  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.
