  The purpose of this study is to apply some new RBF collocation schemes and
recently-developed kernel RBFs to various types of partial differential
equation systems. By analogy with the Fasshauer's Hermite interpolation, we
recently developed the symmetric BKM and boundary particle methods (BPM), where
the latter is based on the multiple reciprocity principle. The resulting
interpolation matrix of them is always symmetric irrespective of boundary
geometry and conditions. Furthermore, the proposed direct BKM and BPM apply the
practical physical variables rather than expansion coefficients and become very
competitive alternative to the boundary element method. On the other hand, by
using the Green integral we derive a new domain-type symmetrical RBF scheme
called as the modified Kansa method (MKM), which differs from the Fasshaure's
scheme in that the MKM discretizes both governing equation and boundary
conditions on the same boundary nodes. Therefore, the MKM significantly reduces
calculation errors at nodes adjacent to boundary with explicit mathematical
basis. Experimenting these novel RBF schemes with 2D and 3D Laplace, Helmholtz,
and convection-diffusion problems will be subject of this study. In addition,
the nonsingular high-order fundamental or general solution will be employed as
the kernel RBFs in the BKM and MKM.
