Variable shape parameter Kansa RBF method for the solution of nonlinear boundary value problems

Abstract

We apply a variable shape parameter Kansa–radial basis function (RBF) collocation method for the numerical solution of second and fourth order nonlinear boundary value problems in two dimensions. In the current approach, each RBF in the solution approximation is associated with a different shape parameter. These shape parameters are considered to be part of the unknowns along with the values of the coefficients of the RBFs in the solution approximation. The system of nonlinear equations resulting from the Kansa–RBF discretization is solved by directly applying a standard nonlinear solver. The proposed method is applied to several numerical examples.