The aim of the present study is to characterize the unknown thermal conductivity of nonlinear functionally graded materials (FGMs) with the method of fundamental solutions (MFS) in conjunction with the Nelder-Mead simplex (NMS) method. The thermal conductivity is assumed to be a function varying exponentially with position and linearly/exponentially with temperature with the unknown coefficients. The determination of the thermal conductivity of FGMs is mathematically complicated due to its multiparameter property, inherent nonlinearity and ill-posedness. The MFS, a well-known meshless collocation method, is used to deal with the two dimensional heat conduction problems. In the meantime, the NMS method is employed to search for an optimal solution by minimizing a functional measuring the difference between observed and the MFS-predicted temperatures under estimated parameters. Numerical examples show the feasibility, robustness and applicability of the proposed scheme. (C) 2015 Elsevier Ltd. All rights reserved.