A Smart Amalgamation of Spectral Neural Algorithm for Nonlinear Lane-Emden Equations with Simulated Annealing

Najeeb Alam Khan 1  and Amber Shaikh 1
  • 1 Department of Mathematics, University of Karachi, Karachi 75270, Pakistan


The actual motivation of this paper is to develop a functional link between artificial neural network (ANN) with Legendre polynomials and simulated annealing termed as Legendre simulated annealing neural network (LSANN). To demonstrate the applicability, it is employed to study the nonlinear Lane-Emden singular initial value problem that governs the polytropic and isothermal gas spheres. In LSANN, minimization of error is performed by simulated annealing method while Legendre polynomials are used in hidden layer to control the singularity problem. Many illustrative examples of Lane-Emden type are discussed and results are compared with the formerly used algorithms. As well as with accuracy of results and tranquil implementation it provides the numerical solution over the entire finite domain.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] A. M. Wazwaz, A new algorithm for solving diferential equations of Lane-Emden type, Appl. Math. Comput, 118, 2001, 287–310

  • [2] M. Dehghan, F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astron, 13(1), 2008, 53–59

  • [3] K. Parand, M. Shahini, M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type, J. Comput. Phys, 228(23), 2009, 8830–8840

  • [4] K. Parand, A. Pirkhedri, Sinc-Collocation method for solving astrophysics equations, New Astron, 15(6), 2010, 533–537

  • [5] K. Boubaker, R. A. Van Gorder, Application of the BPES to Lane-Emden equations governing polytropic and isothermal gas spheres, New Astron, 17(6), 2012, 565–569

  • [6] R.K. Pandey, N. Kumar, A. Bhardwaj, G. Dutta, Solution of Lane-Emden type equations using Legendre operational matrix of differentiation, Appl. Math. Comput, 218(14), 2012, 7629–7637

  • [7] A.M. Rismani, H. Monfared, Numerical solution of singular IVPs of Lane-Emden type using a modified Legendre-spectral method, Appl. Math. Model, 36(10), 2012, 4830–4836

  • [8] E.H. Doha, W.M. Abd-Elhameed, Y.H. Youssri, Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type, New Astron, 23-24, 2013, 113–117

  • [9] H. Kaur, R.C. Mittal, V. Mishra, Haar wavelet approximate solutions for the generalized Lane-Emden equations arising in astrophysics, Comput. Phys. Commun, 184(9), 2013, 2169–2177

  • [10] A. Nazari-Golshan, S.S. Nourazar, H. Ghafoori-Fard, A. Yildirim, A. Campo, A modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane-Emden equations, Appl. Math. Lett, 26, 2013, 1018–1025

  • [11] B. Grbz, M. Sezer, Laguerre polynomial approach for solving Lane-Emden type functional differential equations, Appl. Math. Comput, 242, 2014, 255–264

  • [12] S. Mall, S. Chakraverty, Chebyshev Neural Network based model for solving Lane–Emden type equations, Appl. Math. Comput, 247, 2014, 100–114

  • [13] P. Pablo, C. Alzate, An Iterative Method for Solving Two Special Cases of Lane-Emden Type Equation, AJCM, 4, 2014, 242–253

  • [14] Z. Łmarda, Y. Khan, An efficient computational approach to solving singular initial value problems for Lane–Emden type equations, J. Comput. Appl. Math, 290, 2015, 65–73

  • [15] A. Kazemi Nasab, A. Kılıman, Z. P. Atabakan, W.J. Leong, A numerical approach for solving singular nonlinear Lane–Emden type equations arising in astrophysics, New Astron, 34, 2015, 178–186

  • [16] R. Iacono, M. De Felice, Constructing analytic approximate solutions to the Lane–Emden equation, Phys. Lett. A, 379(32-33), 2015, 1802–1807

  • [17] L.P. Aarts, P. V. Veer, Neural Network Method for Solving Partial Differential Equations, Neural process lett., 14, 2001, 261–271

  • [18] A. J. Meade, The Numerical Solution of Linear Ordinary Differential Equations by Feedforward Neural Networks, Mathl. Comput Modelling, 19(12), 1994, 1-25

  • [19] D. R. Parisi, C. Mariani, and M. A. Laborde, Solving differential equations with unsupervised neural networks, Chem. Eng. Process. Process Intensif., 42, 2003, 715–721

  • [20] I. E. Lagaris, A. C. Likas, and D. I. Fotiadis, Artifical Neural Networks for Solving Ordinary and Partial Differential Equations, Neural Networks, IEEE Trans., 9(5), 1998, 1–26

  • [21] L. Jianyu, L. Siwei, Q. Yingjian, H. Yaping, Numerical solution of elliptic partial differential equation using radial basis function neural networks, Neural Networks, 16, 2003, 729–734

  • [22] A. Malek, R. S. Beidokhti, Numerical solution for high order differential equations using a hybrid neural network – Optimization method, Appl. Math. Comput., 183, 2006, 260–271

  • [23] F. Fazayeli, L. P. Wang, and W. Liu, Back-propagation with chaos, Proc. 2008IEEE International Conference on Neural Networks and Signal Processing(ICNNSP2008), Zhenjiang, China, June 7-11, pp.5-8, 2008

  • [24] K. Parand, Z. Roozbahani, Solving nonlinear Lane-Emden type equations with unsupervised combined artificial neural networks, Int. J. of Industrial Math., 5, 2013, 355–366

  • [25] V. Kecman, Learning and Soft Computing, Support Vector machines, NeuralNetworks and Fuzzy Logic Models, The MIT Press, Cambridge, MA, 2001

  • [26] L.P. Wang and X.J. Fu, Data Mining with Computational Intelligence, Springer, Berlin, 2005

  • [27] J.C. Patra, P.K. Meher, G. Chakraborty, Nonlinear channel equalization for wireless communication systems using Legendre neural networks, Signal Processing, 89(11), 2009, 2251–2262

  • [28] S.K. Nanda, D.P. Tripathy, Application of Functional Link Artificial Neural Network for Prediction of Machinery Noise in Opencast Mines, Adv. Fuzzy Syst, 2011, 2011, 1–11

  • [29] K.K. Das, J.K. Satapathy, Novel Algorithms Based on Legendre Neural Network for Nonlinear Active Noise Control with Nonlinear Secondary Path, IJCSIT, 3(5), 2012, 5036–5039

  • [30] Y. H. Pao, Y. Takefuji, Functional-Link Net Computing: Theory, System, Architecture and Functionalities, Computer, 25(5), 1992, 76–79

  • [31] K. A. Dowsland, Hill-Climbing, Simulated Annealing and the Steiner Problem in Graphs, Eng. Optim, 17, 1991, 91–107

  • [32] B. Caruntu, C. Bota, Approximate polynomial solutions of the nonlinear Lane-Emden type equations arising in astrophysics using the squared remainder minimization method, Comput. Phys. Commun, 184, 2013, 1643–1648

  • [33] J. A. Khan, M. A. Z. Raja, I. M. Qureshi, Numerical treatment of nonlinear Emden–Fowler equation using stochastic technique, Ann Math Artif Intell, 63(2), 2011, 185–207


Journal + Issues