Full Text:   <2521>

Summary:  <1512>

CLC number: TP183; O175

On-line Access: 2017-04-12

Received: 2015-11-10

Revision Accepted: 2016-02-17

Crosschecked: 2017-03-14

Cited: 1

Clicked: 6031

Citations:  Bibtex RefMan EndNote GB/T7714


Muhammad Asif Zahoor Raja


-   Go to

Article info.
Open peer comments

Frontiers of Information Technology & Electronic Engineering  2017 Vol.18 No.4 P.464-484


Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

Author(s):  Muhammad Asif Zahoor Raja, Iftikhar Ahmad, Imtiaz Khan, Muhammed Ibrahem Syam, Abdul Majid Wazwaz

Affiliation(s):  Department of Electrical Engineering, COMSATs Institute of Information Technology, Attock 43200, Pakistan; more

Corresponding email(s):   rasifzahoor@yahoo.com, Muhammad.asif@ciit-attock.edu.pk

Key Words:  Neural networks, Initial value problems (IVPs), Functional differential equations (FDEs), Unsupervised learning, Genetic algorithms (GAs), Interior-point technique (IPT)

Muhammad Asif Zahoor Raja, Iftikhar Ahmad, Imtiaz Khan, Muhammed Ibrahem Syam, Abdul Majid Wazwaz. Neuro-heuristic computational intelligence for solving nonlinear pantograph systems[J]. Frontiers of Information Technology & Electronic Engineering, 2017, 18(4): 464-484.

@article{title="Neuro-heuristic computational intelligence for solving nonlinear pantograph systems",
author="Muhammad Asif Zahoor Raja, Iftikhar Ahmad, Imtiaz Khan, Muhammed Ibrahem Syam, Abdul Majid Wazwaz",
journal="Frontiers of Information Technology & Electronic Engineering",
publisher="Zhejiang University Press & Springer",

%0 Journal Article
%T Neuro-heuristic computational intelligence for solving nonlinear pantograph systems
%A Muhammad Asif Zahoor Raja
%A Iftikhar Ahmad
%A Imtiaz Khan
%A Muhammed Ibrahem Syam
%A Abdul Majid Wazwaz
%J Frontiers of Information Technology & Electronic Engineering
%V 18
%N 4
%P 464-484
%@ 2095-9184
%D 2017
%I Zhejiang University Press & Springer
%DOI 10.1631/FITEE.1500393

T1 - Neuro-heuristic computational intelligence for solving nonlinear pantograph systems
A1 - Muhammad Asif Zahoor Raja
A1 - Iftikhar Ahmad
A1 - Imtiaz Khan
A1 - Muhammed Ibrahem Syam
A1 - Abdul Majid Wazwaz
J0 - Frontiers of Information Technology & Electronic Engineering
VL - 18
IS - 4
SP - 464
EP - 484
%@ 2095-9184
Y1 - 2017
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/FITEE.1500393

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.


概要: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。


Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1]Agarwal, R.P., Chow, Y.M., 1986. Finite difference methods for boundary-value problems of differential equations with deviating arguments. Comput. Math. Appl., 12(11): 1143-1153.

[2]Arqub, O.A., Zaer, A.H., 2014. Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inform. Sci., 279:396-415.

[3]Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F., 2007. Intoduction to the Theory of Functional Differential Equations: Methods and Applications. Hindawi Publishing Corporation, New York, USA.

[4]Barro, G., So, O., Ntaganda, J.M., et al., 2008. A numerical method for some nonlinear differential equation models in biology. Appl. Math. Comput., 200(1):28-33.

[5]Chakraverty, S., Mall, S., 2014. Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems. Neur. Comput. Appl., 25(3):585-594.

[6]Dehghan, M., Salehi, R., 2010. Solution of a nonlinear time-delay model in biology via semi-analytical approaches. Comput. Phys. Commun., 181:1255-1265.

[7]Derfel, G., Iserles, A., 1997. The pantograph equation in the complex plane. J. Math. Anal. Appl., 213(1):117-132.

[8]Evans, D.J., Raslan, K.R., 2005. The Adomian decomposition method for solving delay differential equation. Int. J. Comput. Math., 82(1):49-54.

[9]Holland, J.H., 1975. Adaptation in Natural and Artificial Systems: an Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. The University of Michigan Press, Ann Arbor, USA.

[10]Iserles, A., 1993. On the generalized pantograph functional-differential equation. Eur. J. Appl. Math., 4(1):1-38.

[11]Khan, J.A., Raja, M.A.Z., Qureshi, I.M., 2011. Novel approach for van der Pol oscillator on the continuous time domain. Chin. Phys. Lett., 28:110205.

