Skip to main content
SpringerLink
Log in

Solving Emden–Fowler Equations Using Improved Extreme Learning Machine Algorithm Based on Block Legendre Basis Neural Network

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

Abstract

In this paper, a numerical method to solve second order nonlinear ordinary differential equations with singular initial value problems was proposed. Here, block Legendre basis neural network (B-LBNN) was developed to create approximate solution and its derivatives. We use Legendre polynomials to expand the input pattern and eliminate hidden layer. By constructing block subnetwork and transforming variables, the problem of Emden–Fowler type equations was converted to solving the algebraic equation systems. The improved extreme learning machine algorithm was used to train network weights. A series of homogeneous or nonhomogeneous Emden–fowler equations were test to validate the accuracy and efficiency of B-LBNN model. Experimental results showed that the proposed algorithm provides a new tool for this equations and can outperform other approaches in literature in terms of accuracy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Chandrasekhar S (1967) Introduction to the study of Stellar structure. Dover Publications, New York

    MATH  Google Scholar 

  2. Davis HT (2010) Introduction to nonlinear differential and integral equations, Dover Publications, New York

  3. Richardson OU (1921) The emission of electricity from hot bodies, London

  4. Wazwaz AM (2001) A new algorithm for solving differential equations of Lane-Emden type. Appl Math Comput 118:287–310

    Article  MathSciNet  MATH  Google Scholar 

  5. Wazwaz AM (2006) The modified decomposition method for analytic treatment of differential equations. Appl Math Comput 173:165–176

    Article  MathSciNet  MATH  Google Scholar 

  6. Wazwaz AM (2010) The numerical solution of special fourth-order boundary value problems by the modified decomposition method. Int J Comput Math 79(3):345–356

    Article  MathSciNet  MATH  Google Scholar 

  7. Bengochea G, Verde-Star L (2016) An operational approach to the Emden-Fowler equation. Math Methods Appl Sci 38(18):4630–4637

    Article  MathSciNet  MATH  Google Scholar 

  8. Chowdhury MSH, Hashim I (2009) Solution of Emden-Fowler equations by homotopy-perturbation method. Nonlinear Anal Real Word Appl 10:104–115

    Article  MathSciNet  MATH  Google Scholar 

  9. Singh OP, Pandey RK, Singh VK (2009) An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified homotopy analysis method. Comput Phys Commun 180:1116–1124

    Article  MathSciNet  MATH  Google Scholar 

  10. Lakestani M, Saray BN (2012) Numerical solution of singular IVPs of Emden-Fowler type using Legendre scaling functions. Int J Nonlinear Sci. 13(2):211–219

    MathSciNet  MATH  Google Scholar 

  11. Rismani AM, Monfared H (2012) Numerical solution of singular IVPs of Lane-Emden type using a modified Legendre-spectral method. Appl Math Model 36:4830–4836

    Article  MathSciNet  MATH  Google Scholar 

  12. Mall S, Chakraverty S (2014) Chebyshev neural network based model for solving Lane-Emden type equations. Appl Math Comput 247:100–114

    Article  MathSciNet  MATH  Google Scholar 

  13. Khalid M, Sultana M, Zaidi F (2014) Numerical solution of six-order differential equations arising in astrophysics by neural network. Int J Comput Appl 107(6):1–6

    Google Scholar 

  14. Zhao TG, Wu Y (2016) Numerical solution to singular ordinary differential equations of Lane-Emden type by Legendre collocation method, In: Proceedings of the 3rd international conference on material engineering and application (ICMEA 2016), Advances in Engineering Research. 103: 496–501.

  15. Verma A, Kumar M (2020) Numerical solution of third-order Emden-Fowler type equations using artificial neural network technique. Eur Phys J Plus 135(9):1–14

    Article  Google Scholar 

  16. Sabir Z, Baleanu D, Raja MAZ, Hincal E (2022) A hybrid computing approach to design the novel second order singular perturbed delay differential Lane-Emden model. Phys Scr 97(8):085002

    Article  Google Scholar 

  17. Armaghani DJ, Hasanipanah M, Mahdiyar A et al (2018) Airblast prediction through a hybrid genetic algorithm -ANN model. Neural Comput Appl 29(9):619–629

    Article  Google Scholar 

  18. Hernández-Travieso JG, Ravelo-García AG, Alonso-Hernández JB et al (2020) Neural networks fusion for temperature forecasting. Neural Comput Appl 32(20):15699–15710

    Article  Google Scholar 

  19. Muzhou H, Taohua L, Yunlei Y (2017) A new hybrid constructive neural network method for impacting and its application on tungsten price prediction. Appl Intell 47(1):28–43

    Article  Google Scholar 

  20. Lu X, Muzhou H, Moonho L et al (2014) A new constructive neural network method for noise processing and its application on stock market prediction. Appl Soft Comput 15:57–66

    Article  Google Scholar 

  21. Rizk Y, Awad M (2019) On extreme learning machines in sequential and time series prediction: a non-iterative and approximate training algorithm for recurrent neural networks. Neurocomputig 345:1–19

    Google Scholar 

  22. Dwivedi AK (2018) Artificial neural network model for effective cancer classification using microarray gene expression data. Neural Comput Appl 29(12):1545–1554

    Article  Google Scholar 

  23. Jiao Y, Pan X, Zhao Z et al (2018) Learning sparse partial differential equations for vector-valued images. Neural Comput Appl 29(11):1205–1216

    Article  Google Scholar 

  24. Abdulla MB, Costa AL, Sousa RL (2018) Probabilistic identification of subsurface gypsum geohazards using artificial neural networks. Neural Comput Appl 29(12):1377–1391

