The strong-form type collocation method, utilizing reproducing kernel (RK) approximation, can provide precise solutions but has been criticized for its complex derivative calculations. This method becomes particularly intricate and time-consuming when applied in the high-order partial differential equations (PDEs), making it challenging to use in engineering practices. Additionally, attempts to improve computational efficiency by using low-order basis functions often prove to be impractical, as these functions struggle to fulfill the completeness and consistency requirements necessary for approximating high-order equations.

This work introduces an efficient meshfree stabilized collocation method based on gradient reproducing kernel approximations (GRKSCM), which meets the high-order consistency requirements while also circumventing the complexity associated with high-order derivatives within traditional RK approximations. This results in enhanced accuracy and improved efficiency. Furthermore, GRKSCM retains the advantages of subdomain integration found in traditional stabilized collocation methods (SCM), which fulfills the high-order integration constraints efficiently and achieves accurate subdomain integration. This guarantees high accuracy and optimal convergence. Additionally, subdomain integration helps to lower the condition number of the discrete matrix, which in turn boosts the algorithm's stability. The subdomains for integration are determined by the positions of particles and the subdomains can remain regular without undergoing any deformation. Several numerical results indicate that the proposed method surpasses traditional radial basis function collocation method (RBCM) in terms of accuracy and stability.