Kansa radial basis function method with fictitious centres for solving nonlinear boundary value problems

Abstract

A Kansa–radial basis function (RBF) collocation method is applied to two–dimensional second and fourth order nonlinear boundary value problems. The solution is approximated by a linear combination of RBFs, each of which is associated with a centre and a different shape parameter. As well as the RBF coefficients in the approximation, these shape parameter values are taken to be among the unknowns. In addition, the centres are distributed within a larger domain containing the physical domain of the problem. The size of this larger domain is controlled by a dilation parameter which is also included in the unknowns. In fourth order problems where two boundary conditions are imposed, two sets of (different) boundary centres are selected. The Kansa–RBF discretization yields a system of nonlinear equations which is solved by standard software. The proposed technique is applied to four problems and the numerical results are analyzed and discussed.