## 1 Introduction
The partition of unity interpolation approach is first used in the pioneer work of [PERSON] [54], where it is used to smoothly interpolate irregular data. [PERSON] [39] combined the partition of unity method with respect to space and RBF interpolation in order to form the RBF-PU method that can resolve large-scale scattered data problems. [PERSON] et al. [40] proposed a RBF-PU based method to suitably select variable shape parameters and sub-domain sizes that improve the accuracy by minimizing the error. [PERSON] et al. [41] used the RBF-PU with a stabilized weighted singular value decomposition (WSVD) basis. This algorithm is stable, accurate, and efficient given that it is local and has a stabilized basis. [PERSON] et al. [42] proposed the RBF-QR method to eliminate ill-conditioning of the interpolation matrix for near-flat basis function. They applied the method for a particular case of points on a sphere surface. [PERSON] et al. [43] compared different types of RBFs in the framework of RBF-QR to solve convective PDE over a wide range of shape parameter values. They found that the RBF-QR method is accurate and stable even for a shape parameter value approaching zero. Additionally, RBF-QR is extended and applied to two-dimensional problems [44]. Furthermore, RBF-QR is extended to work on an arbitrary node-set in 1D, 2D, and 3D problems [44]. [PERSON] et al. [45] offered a simpler algorithm for RBF-QR in arbitrary dimensions by using eigen-decomposition parametrized by the scale of the problem. [PERSON] et al. [46] extended RBF-QR to calculate the weights and differentiation matrix in order to solve PDEs. The proposed model has better accuracy and stability than the direct approach. [PERSON] et al. [47] introduced the RBF-GA algorithm. The algorithm helps in numerically determining a well-conditioned basis utilizing the incomplete gamma function, which allows changing the basis without using infinite expansion. Based on the use of a rational approximation of a vector-valued function, the RBF-RA algorithm is adapted for a stable computation of the RBF interpolant when the shape parameter approaches zero [48; 49]. In the current work, a novel approach for improving the accuracy of the RBF method is presented. Based on the novel approach, the Taylor series expansion is used to expand the RBF basis with respect to shape parameter around a predefined reference value to find a modified stable basis. According to the preliminary findings, this approach has weak coupling between shape parameter and distances near the reference shape parameter values. The further away the reference shape-parameter from the shape parameter, the weaker the coupling. In order to resolve this issue, the partition of unity with respect to the shape parameter is applied to the resultant equation. The current work uses PU with respect to the shape parameter, whereas previous literature works used it with respect to spatial coordinates. This modified approach improves accuracy and provides reliable and consistent results, even when the shape parameter approaches zero. The paper is structured as follows. Section 2 explains using RBF for interpolation and approximating derivatives. In Section 3, the expansion of RBF using the Taylor series with respect to a fixed shape parameter value is introduced. Section 4 proposes applying the partition of unity approach to the Taylor series expansion of RBF. The results are discussed in Section 5. Finally, the conclusions are presented in Section 6.