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In this work we explore the applicability of the RBF method to laminar flame propagation modeling. This problem is an interesting challenge for the RBF method since it involves the solution of two coupled nonlinear parabolic equations in temperature and mass fraction. We show the suitability of the method by solving unsteady flame propagation problems in one and two dimensions. We also apply the method to compute the shape of an anchored flame using both equispaced and non-equispaced nodes.