    Article  Google Scholar 

  25. Vargas JA, Pedrycz W, Hemerly EM (2019) Improved learning algorithm for two-layer neural networks for identification of nonlinear systems. Neurocomputing 349:86–96

    Article  Google Scholar 

  26. Wang Y, Liu M, Bao Z et al (2019) Stacked sparse autoencoder with PCA and SVM for data-based line trip fault diagnosis in power systems. Neural Comput Appl 31(10):6719–6731

    Article  Google Scholar 

  27. Qiao J, Zhang W (2018) Dynamic multi-objective optimization control for wastewater treatment process. Neural Comput Appl 29(11):1261–1271

    Article  Google Scholar 

  28. Luo D, Wang JR, Shen D (2018) Learning formation control for fractional-order multi-agent systems. Math Methods Appl Sci 41:5003–5014

    Article  MathSciNet  MATH  Google Scholar 

  29. Luo D, Wang JR, Shen D (2019) $PD^{\alpha}$-type distributed learning control for nonlinear fractional-order multi-agent systems. Math Methods Appl Sci 42:4543–4553

    Article  MathSciNet  MATH  Google Scholar 

  30. Luo D, Wang JR, Shen D (2020) Consensus tracking problem for linear fractional multi-agent systems with initial state error. Nonlinear Anal Model Control 25:766–785

    MathSciNet  MATH  Google Scholar 

  31. Si Y, Wang JR (2022) Relative controllability multi agent systems with pairwise different delays in states. Nonlinear Anal Model Control 27:289–307

    MathSciNet  MATH  Google Scholar 

  32. Cao X, Fekan M, Shen D, Wang JR (2021) Iterative learning control for multi-agent systems with impulsive consensus tracking. Nonlinear Anal Model Control 26:130–150

    Article  MathSciNet  MATH  Google Scholar 

  33. Sahoo AK, Chakraverty S (2022) Machine intelligence in dynamical systems: a state-of-art review. Wiley Interdiscip Rev Data Min Knowl Discov. 12(4):e1461

    Article  Google Scholar 

  34. Chakraverty S (ed) (2022). Springer, Germany

    Google Scholar 

  35. Sahoo AK, Chakraverty S (2021) Multilayer unsupervised symplectic artificial neural network model for solving Duffing and Van der Pol–Duffing oscillator equations arising in engineering problems, modeling and computation in vibration problems, Volume 2: Soft computing and uncertainty. IOP Publishing

  36. Muzhou H, Xuli H (2010) Constructive approximation to multivariate function by decay RBF neural network. IEEE Trans Neural Netw 21(9):1517–1523

    Article  Google Scholar 

  37. Muzhou H, Xuli H (2011) The multidimensional function approximation based on constructive wavelet RBF neural network. Appl Soft Comput 11(2):2173–2177

    Article  Google Scholar 

  38. Muzhou H, Xuli H (2012) Multivariate numerical approximation using constructive L-2(R) RBF neural network. Neural Comput Appl 21(1):25–34

    Article  Google Scholar 

  39. Huang G-B, Zhu Q-Y, Siew C-K (2006) Extreme learning machine: theory and applications. Neurocomputing 70:489–501

    Article  Google Scholar 

  40. Hua D (2001) Matrix theory. Science Press, Beijing

    Google Scholar 

  41. Zeng X, Liang S, Hong Y, Chen J (2019) Distributed computation of linear matrix equations: an optimization perspective. IEEE Trans Autom Control 64(5):1858–1873

    Article  MathSciNet  MATH  Google Scholar 

  42. Liang X (2001) A recurrent neural network for nonlinear continuously differentiable optimization over a compact convex subset. IEEE Trans Neural Netw 12(6):1487–1490

    Article  Google Scholar 

  43. Cheng L, Hou ZG, Lin Y, Tan M, Zhang WC, Wu FX (2011) Recurrent neural network for non-smooth convex optimization problems with application to the identification of genetic regulatory networks. IEEE Trans Neural Netw 22(5):714–726

    Article  Google Scholar 

  44. Xia Z, Liu Y, Kou KI, Wang J (2022) Clifford-valued distributed optimization based on recurrent neural networks, IEEE Trans Neural Netw Learn Syst, 1–12

  45. Cao K, Zeng X, Hong Y (2017) Continuous-time distributed algorithms for solving linear algebraic equation, In: Proceedings of the 36th Chinese Control Conference, July 26–28, Dalian, China

Download references

Acknowledgements

This work was supported by Guizhou Provincial Science and Technology Projects (No. QKHJC-ZK[2021]YB017), Guizhou Provincial Education Department Higher Education Institution Youth Science Research Projects (QJJ[2022]098), Guizhou Provincial Science and Technology Projects (No. QKHJC-ZK[2023]YB036), Hunan Province Natural Science Foundation (No. 2022JJ30673).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, China

    Yunlei Yang & Yang Wu

  2. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Muzhou Hou

  3. High-Tech Research Institute, Hunan Institute of Traffic Engineering, Hengyang, 421001, China

    Jianshu Luo

  4. School of Mathematics and Statistics, and Mobile E-Business Collaborative Innovation Center of Hunan Province, Hunan University of Commerce, Changsha, 410205, China

    Xiaoliang Xie

Authors
  1. Yunlei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muzhou Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianshu Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiaoliang Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muzhou Hou.

Ethics declarations

Conflict of interests

The authors declare that they have no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, Y., Wu, Y., Hou, M. et al. Solving Emden–Fowler Equations Using Improved Extreme Learning Machine Algorithm Based on Block Legendre Basis Neural Network. Neural Process Lett 55, 7135–7154 (2023). https://doi.org/10.1007/s11063-023-11254-9

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-023-11254-9

Keywords

Search

Navigation