An ODE--based approach to some Riemann--Hilbert problems motivated by wave diffraction

2012/10/06 by А. В. Шанин, Andrey V. Shanin, Shanin, Andrey V.
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Electromagnetic Scattering and Analysis #FOS: Mathematics #Numerical methods in engineering #Numerical methods in inverse problems #math.AP #math.CA #math.CV

paper · pdf · doi:10.48550/arxiv.1210.1964

17 pages, 7 figures

arxiv created 2012/10/06 · openalex publication_date 2012/10/06 · arxiv updated 2012/10/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

A novel approach to Riemann--Hilbert problems of particular class is introduced. The approach is applicable to problems in which the multiplicative jump is set on a half-line. Such problems are linked to some Wiener--Hopf problems motivated by diffraction theory. The new approach is based on ordinary differential equations: the Riemann--Hilbert problem is reduced to finding a coefficient of an ordinary differential equation and solving this equation. The new method leads to an efficient numerical algorithm and opens a road to new asymptotical and analytical advances.

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