ИСТИНА

The parabolic equation approximation of diffraction problems is well known. This approximation describes only ``paraxial'' waves having a narrow angular spectrum. Recently, this method has been applied to the variety of problems: the problem of diffraction by a strip, thin cone, curved surfaces, and waveguide problems. New analytical and numerical results have been obtained, and some existing formulae have been re-derived. Unfortunately, in almost all cases it is difficult to prove in a strong mathematical way that the parabolic equation provides an approximation to the solution of an exact problem for the Helmholtz equation and to define a domain in which parabolic approximation is valid. Nevertheless, numerical experiments show that the parabolic equation method works well even in cases when it isn't expected from the physical point of view. In the current work authors present a rigorous mathematical explanation of this surprising fact. The classical half-plane diffraction problem is taken as an example. A parabolic equation with appropriate boundary condition is formulated. A boundary integral equation is formulated for the problem. This equation is analogous to the corresponding integral equation for the initial (Helmholtz) problem. We are trying to answer the following question: in what sense the kernel of the ``parabolic'' (approximate) integral equation is close to the kernel of the ``Helmholtz'' (exact) integral equation? The answer is as follows. The kernel of the ``parabolic'' equation is one of the factors of the factorization (in the Wiener-Hopf sense) of the ``Helmholtz'' equation kernel. That is why, surprisingly, the parabolic method gives an almost exact solution even near the edge of the half-plane, where, seemingly, it should work poorly. The work is supported by the RSF grant 14-22-00042.