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

摘要

The vector Riemann-Hilbert problem is analysed 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 a 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 analysed.