Functional-difference equations and applications
Yuri Antipov
The analysis of canonical problems in acoustic and electromagnetic diffraction
by half-planes and wedges is frequently carried out using the Sommerfeld
integral representation of the corresponding physical field. This approach
ultimately yields difference equations with periodic coefficients of the
second order. In spite of the importance of such equations for the geometric
theory of diffraction, a general exact method for their solution is still
unavailable in the literature. The main objectives of this talk are
(i) to present a general method for second order functional-difference
equations with periodic coefficients;
(ii) to illustrate the technique by solving in closed form a problem on
electromagnetic scattering by a right-angled magnetically conductive wedge.
The method to be proposed involves reducing the governing equation to a
scalar Riemann-Hilbert problem for two finite segments on a hyper-elliptic
surface and its solution in terms of Weierstrass integrals. The final step
of the procedure is solving the classical nonlinear Jacobi inversion
problem.