[12]Khan, J.A., Raja, M.A.Z., Syam, M.A., et al., 2015. Design and application of nature inspired computing approach for non-linear stiff oscillatory problems. Neur. Comput. Appl., 26(7):1763-1780.

[13]Mall, S., Chakraverty, S., 2014a. Chebyshev neural network based model for solving Lane–Emden type equations. Appl. Math. Comput., 247:100-114.

[14]Mall, S., Chakraverty, S., 2014b. Numerical solution of nonlinear singular initial value problems of Emden–Fowler type using Chebyshev neural network method. Neurocomputing, 149(B):975-982.

[15]McFall, K.S., 2013. Automated design parameter selection for neural networks solving coupled partial differential equations with discontinuities. J. Franklin Inst., 350(2): 300-317.

[16]Ockendon, J.R., Tayler, A.B., 1971. The dynamics of a current collection system for an electric locomotive. Proc. R. Soc. A, 322(1551):447-468.

[17]Pandit, S., Kumar, M., 2014. Haar wavelet approach for numerical solution of two parameters singularly perturbed boundary value problems. Appl. Math. Inform. Sci., 8(6): 2965-2974.

[18]Peng, Y.G., Jun, W., Wei, W., 2014. Model predictive control of servo motor driven constant pump hydraulic system in injection molding process based on neurodynamic optimization. J. Zhejiang Univ.-Sci. C (Comput. & Electron.), 15(2):139-146.

[19]Potra, F.A., Wright, S.J., 2000. Interior-point methods. J. Comput. Appl. Math., 124(1-2):281-302.

[20]Raja, M.A.Z., 2014a. Numerical treatment for boundary value problems of pantograph functional differential equation using computational intelligence algorithms. Appl. Soft Comput., 24:806-821.

[21]Raja, M.A.Z., 2014b. Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP. Connect. Sci., 26(3):195-214.

[22]Raja, M.A.Z., 2014c. Stochastic numerical techniques for solving Troesch’s problem. Inform. Sci., 279:860-873.

[23]Raja, M.A.Z., 2014d. Unsupervised neural networks for solving Troesch’s problem. Chin. Phys. B, 23(1):018903.

[24]Raja, M.A.Z., Ahmad, S.I., 2014. Numerical treatment for solving one-dimensional Bratu problem using neural networks. Neur. Comput. Appl., 24(3):549-561.

[25]Raja, M.A.Z., Samar, R., 2014a. Numerical treatment for nonlinear MHD Jeffery–Hamel problem using neural networks optimized with interior point algorithm. Neurocomputing, 124:178-193.

[26]Raja, M.A.Z., Samar, R., 2014b. Numerical treatment of nonlinear MHD Jeffery–Hamel problems using stochastic algorithms. Comput. Fluids, 91:28-46.

[27]Raja, M.A.Z., Khan, J.A., Qureshi, I.M., 2010a. Evolutionary computational intelligence in solving the fractional differential equations. Asian Conf. on Intelligent Information and Database Systems, p.231-240.

[28]Raja, M.A.Z., Khan, J.A., Qureshi, I.M., 2010b. Heuristic computational approach using swarm intelligence in solving fractional differential equations. Proc. 12th Annual Conf. Companion on Genetic and Evolutionary Computation, p.2023-2026.

[29]Raja, M.A.Z., Khan, J.A., Qureshi, I.M., 2010c. A new stochastic approach for solution of Riccati differential equation of fractional order. Ann. Math. Artif. Intell., 60(3):229-250.

[30]Raja, M.A.Z., Khan, J.A., Qureshi, I.M., 2011a. Solution of fractional order system of Bagley-Torvik equation using evolutionary computational intelligence. Math. Prob. Eng., 2011:765075.

[31]Raja, M.A.Z., Khan, J.A., Qureshi, I.M., 2011b. Swarm intelligence optimized neural network for solving fractional order systems of Bagley-Tervik equation. Eng. Intell. Syst., 19(1):41-51.

[32]Raja, M.A.Z., Khan, J.A., Ahmad, S.I., et al., 2012. A new stochastic technique for Painlevé equation-I using neural network optimized with swarm intelligence. Comput. Intell. Neur., 2012:721867.

[33]Raja, M.A.Z., Ahmad, S.I., Samar, R., 2013. Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem. Neur. Comput. Appl., 23(7):2199-2210.

[34]Raja, M.A.Z., Samar, R., Rashidi, M.M., 2014a. Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation. Neur. Comput. Appl., 25(7):1585-1601.

