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Journal of Artificial Intelligence and Soft Computing Research

The Journal of Polish Neural Network Society, the University of Social Sciences in Lodz & Czestochowa University of Technology

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2083-2567
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A Smart Amalgamation of Spectral Neural Algorithm for Nonlinear Lane-Emden Equations with Simulated Annealing

Najeeb Alam Khan / Amber Shaikh
Published Online: 2017-03-20 | DOI: https://doi.org/10.1515/jaiscr-2017-0015

Abstract

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.

Keywords: Lane-Emden equations; simulated annealing; legendre polynomials; neural network

References

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

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

  • [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, 8830–8840Web of ScienceGoogle Scholar

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

  • [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, 565–569CrossrefWeb of ScienceGoogle Scholar

  • [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, 7629–7637Web of ScienceGoogle Scholar

  • [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, 4830–4836CrossrefWeb of ScienceGoogle Scholar

  • [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, 113–117Web of ScienceCrossrefGoogle Scholar

  • [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, 2169–2177Web of ScienceGoogle Scholar

  • [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, 1018–1025Web of ScienceCrossrefGoogle Scholar

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

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

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

  • [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, 65–73Web of ScienceGoogle Scholar

  • [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, 178–186Web of ScienceCrossrefGoogle Scholar

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

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

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

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

  • [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, 1–26Google Scholar

  • [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, 729–734CrossrefGoogle Scholar

  • [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, 260–271Google Scholar

  • [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, 2008Google Scholar

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

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

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

  • [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, 2251–2262Web of ScienceGoogle Scholar

  • [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, 1–11CrossrefGoogle Scholar

  • [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, 5036–5039Google Scholar

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

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

  • [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, 1643–1648Web of ScienceCrossrefGoogle Scholar

  • [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, 185–207CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2016-01-01

Accepted: 2016-07-04

Published Online: 2017-03-20

Published in Print: 2017-07-01


Citation Information: Journal of Artificial Intelligence and Soft Computing Research, Volume 7, Issue 3, Pages 215–224, ISSN (Online) 2083-2567, DOI: https://doi.org/10.1515/jaiscr-2017-0015.

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© 2017 Academy of Management (SWSPiZ), Lodz. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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