  This paper aims to survey our recent work relating to the radial basis
function (RBF) from some new views of points. In the first part, we established
the RBF on numerical integration analysis based on an intrinsic relationship
between the Green's boundary integral representation and RBF. It is found that
the kernel function of integral equation is important to create efficient RBF.
The fundamental solution RBF (FS-RBF) was presented as a novel strategy
constructing operator-dependent RBF. We proposed a conjecture formula featuring
the dimension affect on error bound to show the independent-dimension merit of
the RBF techniques. We also discussed wavelet RBF, localized RBF schemes, and
the influence of node placement on the RBF solution accuracy. The
centrosymmetric matrix structure of the RBF interpolation matrix under
symmetric node placing is proved.
  The second part of this paper is concerned with the boundary knot method
(BKM), a new boundary-only, meshless, spectral convergent, integration-free RBF
collocation technique. The BKM was tested to the Helmholtz, Laplace, linear and
nonlinear convection-diffusion problems. In particular, we introduced the
response knot-dependent nonsingular general solution to calculate
varying-parameter and nonlinear steady convection-diffusion problems very
efficiently. By comparing with the multiple dual reciprocity method, we
discussed the completeness issue of the BKM.
  Finally, the nonsingular solutions for some known differential operators were
given in appendix. Also we expanded the RBF concepts by introducing time-space
RBF for transient problems.

Computational Engineering, Finance, and Science;Computational Geometry