[35]Raja, M.A.Z., Ahmad, S.I., Samar, R., 2014b. Solution of the 2-dimensional Bratu problem using neural network, swarm intelligence and sequential quadratic programming. Neur. Comput. Appl., 25(7):1723-1739.

[36]Raja, M.A.Z., Khan, J.A., Shah, S.M., et al., 2015a. Comparison of three unsupervised neural network models for first Painlevé transcendent. Neur. Comput. Appl., 26(5):1055-1071.

[37]Raja, M.A.Z., Sabir, Z., Mahmood, N., et al., 2015b. Design of stochastic solvers based on genetic algorithms for solving nonlinear equations. Neur. Comput. Appl., 26(1):1-23.

[38]Raja, M.A.Z., Manzar, M.A., Samar, R., 2015c. An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP. Appl. Math. Model., 39(10-11):3075-3093.

[39]Raja, M.A.Z., Khan, J.A., Behloul, D., et al., 2015d. Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation. Appl. Soft Comput., 26:244-256.

[40]Raja, M.A.Z., Khan, J.A., Haroon, T., 2015e. Numerical treatment for thin film flow of third grade fluid using unsupervised neural networks. J. Taiw. Inst. Chem. Eng., 48:26-39.

[41]Saadatmandi, A., Dehghan, M., 2009. Variational iteration method for solving a generalized pantograph equation. Comput. Math. Appl., 58(11-12):2190-2196.

[42]Sedaghat, S., Ordokhani, Y., Dehghan, M., 2012. Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials. Commun. Nonl. Sci. Numer. Simul., 17(12):4815-4830.

[43]Shakeri, F., Dehghan, M., 2010. Application of the decomposition method of Adomian for solving the pantograph equation of order m. J. Phys. Sci., 65(5):453-460.

[44]Srinivasan, S., Saghir, M.Z., 2014. Predicting thermodiffusion in an arbitrary binary liquid hydrocarbon mixtures using artificial neural networks. Neur. Comput. Appl., 25(5): 1193-1203.

[45]Tang, L., Ying, G., Liu, Y.J., 2014. Adaptive near optimal neural control for a class of discrete-time chaotic system. Neur. Comput. Appl., 25(5):1111-1117.

[46]Tohidi, E., Bhrawy, A.H., Erfani, K.A., 2013. A collocation method based on Berneoulli operational matrix for numerical solution of generalized pantograph equation. Appl. Math. Model, 37(6):4283-4294.

[47]Troiano, L., Cosimo, B., 2014. Genetic algorithms supporting generative design of user interfaces: examples. Inform. Sci., 259:433-451.

[48]Uysal, A., Raif, B., 2013. Real-time condition monitoring and fault diagnosis in switched reluctance motors with Kohonen neural network. J. Zhejiang Univ.-Sci. C (Comput. & Electron.), 14(12):941-952.

[49]Wright, S.J., 1997. Primal-Dual Interior-Point Methods. SIAM, Philadelphia, USA.

[50]Xu, D.Y., Yang, S.L., Liu, R.P., 2013. A mixture of HMM, GA, and Elman network for load prediction in cloud-oriented data centers. J. Zhejiang Univ.-Sci. C (Comput. & Electron.), 14(11):845-858.

[51]Yusufoğlu, E., 2010. An efficient algorithm for solving gener alized pantograph equations with linear functional argument. Appl. Math. Comput., 217(7):3591-3595.

[52]Yüzbaşı, Ş., Mehmet, S., 2013. An exponential approximation for solutions of generalized pantograph-delay differential equations. Appl. Math. Model., 37(22):9160-9173.

[53]Yüzbaşı, Ş., Sahin, N., Sezer, M., 2011. A Bessel collocation method for numerical solution of generalized pantograph equations. Numer. Meth. Part. Diff. Eq., 28(4):1105-1123.

[54]Zhang, H.G., Wang, Z., Liu, D., 2008. Global asymptotic stability of recurrent neural networks with multiple time-varying delays. IEEE Trans. Neur. Netw., 19(5): 855-873.

[55]Zhang, Y.T., Liu, C.Y., Wei, S.S., et al., 2014. ECG quality assessment based on a kernel support vector machine and genetic algorithm with a feature matrix. J. Zhejiang Univ.-Sci. C (Comput. & Electron.), 15(7):564-573.

[56]Zoveidavianpoor, M., 2014. A comparative study of artificial neural network and adaptive neurofuzzy inference system for prediction of compressional wave velocity. Neur. Comput. Appl., 25(5):1169-1176.

Open peer comments: Debate/Discuss/Question/Opinion


Please provide your name, email address and a comment

Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - 2024 Journal of Zhejiang University-SCIENCE