In this work, the association of the Asymptotic Numerical Method (ANM) with the Method of Fundamental Solutions (MFS) to solve the Von Karman equation is investigated. The Von Karman equation introduced a system of two fourth order elliptic nonlinear partial differential equations which can be used to describe the large deflections and stresses produced in a thin elastic plate subjected to external loads. The MFS, that has attracted the attention of many searchers in recent years, is one of the most developed meshless methods and is probably the most used Trefftz method. Indeed, the approximation of the solution is constructed by a linear combination of Green functions for the homogeneous equations and combined to the radial basis functions (RBF). The method is ill-conditioned and has no longer this disadvantage in comparison with other numerical methods. Furthermore, it preserves its advantages, such as the lack of any mesh, the high precision of the numerical results, the reduction of the number of unknowns, simple theory etc....

Thanks to the development in Taylor series, the system of two fourth order elliptic nonlinear partial differential equations are transformed into a succession of a system of two linear bi-harmonic equations. Knowing that the fundamental solution is not always available, the MFS-RBF (Radial Basic Functions) method is combined with the Analog Equation Method (AEM) to solve these resulting system of two linear bi-harmonic equations and computes the nonlinear branch solutions. As the series solution is limited by a radius of convergence, a continuation procedure allows one to obtain the whole solution branch in a step by step manner. The step length is computed a posteriori by exploiting the terms of these series. The efficiency of the method is verified through a numerical example.