## 1 Introduction  
Over the past few decades, radial basis function (RBF) has attracted researchers' attention as a method for interpolation, differentiation, and solving partial differential equations (PDEs). RBF-based methods have several advantages over the conventional methods including their ability to produce spectral accuracy [1] and their simplicity in implementation regardless of the problem's dimension [2]. Their accuracy and stability are strongly predicated on the condition of the generated matrix. The improvement on the RBF can be classified into matrix solvers, shape parameter, and basis functions. In the following, a brief literature review of each methodology is provided. As known, the preconditioning of a matrix improves the stability of the solver. So, dealing with the matrix solvers could improve the condition of the RBF matrix. Thereare several studies that investigated and evaluated the preconditioning of the RBF matrix [3; 4; 5; 6; 7; 8; 9; 10; 11; 12]. [PERSON] et al. [11; 12] used extended precision for the calculation in order to remove the computational singularity and enhance the condition of the interpolation matrix. Until now, the matrix solvers are not able to improve accuracy to the desired level. The studies on the shape parameter can be categorized as modifying the shape parameter [13], using variable shape parameter [14; 15; 16; 17], and finding optimal shape parameter [18; 19; 20; 21; 22; 23; 24; 25]. Despite the success of identifying the optimal value of the shape parameter, the proposed methods are neither universal nor consistent. Furthermore, resolving different kinds of problems need fine-tuning. Consequently, the researchers investigated changing or transforming the RBFs to find a more general approach for a wide range of applications. The modifications to the basis of the interpolant of the RBF can be classified into either single contributions [26; 27; 28; 29; 30; 31] or classes of methods (pseudo-spectral radial basis function (RBF-PS) [32; 33], finite difference radial basis function (RBF-FD) [34; 35; 36; 37; 38], partition of unity radial basis function (RBF-PU) [39; 40; 41], radial basis function using QR factorization (RBF-QR) [42; 43; 44; 45], stabilized expansion of Gaussian radial basis function (RBF-GA) [47], rational radial basis function (RBF-RA) [48; 49], and variably scaled kernels (VSK) [50; 51; 52; 53]. [PERSON] et al. [30] proposed a stable algorithm for evaluating the interpolation matrix based on the eigen-function expansion of the Gaussian function. The accuracy and computation cost of the proposed method are verified for 1D and 2D cases. [PERSON] et al. [31] stabilized the Gaussian RBF by using the generating function of the Hermite polynomial. The proposed algorithm is found to be stable and can be generalized to higher dimensions. [PERSON] et al. [32; 33] demonstrated that RBF could be interpreted in the framework of pseudo-spectral methods, which is more efficient than the regular ([PERSON]'s) RBF method for time-dependent PDEs. [PERSON] et al. [34] proposed the RBF interpolant with local supports, which is similar to the stencils of finite difference methods, and hence named RBF-FD. The method has reasonable accuracy and convergence. [PERSON] et al. [35] combined the RBF-FD with the Hermite RBF interpolation to involve a smaller number of stencil nodes without sacrificing accuracy. [PERSON] et al. [23; 36] proposed an analytical formula for the RBF-FD weights to avoid the ill-condition in the computation of the RBF-FD weights. The formula is based on the semi-analytical computation of the Laurent series of the inverse of the RBF interpolation matrix. [PERSON] et al. [37] and [PERSON] et al. [38] observed that polyharmonic splines (PHS) padded with a polynomial provide a simple way to overcome stagnation error and produce good accuracy for interpolation and derivative approximation without the need for the shape parameter.