Coordinate equations for a problem on a sphere with a cut associated with diffraction by an ideal quarter-plane

In our prior work, a set of modified Smyshlyaev formulae was derived for the problem of diffraction by a Dirichlet flat cone (quarter-plane). The problem was reduced to finding the so-called edge Green's functions on a unit sphere. Here we explain how these edge Green's functions can be calculated. The new method proposed for this can be considered as a generalization of the method of separation of variables, since it reduces the partial differential equation to a system of ordinary differential equations in some sense. We demonstrate also a numerical application of the new method.