On the Selection of a Good Shape Parameter of the Localized Method of Approximated Particular Solutions (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we propose a new approach for selecting suitable shape parametersof radial basis functions (RBFs) in the context of the localized method ofapproximated particular solutions. Traditionally, there are no direct connections onchoosing good shape parameters and choosing interior and boundary nodes using thelocal collocation methods. As a result, the approximations of derivative functions areless accurate and the stability is also an issue. One of the focuses of this study is to selectthe 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 goodshape parameter for the given differential equation can be selected through a goodshape parameter for the test differential equation. Three numerical examples, includinga Poison's equation and an eigenvalue problem, are tested. Uniformly distributednode arrangement is compared with the proposed cross knot distribution with Dirichletboundary conditions and mixed boundary conditions. The numerical results showsome potentials for the proposed node arrangements and shape parameter selections.