Geodesic
Boundary integral equation and the problem of diffraction on a~curved surface for the parabolic equation of the diffraction theory
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 46, Tome 451 (2016), pp. 188-207

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The two-dimensional problem of diffraction on a curved surface for the parabolic equation of the diffraction theory is considered. Ideal boundary conditions is satisfied on the surface. The boundary integral equation of Volterra type is introduced. Using the latter the problem of diffraction on parabola is analyzed. It is shown that solution of this problem coincides with the Fock asymptotic solution for cylinder. Also the iterative solution of the boundary integral equation is constructed. The problem of diffraction on a perturbation of a straight line is solved numerically using the boundary integral equation. It is showed that this numerical approach is relatively cheap. 
@article{ZNSL_2016_451_a11,
     author = {A. V. Shanin and A. I. Korol'kov},
     title = {Boundary integral equation and the problem of diffraction on a~curved surface for the parabolic equation of the diffraction theory},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {188--207},
     publisher = {mathdoc},
     volume = {451},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_451_a11/}
}
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A. V. Shanin; A. I. Korol'kov. Boundary integral equation and the problem of diffraction on a~curved surface for the parabolic equation of the diffraction theory. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 46, Tome 451 (2016), pp. 188-207. http://geodesic.mathdoc.fr/item/ZNSL_2016_451_a11/