Skip to main content
SpringerLink
Log in

Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

In the present study, stochastic numerical computing approach is developed by applying artificial neural networks (ANNs) to compute the solution of Lane–Emden type boundary value problems arising in thermodynamic studies of the spherical gas cloud model. ANNs are used in an unsupervised manner to construct the energy function of the system model. Strength of efficient local optimization procedures based on active-set (AS), interior-point (IP) and sequential quadratic programming (SQP) algorithms is used to optimize the energy functions. The performance of all three design methodologies ANN-AS, ANN-IP and ANN-SQP is evaluated on different nonlinear singular systems. The effectiveness of the proposed schemes in terms of accuracy and convergence is established from the results of statistical indicators.

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
Fig. 13

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Parand K, Dehghan M, Rezaei AR, Ghaderi SM (2010) An approximation algorithm for the solution of the nonlinear Lane–Emden type equations arising in astrophysics using Hermite functions collocation method. Comput Phys Commun 181(6):1096–1108

    Article  MathSciNet  MATH  Google Scholar 

  3. Parand K, Rezaei AR, Taghavi A (2010) Lagrangian method for solving Lane–Emden type equation arising in astrophysics on semi-infinite domains. Acta Astronaut 67:673–680

    Article  Google Scholar 

  4. Mukherjee S, Roy B, Chaterjee PK (2011) Solution of Lane–Emden equation by differential transform method. Int J Nonlinear Sci 12(4):478

    MathSciNet  Google Scholar 

  5. Horedt GP (1986) Seven-digit tables of Lane–Emden functions. Astrophys Space Sci 126:357

    Article  Google Scholar 

  6. Bender CM, Milton KA, Pinsky SS, Simmons LM Jr (1989) A new perturbative approach to nonlinear problems. J Math Phys 30:1447

    Article  MathSciNet  MATH  Google Scholar 

  7. Shawagfeh NT (1993) Nonperturbative approximate solution for Lane–Emden equation. J Math Phys 34:4364

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Mandelzweig VB, Tabakin F (2001) Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput Phys Commun 141:268

    Article  MathSciNet  MATH  Google Scholar 

  10. He JH (2003) Variational approach to the Lane–Emden equation. Appl Math Comput 143:539

    MathSciNet  MATH  Google Scholar 

  11. Liao SJ (2003) A new algorithm for solving singular IVPs of Lane–Emden type. Appl Math Comput 142:1

    MathSciNet  Google Scholar 

  12. 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(7):1116–1124

    Article  MathSciNet  MATH  Google Scholar 

  13. Yousefi SA (2006) Legendre wavelets method for solving differential equations of Lane–Emden type. Appl Math Comput 181(2):1417

    MathSciNet  MATH  Google Scholar 

  14. Chowdhury MSH, Hashim I (2007) Solutions of a class of singular second-order IVPs by homotopy-perturbation method. Phys Lett A 365(5):439

    Article  MathSciNet  MATH  Google Scholar 

  15. Yildirim A, Özis T (2007) Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method. Phys Lett A 369:70–76

    Article  MATH  Google Scholar 

  16. Marzban HR, Tabrizidooz HR, Razzaghi M (2008) Hybrid functions for nonlinear initial-value problems with applications to Lane–Emden type equations. Phys Lett A 372(37):5883–5886

    Article  MathSciNet  MATH  Google Scholar 

  17. Ramos JI (2008) Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method. Chaos Solitons Fractals 38(2):400–408

    Article  MathSciNet  MATH  Google Scholar 

  18. VanGorder RA, Vajravelu K (2008) Analytic and numerical solutions to the Lane–Emden equation. Phys Lett A 372:6060–6065

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  20. Kumar N, Pandey RK, Cattani C (2011) Solution of Lane–Emden type equations Bernstein operational matrix of integration. ISRN Astron Astrophys 2011(2011):351747. doi:10.5402/2011/351747

    Google Scholar 

  21. Pandey RK, Kumar N (2012) Solution of Lane–Emden type equations using Bernstein operational matrix of differentiation. New Astron 17(3):303–308

    Article  MathSciNet  Google Scholar 

  22. Iqbal S, Javed A (2011) Application of optimal homotopy asymptotic method for the analytic solution of singular Lane–Emden type equation. Appl Math Comput 217(19):7753–7761

    MathSciNet  MATH  Google Scholar 

  23. Eslahchi MR, Dehghan M, Ahmadi-Asl S (2012) The general shifted Jacobi matrix method for solving the general high order linear differential difference equations with variable coefficients. Appl Math Model 36:3387

    Article  MathSciNet  MATH  Google Scholar 

  24. Boubaker K, VanGorder RA (2012) Application of the BPES to Lane–Emden equations governing polytropic and isothermal gas spheres. New Astron 17:565–569

