Analysis of the extended boundary condition method: an electrostatic problem for Chebyshev particles

The extended boundary condition method (EBCM) in solving the electrostatic problem for axisymmetric Chebyshev particles the surface of which is described by the equation r(θ) = a(1 + ɛcosnθ) is studied. The main attention is paid to the case n = 1. The problem is reduced to solving infinite systems of linear algebraic equations (ISLAEs) for expansion coefficients of internal and “scattered” fields in terms of a spherical basis where matrix elements of the fields are integrals of products of Legendre functions and power functions. Radii of convergence R2 and R1 of these expansions, respectively, have been found analytically. For the considered particles, depending on perturbation parameter ɛ, conditions of the applicability of the EBCM have been obtained, i.e., conditions of correct construction of the T matrix (R1 < R2), as well as conditions of the validity of the external (R1 < minr(θ)) and internal (R2 > maxr(θ)) Rayleigh hypotheses under which expansions of the “scattered” and internal fields in terms of the spherical basis converge up to the surface of the particle beyond and inside it, respectively. In the particular case n = 1, numerical calculations have been performed; in this process, integral ISLAE elements have been represented as finite sums the summands of which depend on gamma functions. Calculations of matrix elements by explicit formulas have made it possible to considerably increase the dimension of the solved reduced ISLAE with preservation of the necessary accuracy. Analysis of results of numerical calculations verified their agreement with theoretical conclusions.