Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

Muhammad Asif Zahoor RAJA; Iftikhar AHMAD; Imtiaz KHAN; Muhammed Ibrahem SYAM; Abdul Majid WAZWAZ

doi:10.1631/FITEE.1500393

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Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

Regular Papers | Updated：2022-05-19

- Neuro-heuristic computational intelligence for solving nonlinear pantograph systems
- 用于解决非线性受电弓系统的启发式神经网络计算
- Frontiers of Information Technology & Electronic Engineering Vol. 18, Issue 4, Pages: 464-484(2017)
- Affiliations：
  
  1Department of Electrical Engineering, COMSATs Institute of Information Technology, Attock 43200, Pakistan
  2Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan
  3Department of Mathematics, Preston University, Islamabad Campus, Kohat, Islamabad 44000, Pakistan
  4Department of Mathematical Sciences, United Arab Emirates University, Al-Ain Box 15551, UAE
  5Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA
- Author bio：
  
  E-mail:rasifzahoor@yahoo.com;Muhammad.asif@ciit-attock.edu.pk
- Funds：
- DOI：10.1631/FITEE.1500393
  CLC： TP183;O175
- Received：10 November 2015，
  
  Accepted：17 February 2016，
  
  Published：2017-04
- Accepted：
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Muhammad Asif Zahoor RAJA, Iftikhar AHMAD, Imtiaz KHAN, et al. Neuro-heuristic computational intelligence for solving nonlinear pantograph systems[J]. Frontiers of Information Technology & Electronic Engineering, 2017, 18(4): 464-484.
DOI：

Muhammad Asif Zahoor RAJA, Iftikhar AHMAD, Imtiaz KHAN, et al. Neuro-heuristic computational intelligence for solving nonlinear pantograph systems[J]. Frontiers of Information Technology & Electronic Engineering, 2017, 18(4): 464-484. DOI： 10.1631/FITEE.1500393.

摘要

本文提出了一种启发式神经网络计算平台，用于解决基于不同阶数泛函微分方程的非线性受电弓系统（Pantograph systems based on functional differential equations

P-FDEs）中的初值问题（Initial value problems

IVPs）。该方案利用了前馈人工神经网络（Artificial neural networks

ANNs）、基于遗传算法（Genetical gorithms

GAs）的进化计算技术，以及内点技术（Interior-point technique

IPT）。通过设定一个无监督学习误差，针对完全和不完全满足初始条件两种情况，利用ANNs创建了系统的两种数学模型。采用GA-IPT混合算法，对ANN模型的设计参数进行了优化。在GA-IPT中，GA是有效的全局搜索工具，IPT则用于快速的局部收敛。针对三种不同类型的1-3阶P-FDEs的IVPs对该方案进行了测试。通过对比现有的精确解，确认了该方案的正确性。通过采用不同数量神经元的ANN模型进行了大量的数值实验，进一步验证了该方案的准确性和收敛性。

Abstract

We present a neuro-heuristic computing platform for finding the solution for initial value problems (IVPs) of nonlinear pantograph systems based on functional differential equations (P-FDEs) of different orders. In this scheme

the strengths of feed-forward artificial neural networks (ANNs)

the evolutionary computing technique mainly based on genetic algorithms (GAs)

and the interior-point technique (IPT) are exploited. Two types of mathematical models of the systems are constructed with the help of ANNs by defining an unsupervised error with and without exactly satisfying the initial conditions. The design parameters of ANN models are optimized with a hybrid approach GA–IPT

where GA is used as a tool for effective global search

and IPT is incorporated for rapid local convergence. The proposed scheme is tested on three different types of IVPs of P-FDE with orders 1–3. The correctness of the scheme is established by comparison with the existing exact solutions. The accuracy and convergence of the proposed scheme are further validated through a large number of numerical experiments by taking different numbers of neurons in ANN models.

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references

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