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

TL;DR

This paper introduces an ODE-based method for solving specific Riemann--Hilbert problems related to wave diffraction, enabling efficient numerical solutions and potential analytical advancements.

Contribution

The paper presents a novel ODE-based approach for Riemann--Hilbert problems with half-line jumps, connecting them to diffraction theory and improving solution techniques.

Findings

Developed an ODE reduction for Riemann--Hilbert problems

Created an efficient numerical algorithm based on the approach

Opened possibilities for new asymptotic and analytical results

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.