  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.