    Article  Google Scholar 

  25. Rismani AM, Monfared H (2012) Numerical solution of singular IVPs of Lane–Emden type using a modified Legendre-spectral method. Appl Math Model 36(10):4830–4836

    Article  MathSciNet  MATH  Google Scholar 

  26. Pandey RK, Kumar N, Bhardwaj A, Dutta G (2012) Solution of Lane–Emden type equations using Legendre operational matrix of differentiation. Appl Math Comput 218(14):7629–7637

    MathSciNet  MATH  Google Scholar 

  27. Dehghan M, Aryanmehr S, Eslahch MR (2013) A technique for the numerical solution of initial-value problems based on a class of Birkhoff-type interpolation method. J Comput Appl Math 244:125–139

    Article  MathSciNet  MATH  Google Scholar 

  28. Lakestani M, Dehghan M (2013) Four techniques based on the B-spline expansion and the collocation approach for the numerical solution of the Lane–Emden equation. Math Methods Appl Sci 36(16):2243–2253

    Article  MathSciNet  MATH  Google Scholar 

  29. Căruntu B, Bota C (2013) Approximate polynomial solutions of the nonlinear Lane–Emden type equations arising in astrophysics using the squared remainder minimization method. Comput Phys Commun 184:1643–1648

    Article  MathSciNet  Google Scholar 

  30. Kumar M, Yadav N (2011) Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: a survey. Comput Math Appl 62(10):3796–3811

    Article  MathSciNet  MATH  Google Scholar 

  31. Shirvany Y, Hayati M, Moradian R (2009) Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations. Appl Soft Comput 9(1):20–29

    Article  Google Scholar 

  32. Shirvany Y, Hayati M, Moradian R (2008) Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks. Commun Nonlinear Sci Numer Simul 13(10):2132–2145

    Article  MathSciNet  MATH  Google Scholar 

  33. Hayati M, Karami B (2007) Feedforward neural network for solving partial differential equations. J Appl Sci 7(19):2812–2817

    Article  Google Scholar 

  34. Raja MAZ, Samar R (2014) Numerical treatment of nonlinear MHD Jeffery–Hamel problems using stochastic algorithms. Comput Fluids 91:28–46

    Article  MathSciNet  Google Scholar 

  35. Khan JA, Raja MAZ, Rashidi MM, Syam MI, Wazwaz AM (2015) Nature-inspired computing approach for solving non-linear singular Emden–Fowler problem arising in electromagnetic theory. Connect Sci 27(4):377–396

  36. Raja MAZ, Samar R, Alaidarous ES, Shivanian E (2016) Bio-inspired computing platform for reliable solution of Bratu-type equations arising in the modeling of electrically conducting solids. Appl Math Model 40(11–12):5964–5977

    Article  MathSciNet  Google Scholar 

  37. Raja MAZ (2014) 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

    Article  Google Scholar 

  38. Raja MAZ, Samar R, Haroon T, Shah SM (2015) Unsupervised neural network model optimized with evolutionary computations for solving variants of nonlinear MHD Jeffery–Hamel problem. Appl Math Mech 36(12):1611–1638

    Article  MathSciNet  Google Scholar 

  39. Raja MAZ, Khan JA, Haroon T (2015) Stochastic numerical treatment for thin film flow of third grade fluid using unsupervised neural networks. J Taiwan Inst Chem Eng 48:26–39

    Article  Google Scholar 

  40. Raja MAZ, Shah FH, Khan AA, Khan NA (2016) Design of bio-inspired computational intelligence technique for solving steady thin film flow of Johnson–Segalman fluid on vertical cylinder for drainage problems. J Taiwan Inst Chem Eng 60:59–75

    Article  Google Scholar 

  41. Raja MAZ (2013) Unsupervised neural networks for solving Troesch’s problem. Chin Phys B 23(1):018903

    Article  Google Scholar 

  42. Raja MAZ, Samar R, Rashidi MM (2014) Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation. Neural Comput Appl 25(7–8):1585–1601

    Article  Google Scholar 

  43. Khan JA, Raja MAZ, Syam MI, Tanoli SAK, Awan SE (2015) Design and application of nature inspired computing approach for nonlinear stiff oscillatory problems. Neural Comput Appl 26(7):1763–1780

    Article  Google Scholar 

  44. Raja MAZ (2014) Stochastic numerical treatment for solving Troesch’s problem. Inf Sci 279:860–873

    Article  MathSciNet  MATH  Google Scholar 

  45. Raja MAZ, Khan MAR, Mahmood T, Farooq U, Chaudhary NI (2016) Design of bio-inspired computing technique for nanofluidics based on nonlinear Jeffery–Hamel flow equations. Can J Phys 94(999):1–16

