Hybrid evolutionary padé approximation approach for numerical treatment of nonlinear partial differential equations

Abstract

This study proposes a Padé approximation based hybrid mesh free framework for numerical solutions of nonlinear partial differential equations. The proposed framework involves three novel aspects. Firstly, the conjunction of Padé approximation based residual functional and penalty function approach for handling boundary/Initial conditions is employed to design an equivalent optimization problem. Secondly, the notion of cooperative multi-simplex searches in the environment of Nelder-Mead Simplex (NMS) algorithm is introduced. Finally, a hybrid of multi-simplex search and Differential Evolution (DE) is proposed to efficiently cope with the modelled problem. The developed method is named as Padé Evolutionary Cooperative Multi-Simplex (PECMS) algorithm. The results of the developed PECMS scheme on illustrative examples are compared with three well-known methods, namely, Differential Transform Method (DTM), Adomian Decomposition Method (ADM) and the Variational Iteration Method (VIM). The comparisons demonstrate that the current approach possesses high accuracy when compared to the competing methods.