The aim of this study is to design a singular fractional order pantograph differential model by using the typical form of the Lane-Emden model. The necessary details of the singular-point, fractional order and shape factor of the designed model are also provided. The numerical solutions of the designed model have been presented using the combination of the fractional Meyer wavelet (FMW) neural networks (NNs) modeling and optimization of global search with genetic algorithm (GA) supported with local search of sequential quadratic programming (SQP), i.e., FMWNN-GASQP. The strength of FMWNN is employed to design an objective function using the differential model along with its initial conditions of the singular fractional order pantograph model. The optimization of this objective function is performed using the integrated competence of GA-SQP. The verification, perfection and authentication of the singular fractional order pantograph model using fractional Meyer computing solver is observed for different cases through comparative studies from the available exact solutions which endorsed its robustness, convergence and stability. Moreover, the statistics observation with necessary explanations further authenticate the performance of the FMWNN-GASQP in terms of accuracy and reliability.

本研究的目的是利用Lane-Emden模型的典型形式设计一个奇异分数阶受电弓微分模型。还提供了设计模型的奇异点、分数阶和形状因子的必要细节。设计模型的数值解已经使用分数迈耶小波 (FMW) 神经网络 (NN) 建模和全局搜索优化与遗传算法 (GA) 相结合，并支持序列二次规划 (SQP) 的局部搜索，即，FMWNN-GASQP。FMWNN 的强度用于使用微分模型及其奇异分数阶受电弓模型的初始条件来设计目标函数。该目标函数的优化是使用 GA-SQP 的综合能力进行的。通过对可用精确解的比较研究，观察了不同情况下使用分数阶Meyer计算求解器对奇异分数阶受电弓模型的验证、完善和验证，证明其鲁棒性、收敛性和稳定性。此外，带有必要解释的统计观察进一步验证了 FMWNN-GASQP 在准确性和可靠性方面的性能。