In order to improve the approximation of spatial derivatives without meshes

a set of mesh free method for spatial derivatives was developed

which agreed well with the Cartesian

cylindrical and spherical coordinates. Based on the comparisons between numerical and theoretical solutions

the errors and convergences were assessed by a posteriori method

showing that the approximations for functions and derivatives were of the second order accurate

and the scale of the support domain had some influences on numerical errors but not on accuracy orders. With a discrete scale h being 0.01

the relative errors of the numerical simulation for the selected functions and their derivatives were within 0.65%.