On the Selection of a Good Shape Parameter of the Localized Method of Approximated Particular Solutions
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.
