(untitled) (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we propose a new approach for selecting suitable shape parameters of radial basis functions (RBFs) in the context of the localized method of approximated particular solutions. Traditionally, there are no direct connections on choosing good shape parameters and choosing interior and boundary nodes using the local collocation methods. As a result, the approximations of derivative functions are less accurate and the stability is also an issue. One of the focuses of this study is to select the interior and boundary nodes in a special way so that they are correlated. Furthermore, a test differential equation with known exact solution is selected and a good shape parameter for the given differential equation can be selected through a good shape parameter for the test differential equation. Three numerical examples, including a Poison's equation and an eigenvalue problem, are tested. Uniformly distributed node arrangement is compared with the proposed cross knot distribution with Dirichlet boundary conditions and mixed boundary conditions. The numerical results show some potentials for the proposed node arrangements and shape parameter selections.