Vector Riemann-Hilbert problem with almost periodic and meromorphic coefficients and applications

TL;DR

This paper investigates the vector Riemann-Hilbert problem with almost periodic and meromorphic coefficients, providing explicit solutions and factorizations for different pole-zero configurations, with applications to boundary value problems in physics and engineering.

Contribution

It introduces new methods for solving the vector Riemann-Hilbert problem with various meromorphic coefficient structures, including explicit solutions and factorizations.

Findings

Explicit solutions for rational coefficient cases.

Wiener-Hopf factorization using hypergeometric functions.

Reduction to infinite linear systems with exponential convergence.

Abstract

The vector Riemann-Hilbert problem is analyzed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros (rational functions), (ii) periodic poles and zeros, and (iii) an infinite number of non periodic zeros and poles, are considered. The first case is illustrated by the heat equation for a composite rod with a finite number of discontinuities and a system of convolution equations; both problems are solved explicitly. In the second case, a Wiener-Hopf factorization is found in terms of the hypergeometric functions, and the exact solution of a mixed boundary value problem for the Laplace equation in wedge is derived. In the last case, the Riemann-Hilbert problem reduces to an infinite system of linear algebraic equations with the exponential rate of convergence. As an example, the Neumann boundary value problem for the Helmholtz equation in a strip with a slit is analyzed.