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A meshless local Galerkin method for the numerical solution of Hammerstein integral equations based on the moving least squares technique
Pouria Assari
Keywords:Hammerstein integral equation, Discrete Galerkin method, Moving least squares (MLS), Meshless Method, Error analysis
Abstract:
      In this paper, a computational scheme is proposed to estimate the solution of one- and two-dimensional Fredholm-Hammerstein integral equations of the second kind. The method approximates the solution using the discrete Galerkin method based on the moving least squares (MLS) approach as a locally weighted least square polynomial fitting. The discrete Galerkin technique for integral equations results from the numerical integration of all integrals in the method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The implication of the scheme for solving two-dimensional integral equations is independent of the geometry of the domain. The new method is simple, efficient and more flexible for most classes of nonlinear integral equations. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.