  This paper aims to survey our recent work relating to the radial basis
function (RBF) and its applications to numerical PDEs. We introduced the kernel
RBF involving general pre-wavelets and scale-orthogonal wavelets RBF. A
dimension-independent RBF error bound was also conjectured. The centrosymmetric
structure of RBF interpolation matrix under symmetric sample knots was pointed
out. On the other hand, we introduced the boundary knot method via nonsingular
general solution and dual reciprocity principle and the boundary particle
method via multiple reciprocity principle. By using the Green integral we
developed a domain-type Hermite RBF scheme called the modified Kansa method,
which significantly reduces calculation errors around boundary. To circumvent
the Gibbs phenomenon, the least square RBF collocation scheme was presented.
All above discretization schemes are meshfree, symmetric, spectral convergent,
integration-free and mathematically very simple. The numerical validations are
also briefly presented.
