Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

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.

Keywords

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

References

[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. http://dx.doi.org/10.1016/0898-1221(86)90018-0

[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. http://dx.doi.org/10.1016/j.ins.2014.03.128

[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. http://dx.doi.org/10.1155/9789775945495

[4] Barro,G., So,O., Ntaganda,J.M. , , 2008. A numerical method for some nonlinear differential equation models in biology. Appl. Math. Comput., 200(1):28–33. http://dx.doi.org/10.1016/j.amc.2007.10.041

[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. http://dx.doi.org/10.1007/s00521-013-1526-4

[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. http://dx.doi.org/10.1016/j.cpc.2010.03.014

[7] Derfel,G., Iserles, A., 1997. The pantograph equation in the complex plane. J. Math. Anal. Appl., 213(1):117–132. http://dx.doi.org/10.1006/jmaa.1997.5483

[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. http://dx.doi.org/10.1080/00207160412331286815

[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 functionaldifferential equation. Eur. J. Appl. Math., 4(1):1–38. http://dx.doi.org/10.1017/S0956792500000966

[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. http://dx.doi.org/10.1088/0256-307X/28/11/110205

[12] Khan,J.A., Raja,M.A.Z., Syam,M.A. , , 2015. Design and application of nature inspired computing approach for non-linear stiff oscillatory problems. Neur. Comput. Appl., 26(7):1763–1780. http://dx.doi.org/10.1007/s00521-015-1841-z

[13] Mall,S., Chakraverty, S., 2014a. Chebyshev neural network based model for solving Lane–Emden type equations. Appl. Math. Comput., 247:100–114. http://dx.doi.org/10.1016/j.amc.2014.08.085

[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. http://dx.doi.org/10.1016/j.neucom.2014.07.036

[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. http://dx.doi.org/10.1016/j.jfranklin.2012.11.003

[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. http://dx.doi.org/10.1098/rspa.1971.0078

[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. http://dx.doi.org/10.1631/jzus.C1300182

[19] Potra,F.A., Wright, S.J., 2000. Interior-point methods. J. Comput. Appl. Math., 124(1-2):281–302. http://dx.doi.org/10.1016/S0377-0427(00)00433-7

[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. http://dx.doi.org/10.1016/j.asoc.2014.08.055

[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. http://dx.doi.org/10.1080/09540091.2014.907555

[22] Raja,M.A.Z., 2014c. Stochastic numerical techniques for solving Troesch’s problem. Inform. Sci., 279:860–873. http://dx.doi.org/10.1016/j.ins.2014.04.036

[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. http://dx.doi.org/10.1007/s00521-012-1261-2

[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. http://dx.doi.org/10.1016/j.neucom.2013.07.013

[26] Raja,M.A.Z., Samar, R., 2014b. Numerical treatment of nonlinear MHD Jeffery–Hamel problems using stochastic algorithms. Comput. Fluids, 91:28–46. http://dx.doi.org/10.1016/j.compfluid.2013.12.005

[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. http://dx.doi.org/10.1007/978-3-642-12145-6_24

[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. http://dx.doi.org/10.1145/1830761.1830850

[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. http://dx.doi.org/10.1007/s10472-010-9222-x

[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. http://dx.doi.org/10.1155/2011/675075

[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. , , 2012. A new stochastic technique for Painlevé equation-I using neural network optimized with swarm intelligence. Comput. Intell. Neur., 2012:721867. http://dx.doi.org/10.1155/2012/721867

[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. http://dx.doi.org/10.1007/s00521-012-1170-4

[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. http://dx.doi.org/10.1007/s00521-014-1641-x

[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. http://dx.doi.org/10.1007/s00521-014-1664-3

[36] Raja,M.A.Z., Khan,J.A., Shah,S.M., , 2015a. Comparison of three unsupervised neural network models for first Painlevé transcendent. Neur. Comput. Appl., 26(5):1055–1071. http://dx.doi.org/10.1007/s00521-014-1774-y

[37] Raja,M.A.Z., Sabir, Z., Mahmood,N. , , 2015b. Design of stochastic solvers based on genetic algorithms for solving nonlinear equations. Neur. Comput. Appl., 26(1):1–23. http://dx.doi.org/10.1007/s00521-014-1676-z

[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. http://dx.doi.org/10.1016/j.apm.2014.11.024

[39] Raja,M.A.Z., Khan,J.A., Behloul,D. , , 2015d. Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation. Appl. Soft Comput., 26:244–256. http://dx.doi.org/10.1016/j.asoc.2014.10.009

[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. http://dx.doi.org/10.1016/j.jtice.2014.10.018

[41] Saadatmandi,A., Dehghan, M., 2009. Variational iteration method for solving a generalized pantograph equation. Comput. Math. Appl., 58(11-12):2190–2196. http://dx.doi.org/10.1016/j.camwa.2009.03.017

[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. http://dx.doi.org/10.1016/j.cnsns.2012.05.009

[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. http://dx.doi.org/10.1515/zna-2010-0510

[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. http://dx.doi.org/10.1007/s00521-014-1603-3

[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. http://dx.doi.org/10.1007/s00521-014-1595-z

[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. http://dx.doi.org/10.1016/j.apm.2012.09.032

[47] Troiano,L., Cosimo, B., 2014. Genetic algorithms supporting generative design of user interfaces: examples. Inform. Sci., 259:433–451. http://dx.doi.org/10.1016/j.ins.2012.01.006

[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. http://dx.doi.org/10.1631/jzus.C1300085

[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. http://dx.doi.org/10.1631/jzus.C1300109

[51] Yusufoğlu,E., 2010. An efficient algorithm for solving gener alized pantograph equations with linear functional argument. Appl. Math. Comput., 217(7):3591–3595. http://dx.doi.org/10.1016/j.amc.2010.09.005

[52] Yüzbaşı,Ş. , Mehmet,S., 2013. An exponential approximation for solutions of generalized pantograph-delay differential equations. Appl. Math. Model., 37(22):9160–9173. http://dx.doi.org/10.1016/j.apm.2013.04.028

[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. http://dx.doi.org/10.1002/num.20660

[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. http://dx.doi.org/10.1109/TNN.2007.912319

[55] Zhang,Y.T., Liu,C.Y., Wei,S.S., , 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. http://dx.doi.org/10.1631/jzus.C1300264

[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. http://dx.doi.org/10.1007/s00521-014-1604-2

RIGHTS & PERMISSIONS

2017 Zhejiang University and Springer-Verlag Berlin Heidelberg