Moving least square–one dimensional integrated radial basis function networks for time dependent problems

Abstract

This paper presents a new numerical procedure for time-dependent problems. The partition of unity method is employed to incorporate the moving least square and one-dimensional integrated radial basis function networks techniques in an approach (MLS-1D-IRBFN) that produces a very sparse system matrix and offers as a high order of accuracy as that of global 1D-IRBFN method. Moreover, the proposed approach possesses the Kronecker-$\delta$ property which helps impose the essential boundary condition in an exact manner. Spatial derivatives are discretised using Cartesian grids and MLS-1D-IRBFN, whereas temporal derivatives are discretised using high-order time-stepping schemes, namely standard $\theta$ and fourth-order Runge-Kutta methods. Several numerical examples including two-dimensional diffusion equation, one-dimensional advection-diffusion equation and forced vibration of a beam are considered. Numerical results show that the current methods are highly accurate and efficient in comparison with other published results available in the literature.