Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

  • Muhammad Asif Zahoor Raja
  • Iftikhar Ahmad
  • Imtiaz Khan
  • Muhammed Ibrahem Syam
  • Abdul Majid Wazwaz


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.

Key words

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

CLC number

TP183 O175 


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  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  7. Derfel, G., Iserles, A., 1997. The pantograph equation in the complex plane. J. Math. Anal. Appl., 213(1): 117–132. Scholar
  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. Scholar
  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.zbMATHGoogle Scholar
  10. Iserles, A., 1993. On the generalized pantograph functionaldifferential equation. Eur. J. Appl. Math., 4(1): 1–38. Scholar
  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. Scholar
  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. Scholar
  13. Mall, S., Chakraverty, S., 2014a. Chebyshev neural network based model for solving Lane–Emden type equations. Appl. Math. Comput., 247: 100–114. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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.MathSciNetCrossRefGoogle Scholar
  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. Scholar
  19. Potra, F.A., Wright, S.J., 2000. Interior-point methods. J. Comput. Appl. Math., 124(1-2):281–302. Scholar
  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. Scholar
  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. Scholar
  22. Raja, M.A.Z., 2014c. Stochastic numerical techniques for solving Troesch’s problem. Inform. Sci., 279: 860–873. Scholar
  23. Raja, M.A.Z., 2014d. Unsupervised neural networks for solving Troesch’s problem. Chin. Phys. B, 23(1):018903.CrossRefGoogle Scholar
  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. Scholar
  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. Scholar
  26. Raja, M.A.Z., Samar, R., 2014b. Numerical treatment of nonlinear MHD Jeffery–Hamel problems using stochastic algorithms. Comput. Fluids, 91: 28–46. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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.Google Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  41. Saadatmandi, A., Dehghan, M., 2009. Variational iteration method for solving a generalized pantograph equation. Comput. Math. Appl., 58(11-12):2190–2196. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar
  47. Troiano, L., Cosimo, B., 2014. Genetic algorithms supporting generative design of user interfaces: examples. Inform. Sci., 259: 433–451. Scholar
  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. Scholar
  49. Wright, S.J., 1997. Primal-Dual Interior-Point Methods. SIAM, Philadelphia, USA.zbMATHCrossRefGoogle Scholar
  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. Scholar
  51. Yusufoglu, E., 2010. An efficient algorithm for solving gener alized pantograph equations with linear functional argument. Appl. Math. Comput., 217(7): 3591–3595. Scholar
  52. Yüzbasi, S., Mehmet, S., 2013. An exponential approximation for solutions of generalized pantograph-delay differential equations. Appl. Math. Model., 37(22): 9160–9173. Scholar
  53. Yüzbasi, S., 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. Scholar
  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. Scholar
  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. Scholar
  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. Scholar

Copyright information

© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Muhammad Asif Zahoor Raja
    • 1
  • Iftikhar Ahmad
    • 2
  • Imtiaz Khan
    • 3
  • Muhammed Ibrahem Syam
    • 4
  • Abdul Majid Wazwaz
    • 5
  1. 1.Department of Electrical EngineeringCOMSATs Institute of Information TechnologyAttockPakistan
  2. 2.Department of MathematicsUniversity of GujratGujratPakistan
  3. 3.Department of MathematicsPreston University, Islamabad CampusKohat, IslamabadPakistan
  4. 4.Department of Mathematical SciencesUnited Arab Emirates UniversityAl-Ain BoxUAE
  5. 5.Department of MathematicsSaint Xavier UniversityChicagoUSA

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