On Choosing a Value of Shape Parameter in Radial Basis Function Collocation Methods

In the paper two approaches to determine a value of shape parameter for radial basis function collocation methods are presented. This value enables to achieve very accurate results in Kansa method and radial basis function-based pseudospectral one. Both approaches are based on the estimation of the condition number of the system matrix. One of them tries to find the value of shape parameter employing a heuristic that relates the condition number to computational precision, the other one searches for the largest value of this parameter that still ensures stable computation using geometrical dependence. The presented algorithms guarantee performing computation in the stable region and are more efficient than commonly used ones. Moreover they allow to automate the computational process for these methods. In the paper the algorithms are validated by a few differential equations using multiquadrics functions. The results show that the approaches lead to high accuracy especially when non-uniform grid is applied.