A higher-order boundary perturbation method (BPM). formulated to treat a class of problems defined in an elliptic domain, is developed to obtain the Green's function due to an eccentric source. The method, based on a dual perturbation, leads to expansion solutions expressed in terms of ellipticity and eccentricity perturbation parameters. General explicit expressions for equivalent boundary conditions on the perturbed boundary are first derived to treat the class of problems for which the associated boundary conditions are of the Dirichlet or Neumann type. The BPM is applied to investigate a clamped elliptic plate subject to eccentric loads. Estimates of the accuracy of the method are given. The BPM is seen to yield reasonably accurate solutions for moderately elliptic domains and moderate ellipticities.