Riemann boundary value problem with piecewise constant matrix

TL;DR

This paper presents a constructive method to solve the Riemann boundary value problem with piecewise constant matrices using functional equations and infinite linear systems, enabling computation of solutions via truncation.

Contribution

It introduces a novel approach employing functional equations and truncation of infinite systems to solve boundary value problems with piecewise constant matrices.

Findings

Method effectively computes solutions and partial indices.

Truncation provides a justified approximation approach.

Solution process is constructive and implementable.

Abstract

The vector-matrix Riemann boundary value problem for the unit disk with piecewise constant matrix is constructively solved by a method of functional equations. By functional equations we mean iterative functional equations with shifts involving compositions of unknown functions analytic in mutually disjoint disks. The functional equations are written as an infinite linear algebraic system on the coefficients of the corresponding Taylor series. The compactness of the shift operators implies justification of the truncation method for this infinite system. The unknown functions and partial indices can be calculated by truncated systems.