A generalization of D. A. Grave's method for plane boundary-value problems in harmonic potential theory

The article describes and proves D. A. Grave's method that solves classical plane boundary-value problems for the Green's function of the Laplace equation in regions whose boundaries are smooth analytical curves defined by finite-order irreducible polynomials. The proposed method has certain advantages compared with the method that constructs the Green's function by conformally mapping the original region onto the unit disk. A class of regions are identified for which Grave's methods produces an explicit analytical solution in convergent-series form. This is a natural generalization of the conformal mapping method for simplest regions. Translated from Prikladnaya Matematika i Informatika, No. 1, pp. 5–19, 1999.