A preliminary study on the meshless local exponential basis functions method for nonlinear and variable coefficient PDEs

Abstract

Purpose

The purpose of this paper is to extend the meshless local exponential basis functions (MLEBF) method to the case of nonlinear and linear, variable coefficient partial differential equations (PDEs).

Design/methodology/approach

The original version of MLEBF method is limited to linear, constant coefficient PDEs. The reason is that exponential bases which satisfy the homogeneous operator can only be determined for this class of problems. To extend this method to the general case of linear PDEs, the variable coefficients along with all involved derivatives are first expanded. This expanded form is evaluated at the center of each cloud, and is assumed to be constant over the entire cloud. The solution procedure is followed as in the former version. Nonlinear problems are first converted to a succession of linear, variable coefficient PDEs using the Newton-Kantorovich scheme and are subsequently solved using the aforementioned approach until convergence is achieved.

Findings

The results obtained show good performance of the method as solution to a wide range of problems. The results are compared with the well-known methods in the literature such as the finite element method, high-order finite difference method or variants of the boundary element method.

Originality/value

The MLEBF method is a simple yet effective tool for analyzing various kinds of problems. It is easy to implement with high parallelization potential. The proposed method addresses the biggest limitation of the method, and extends it to linear, variable coefficient PDEs as well as nonlinear ones.

Citation

Mossaiby, F. and Ghaderian, M. (2016), "A preliminary study on the meshless local exponential basis functions method for nonlinear and variable coefficient PDEs", Engineering Computations, Vol. 33 No. 8, pp. 2238-2263. https://doi.org/10.1108/EC-07-2015-0208