Partial differential equations are a vital component of the study of mathematical
models in science and engineering. There are various tools and techniques developed
by the researchers to solve the differential equations. The radial basis functions have
proven to be an efficient basis function for approximating the solutions to ordinary
and partial differential equations. There are different types of radial basis function
methods that have been developed by the researchers to solve various well known
differential equation. It has been developed for approximation of the solution with various
approaches that lead to the development of hybrid methods. Radial basis function
methods are widely used in numerical analysis and statistics because of their
ability to deal with meshless domain. In this work, the different radial basis function
approaches were investigated along with the focus on the strategies being addressed
to find the shape parameter value. The mathematical formulations of the various radial
basis function methods are discussed along with the available shape parameters to get
the optimal value of the numerical solutions. The present work will lay a foundation
to understand the development of the radial basis functions that could lead to a play
an important role in development of method thereafter.