A collocation mesh-free method based on multiple basis functions

Two dominant shape functions are used to approximate scattered points in mesh-free methods, e.g. the interpolating radial basis function (RBF) and the approximating moving least squares (MLS). In the present paper, a new shape function is developed as a linear interpolating function of both MLS and RBF. This function inherits the properties of both MLS and RBF and is regularized by a control parameter μ, which takes different values in the domain [0,1]. Based on the proposed shape function, the collocation method is applied to solve initial and boundary value problems in one and two dimensions. The present method gives good results and achieves good convergence trends for different values of μ, compared with MLS and RBF individually, for a large number of nodes.

► In the present paper, a new shape function is developed as a linear interpolating function of both MLS and RBF.
► A new control parameter μ takes different values in the domain [0,1] to regularize the interpolating function.
► Collocation method gives good convergence trends for different values of μ, compared with MLS and RBF individually.