article Mathematics, Applied

SCATTERING BY CURVATURES, RADIATIONLESS SOURCES, TRANSMISSION EIGENFUNCTIONS, AND INVERSE SCATTERING PROBLEMS

Emilia L. K. Blasten et al.

Summary: This study explores various topics in the theory of wave propagation, including geometrical characterizations of radiationless sources, nonradiating incident waves, interior transmission eigenfunctions, and their applications in inverse scattering. The major novel discovery is a localization and geometrization property, which allows for explicit bounds between the intensity of an invisible scatterer and its diameter or curvature. These results have significant implications for characterizing radiationless sources or nonradiating waves near high-curvature points, as well as for deriving new intrinsic geometric properties of interior transmission eigenfunctions. Additionally, unique determination results for the single-wave Schiffer's problem are established in scenarios involving collections of well-separated small scatterers.

SIAM JOURNAL ON MATHEMATICAL ANALYSIS (2021)

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Article Mathematics, Applied