    Google Scholar 

  46. Raja MAZ, Farooq U, Chaudhary NI, Wazwaz AM (2016) Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes. Appl Soft Comput 38:561–586

    Article  Google Scholar 

  47. Raja MAZ, Khan JA, Siddiqui AM, Behloul D, Haroon T, Samar R (2015) Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation. Appl Soft Comput 26:244–256

    Article  Google Scholar 

  48. Raja MAZ (2014) Numerical treatment for boundary value problems of pantograph functional differential equation using computational intelligence algorithms. Appl Soft Comput 24:806–821

    Article  Google Scholar 

  49. Arqub OA, Abo-Hammour Z (2014) Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inf Sci 279:396–415

    Article  MathSciNet  MATH  Google Scholar 

  50. Abu Arqub O, Abo-Hammour Z, Momani S, Shawagfeh N (2012) Solving singular two-point boundary value problems using continuous genetic algorithm. In: Abstract and applied analysis, vol 2012. Hindawi Publishing Corporation

  51. Abo-Hammour Z, Abu Arqub O, Momani S, Shawagfeh N (2014) Optimization solution of Troesch’s and Bratu’s problems of ordinary type using novel continuous genetic algorithm. Discrete Dyn Nat Soc 2014(2014):401696. doi:10.1155/2014/401696

    MathSciNet  Google Scholar 

  52. Mall S, Chakraverty S (2015) Numerical solution of nonlinear singular initial value problems of Emden–Fowler type using Chebyshev Neural Network method. Neurocomputing 149:975–982

    Article  Google Scholar 

  53. Mall S, Chakraverty S (2016) Application of Legendre Neural Network for solving ordinary differential equations. Appl Soft Comput 43:347–356

    Article  Google Scholar 

  54. Raja MAZ, Manzar MA, Samar R (2015) An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP. Appl Math Model 39(10):3075–3093

    Article  MathSciNet  Google Scholar 

  55. Raja MAZ, Khan JA, Qureshi IM (2010) A new stochastic approach for solution of Riccati differential equation of fractional order. Ann Math Artif Intell 60(3–4):229–250

    Article  MathSciNet  MATH  Google Scholar 

  56. Karmarkar N (1984) A new polynomial time algorithm for linear programming. Combinatorica 4:373–395

    Article  MathSciNet  MATH  Google Scholar 

  57. Wright SJ (1997) Primal-dual interior-point methods. SIAM, Philadelphia. ISBN 0-89871-382-X

  58. Yana W, Wenb L, Lic W, Chunga CY, Wong KP (2011) Decomposition–coordination interior point method and its application to multi-area optimal reactive power flow. Int J Electrc Power Energy Syst 33(1):55–60

    Article  Google Scholar 

  59. Duvvuru N, Swarup KS (2011) A hybrid interior point assisted differential evolution algorithm for economic dispatch. IEEE Trans Power Syst 26(2):541–549

    Article  Google Scholar 

  60. Hager WW, Zhang H (2006) A new active set algorithm for box constrained optimization. SIAM J Optim 17(02):526–557

    Article  MathSciNet  MATH  Google Scholar 

  61. Sloan SW (2005) A steepest edge active set algorithm for solving sparse linear programming problems. Int J Numer Methods Eng 26(12):2671–2685

    Article  MathSciNet  MATH  Google Scholar 

  62. Judice JJ, Sherali HD, Ribeiro IM, Faustino AM (2007) Complementarity active set algorithm for mathematical programming problems with equilibrium constraints. J Optim Theory Appl 134(3):467–481

    Article  MathSciNet  MATH  Google Scholar 

  63. Wright S, Nocedal J (1999) Numerical optimization. Springer Sci 35:67–68

    MATH  Google Scholar 

  64. Ahmad I, Mukhtar A (2015) Stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model. Appl Math Comput 261:360

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Gujrat, Gujrat, 50700, Pakistan

    Iftikhar Ahmad & Farooq Ashraf

  2. Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock, 43600, Pakistan

    Muhammad Asif Zahoor Raja

  3. Faculty of Science and Technology, University of Malaysia Pahang, Pekan, Malaysia

    Muhammad Bilal

Authors
  1. Iftikhar Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad Asif Zahoor Raja
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muhammad Bilal
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Farooq Ashraf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Iftikhar Ahmad.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmad, I., Raja, M.A.Z., Bilal, M. et al. Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model. Neural Comput & Applic 28 (Suppl 1), 929–944 (2017). https://doi.org/10.1007/s00521-016-2400-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-016-2400-y

Keywords

Search

Navigation