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.

这项研究针对非线性偏微分方程的数值解提出了一种基于Padé近似的混合无网格框架。拟议的框架涉及三个新颖方面。首先，结合基于Padé近似的残差函数和惩罚函数方法来处理边界/初始条件，以设计一个等效的优化问题。其次，介绍了Nelder-Mead Simplex（NMS）算法环境下的协作式多简单搜索概念。最后，提出了一种将多简单搜索与差分进化（DE）相结合的方法，以有效地解决建模问题。所开发的方法被称为Padé进化合作多单形（PECMS）算法。将开发的PECMS方案在说明性示例上的结果与三种众所周知的方法进行比较，即差分变换方法（DTM），阿domian分解方法（ADM）和变分迭代方法（VIM）。比较表明，与竞争方法相比，当前方法具有较高的准确性。