Adomian decomposition method

Physics

Aerospace engineering

Nonlinear system

Mathematics

Partial differential equation

Algorithm

Mathematical optimization

Simplex algorithm

Engineering

Residual

Evolutionary algorithm

Mathematical analysis

Geometry

Linear programming

Simplex

Quantum mechanics

Applied mathematics

Differential evolution

Differential (mechanical device)

Alexandria Engineering Journal

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.

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