A new type of radial basis functions for problems governed by partial differential equations.

The aim of this paper is to introduce a novel category of radial basis functions that incorporate smoothing techniques. Initially, we employ the power augmented and shape parameter schemes to create the radial basis functions. Subsequently, we apply the newly-constructed radial basis functions using the traditional collocation method and singular values decomposition algorithm to solve the corresponding linear system equations. Finally, we analyze several pairs of radial basis functions in depth to address physical problems linked to thermal science that are governed by partial differential equations. The numerical results demonstrate that the radial basis functions constructed using the power augmented and shape parameter schemes exhibit remarkable performance.