Neural Computing and Applications

, Volume 29, Issue 7, pp 449–466 | Cite as

Intelligent computing to solve fifth-order boundary value problem arising in induction motor models

  • Iftikhar Ahmad
  • Fayyaz Ahmad
  • Muhammad Asif Zahoor Raja
  • Hira Ilyas
  • Nabeela Anwar
  • Zarqa Azad
Original Article


In this study, biologically inspired intelligent computing approached based on artificial neural networks (ANN) models optimized with efficient local search methods like sequential quadratic programming (SQP), interior point technique (IPT) and active set technique (AST) is designed to solve the higher order nonlinear boundary value problems arise in studies of induction motor. The mathematical modeling of the problem is formulated in an unsupervised manner with ANNs by using transfer function based on log-sigmoid, and the learning of parameters of ANNs is carried out with SQP, IPT and ASTs. The solutions obtained by proposed methods are compared with the reference state-of-the-art numerical results. Simulation studies show that the proposed methods are useful and effective for solving higher order stiff problem with boundary conditions. The strong motivation of this research work is to find the reliable approximate solution of fifth-order differential equation problems which are validated through strong statistical analysis.


Induction motor model Active set technique Interior point technique Sequential quadratic programming Artificial neural networks 


  1. 1.
    Karageorghis A, Phillips TN, Davies AR (1988) Spectral collocation methods for the primary two-point boundary value problem in modeling viscoelastic flows Int. J Numer Methods Eng 26:805–813CrossRefzbMATHGoogle Scholar
  2. 2.
    Davies AR, Karageorghis A, Phillips TN (1988) Spectral Galerkin methods for the primary two-point boundary value problem in modeling viscoelastic flows Int. J Numer Methods Eng 26:647–662CrossRefzbMATHGoogle Scholar
  3. 3.
    Siddiqi SS, Sadaf M (2015) Application of non-polynomial spline to the solution of fifth-order boundary value problems in induction motor. J Egypt Math Soc 23(1):20–26MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agarwal RP (1986) Boundary value problems for higher order differential equations, vol 181. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  5. 5.
    Caglar HN, Caglar SH, Twizell EH (1999) The numerical solution of fifth-order boundary value problems with sixth degree B-spline functions. Appl Math Lett 12:25–30MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kasi KNS, Viswanadham YS (2012) Raju Quartic B-spline collocation method for fifth-order boundary value problems. Int J Comput Appl 43:1–6Google Scholar
  7. 7.
    Siddiqi SS, Twizell EH (1996) Spline solution of linear sixth-order boundary value problems. Int J Comput Math 60(3):295–304CrossRefzbMATHGoogle Scholar
  8. 8.
    Siddiqi SS, Akram G (2007) Sextic spline solutions of fifth-order boundary value problems. Appl Math Lett 20:591–597MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Siddiqi SS, Akram G, Kanwal A (2011) Solution of fifth-order singularly perturbed boundary value problems using non-polynomial spline technique Euro. J Sci Res 56:415–425Google Scholar
  10. 10.
    Ahmad I, Ahmad S, Bilal M, Anwar N (2016) Stochastic numerical treatment for solving Falkner-Skan equations using feed forward neural networks. Neural Comput Appl. doi: 10.1007/s00521-016-2427-0 Google Scholar
  11. 11.
    Raja MAZ, Ahmad S (2014) Numerical treatment for solving one-dimensional Bratu problem using neural networks. Neural Comput Appl 24:549–561CrossRefGoogle Scholar
  12. 12.
    Ahmad I, Raja MAZ, Bilal M, Ashraf F (2016) Neural network methods to solve the Lane-Emden type equations arising in thermodynamic studies of the spherical gas cloud model. Neural Comput Appl. doi: 10.1007/s00521-016-2400-y Google Scholar
  13. 13.
    Raja MAZ, Khan JA, Chaudhary NI, Shivanian E (2016) Reliable numerical treatment of nonlinear singular Flierl–Petviashivili equations for unbounded domain using ANN GAs, and SQP”. Appl Soft Comput 38:617–636CrossRefGoogle Scholar
  14. 14.
    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):5964–5977MathSciNetCrossRefGoogle Scholar
  15. 15.
    Raja MAZ, Ahmad S, Samar R (2013) Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem. Neural Comput Appl 23(7–8):1–12Google Scholar
  16. 16.
    Raja MAZ, Ahmad S (2014) Numerical treatment for solving one-dimensional Bratu problem using neural networks. Neural Comput Appl 24(3–4):1–13Google Scholar
  17. 17.
    Ahmad I, Mukhtar A (2015) Stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model. Appl Math Comput 261:360MathSciNetGoogle Scholar
  18. 18.
    Raja MAZ, Ahmad I, Khan I, Syam MI, Wazwaz AM (2016) Neuro-heuristic computational intelligence for solving nonlinear pantograph systems. Front Inf Technol Electron Eng. doi: 10.1631/FITEE.1500393 Google Scholar
  19. 19.
    Bender CM, Milton KA, Pinsky SS, Simmons LM Jr (1989) A new perturbative approach to nonlinear problems. J Math Phys 30:1447–1455MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ganesan N, Venkatesh K, Malathi Palani A, Rama MA (2010) Application of neural networks in diagnosing cancer disease using demographic data. Int J Comput Appl 26(1):0975–8887. doi: 10.5120/476-783 Google Scholar
  21. 21.
    Adler I, Karmarkar N, Resende MGC, Veiga G (1989) An implementation of Karmarkar’s algorithm for linear programming. Math Program 44:297–335MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lee P, Swaminathan R (2005) An interior point (Karmarkar) project for solving the global routing problem. Linear algebra with numerical application, vol 2005, pp 4–6Google Scholar
  23. 23.
    Shawagfeh NT (1993) Nonperturbative approximate solution for Lane–Emden equation. J Math Phys 34:4364–4369MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fukushima M, Tseng P (2002) An implementable active-set algorithm for computing a B-stationary point of a mathematical program with linear complementarity constraints. SIAM J Optim 12(3):724–739MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hager WW, Zhang H (2006) A new active set algorithm for box constrained optimization’. SIAM J Optim 17(2):526–557MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  • Iftikhar Ahmad
    • 1
  • Fayyaz Ahmad
    • 2
  • Muhammad Asif Zahoor Raja
    • 3
  • Hira Ilyas
    • 1
  • Nabeela Anwar
    • 1
  • Zarqa Azad
    • 1
  1. 1.Department of MathematicsUniversity of GujratGujratPakistan
  2. 2.Department of StatisticsUniversity of GujratGujratPakistan
  3. 3.Department of Electrical EngineeringCOMSATS Institute of Information Technology, Attock CampusAttockPakistan

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