A meshless collocation method based on the differential reproducing kernel approximation

A differential reproducing kernel (DRK) approximation-based collocation method is developed for solving ordinary and partial differential equations governing the one- and two-dimensional problems of elastic bodies, respectively. In the conventional reproducing kernel (RK) approximation, the shape functions for the derivatives of RK approximants are determined by directly differentiating the RK approximants, and this is very time-consuming, especially for the calculations of their higher-order derivatives. Contrary to the previous differentiation manipulation, we construct a set of differential reproducing conditions to determine the shape functions for the derivatives of RK approximants. A meshless collocation method based on the present DRK approximation is developed and applied to the analysis of one-dimensional problems of elastic bars, two-dimensional potential problems, and plane elasticity problems of elastic solids to validate its accuracy and find the rate of convergence. It is shown that the present method is indeed a fully meshless approach with excellent accuracy and fast convergence rate.