Numerical investigation based on direct meshless local Petrov Galerkin (direct MLPG) method for solving generalized Zakharov system in one and two dimensions and generalized Gross-Pitaevskii equation

In the current investigation, we propose two efficient meshless numerical techniques for solving two models in optic engineering, i.e. the generalized Gross-Pitaevskii equation and the generalized Zakharov system. Two local meshless methods have been employed for solving these models: local radial basis functions collocation method and direct meshless local Petrov-Galerkin method. In this paper, we discrete the space direction using the local radial basis functions collocation and direct meshless local Petrov-Galerkin techniques and to obtain high-order numerical results, we use the fourth-order exponential time differencing Runge-Kutta method for discretizing the temporal direction. The obtained numerical results are compared with some well-known numerical techniques. Moreover, several examples are given that show the acceptable accuracy and efficiency of the proposed scheme.