We investigate both theoretically and numerically the so-called invariance property, see e.g. Sun and Ma (2015a,b), of the solution of boundary value problems associated with the anisotropic heat conduction equation (or Laplace-Beltrami's equation) in two dimensions with respect to elementary transformations of the solution domain, e.g. dilations or contractions. We also show that the standard method of fundamental solutions (MFS) does not satisfy the invariance property. Motivated by these reasons, we introduce, in a natural manner, a modified version of the MFS that remains invariant under elementary transformations of the solution domain and is referred to as the invariant MFS (IMFS). Five two-dimensional examples are thoroughly investigated to assess the numerical accuracy, convergence and stability of the proposed IMFS, in conjunction with the Tikhonov regularization method (Tikhonov and Arsenin, 1986) and Morozov's discrepancy principle (Morozov, 1966), for Laplace-Beltrami's equation with perturbed boundary conditions. (C) 2015 Elsevier Ltd. All rights reserved.