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 fundame...

Radial Basis Function (RBF) collocation methods for time-dependent PDEs, in particular hyperbolic PDEs, are known to be difficult to implement in a way so that they are stable for time integration. It has been hypothesized that the instability is due to the way that boundary conditions are applied and to the relatively large errors in boundary regions. We describe a preconditioning technique th...

This paper investigates the stability of the Radial Basis Function (RBF) collocation method for convective problems on the circle and sphere. We prove that the RBF method is Lax-stable for problems on the circle when the collocation points are equispaced and the transport speed is constant. We also show that the eigenvalues of discretization matrices are purely imaginary in the case of variable...

in this paper, a technique generally known as meshless numerical scheme for solving fractional dierential equations isconsidered. we approximate the exact solution by use of radial basis function(rbf) collocation method. this techniqueplays an important role to reduce a fractional dierential equation to a system of equations. the numerical results demonstrate the accuracy and ability of this me...

The radial basis function (RBF) collocation techniques for the numerical solution of partial differential equation problems are increasingly popular in recent years thanks to their striking merits being inherently meshless, integration-free, and highly accurate. However, the RBF-based methods have markedly been limited to handle isotropic problems due to the use of the isotropic Euclidean dista...

We show how the collocation framework that is prevalent in the radial basis function literature can be modified so that the methods can be interpreted in the framework of standard pseudospectral methods. This implies that many of the standard algorithms and strategies used for solving time-dependent as well as time-independent partial differential equations with (polynomial) pseudospectral meth...

Based on the recently developed Finite Integration Method (FIM) for solving one-dimensional ordinary and partial differential equations, this paper extends the technique to higher dimensional partial differential equations. The main idea is to extend the first order finite integration matrices constructed by using either Ordinary Linear Approach (OLA) (uniform distribution of nodes) or Radial B...

In modeling using diffusion equations, the relationship between fractal geometry and fractional calculus arises by modeling the conditions of the medium as a fractal whose fractional dimension determines the order of this equation. For this reason, it is very useful to have numerical methods that solve them and discretization of the domain is not determinant for the efficiency of the algorithm....