Neural Computing and Applications

, Volume 31, Issue 3, pp 793–812 | Cite as

Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing

  • Muhammad Asif Zahoor RajaEmail author
  • Jabran Mehmood
  • Zulqurnain Sabir
  • A. Kazemi Nasab
  • Muhammad Anwaar Manzar
Original Article


In this paper, a bio-inspired computational intelligence technique is presented for solving nonlinear doubly singular system using artificial neural networks (ANNs), genetic algorithms (GAs), sequential quadratic programming (SQP) and their hybrid GA–SQP. The power of ANN models is utilized to develop a fitness function for a doubly singular nonlinear system based on approximation theory in the mean square sense. Global search for the parameters of networks is performed with the competency of GAs and later on fine-tuning is conducted through efficient local search by SQP algorithm. The design methodology is evaluated on number of variants for two point doubly singular systems. Comparative studies with standard results validate the correctness of proposed schemes. The consistent correctness of the proposed technique is proven through statistics using different performance indices.


Singular systems Artificial neural networks Genetic algorithm Sequential quadratic programming Integrating computing 


Compliance with ethical standards

Conflict of interest

All the authors of the manuscript declared that there is no conflict of interest.


  1. 1.
    Singh R, Kumar J (2014) An efficient numerical technique for the solution of nonlinear singular boundary value problems. Comput Phys Commun 185(4):1282–1289MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Singh R, Kumar J (2014) The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems. J Appl Math Comput 44(1–2):397–416MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Singh R, Kumar J, Nelakanti G (2014) Approximate series solution of singular boundary value problems with derivative dependence using Green’s function technique. Comput Appl Math 33(2):451–467MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Singh R, Kumar J, Nelakanti G (2012) New approach for solving a class of doubly singular two-point boundary value problems using adomian decomposition method. Adv Numer Anal 2012:541083. doi: 10.1155/2012/541083 MathSciNetzbMATHGoogle Scholar
  5. 5.
    Verma AK, Singh M (2015) Singular nonlinear three point BVPs arising in thermal explosion in a cylindrical reactor. J Math Chem 53(2):670–684MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Roul P (2016) An improved iterative technique for solving nonlinear doubly singular two-point boundary value problems. Eur Phys J Plus 131(6):1–15MathSciNetCrossRefGoogle Scholar
  7. 7.
    Roul P, Warbhe U (2016) A novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems. J Comput Appl Math 296:661–676MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Pandey RK, Gupta GK (2012) A note on fourth order method for doubly singular boundary value problems. Adv Numer Anal. doi: 10.1155/2012/349618 MathSciNetzbMATHGoogle Scholar
  9. 9.
    Pandey RK, Verma AK (2010) On solvability of derivative dependent doubly singular boundary value problems. J Appl Math Comput 33(1–2):489–511MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Singh R, Wazwaz AM, Kumar J (2016) An efficient semi-numerical technique for solving nonlinear singular boundary value problems arising in various physical models. Int J Comput Math 93(8):1330–1346MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Malek S (2012) On the summability of formal solutions for doubly singular nonlinear partial differential equations. J Dyn Control Syst 18(1):45–82MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Canalis-Durand M, Mozo-Fernández J, Schäfke R (2007) Monomial summability and doubly singular differential equations. J Differ Equ 233(2):485–511MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mall S, Chakraverty S (2016) Application of legendre neural network for solving ordinary differential equations. Appl Soft Comput 43:347–356CrossRefGoogle Scholar
  14. 14.
    Mall S, Chakraverty S (2015) Numerical solution of nonlinear singular initial value problems of Emden–Fowler type using Chebyshev Neural Network method. Neurocomputing 149:975–982CrossRefGoogle Scholar
  15. 15.
    Mall S, Chakraverty S (2014) Chebyshev Neural Network based model for solving Lane–Emden type equations. Appl Math Comput 247:100–114MathSciNetzbMATHGoogle Scholar
  16. 16.
    Chakraverty S, Mall S (2014) Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems. Neural Comput Appl 25(3–4):585–594CrossRefGoogle Scholar
  17. 17.
    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):5964–5977MathSciNetCrossRefGoogle Scholar
  18. 18.
    Masood Z, Majeed K, Samar R, Raja MAZ (2017) Design of Mexican Hat Wavelet neural networks for solving Bratu type nonlinear systems. Neurocomputing 221:1–14. doi: 10.1016/j.neucom.2016.08.079 CrossRefGoogle Scholar
  19. 19.
    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):195–214MathSciNetCrossRefGoogle Scholar
  20. 20.
    Raja MAZ, Ahmad I, Khan I, Syam MI, Wazwaz AM (2017) Neuro-heuristic computational intelligence for solving nonlinear pantograph systems. Front Inf Technol Electron Eng 18(4):464–484CrossRefGoogle Scholar
  21. 21.
    Raja MAZ, Zameer A, Khan AU, Wazwaz AM (2016) A new numerical approach to solve Thomas–Fermi model of an atom using bio-inspired heuristics integrated with sequential quadratic programming. SpringerPlus 5(1):1400CrossRefGoogle Scholar
  22. 22.
    Effati S, Pakdaman M (2010) Artificial neural network approach for solving fuzzy differential equations. Inf Sci 180(8):1434–1457MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Baymani M, Effati S, Niazmand H, Kerayechian A (2015) Artificial neural network method for solving the Navier–Stokes equations. Neural Comput Appl 26(4):765–773CrossRefGoogle Scholar
  24. 24.
    Effati S, Skandari MHN (2012) Optimal control approach for solving linear Volterra integral equations. Int J Intell Syst Appl (IJISA) 4(4):40Google Scholar
  25. 25.
    Jafarian A, Measoomy S, Abbasbandy S (2015) Artificial neural networks based modeling for solving Volterra integral equations system. Appl Soft Comput 27:391–398CrossRefGoogle Scholar
  26. 26.
    Raja MAZ, Farooq U, Chaudhary NI, Wazwaz AM (2016) Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes. Appl Soft Comput 38:561–586CrossRefGoogle Scholar
  27. 27.
    Effati S, Buzhabadi R (2012) A neural network approach for solving Fredholm integral equations of the second kind. Neural Comput Appl 21(5):843–852CrossRefGoogle Scholar
  28. 28.
    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
  29. 29.
    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
  30. 30.
    Sabouri J, Effati S, Pakdaman M (2017) A neural network approach for solving a class of fractional optimal control problems. Neural Process Lett 45(1):59–74CrossRefGoogle Scholar
  31. 31.
    Effati S, Mansoori A, Eshaghnezhad M (2015) An efficient projection neural network for solving bilinear programming problems. Neurocomputing 168:1188–1197CrossRefGoogle Scholar
  32. 32.
    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(5):474–489CrossRefGoogle Scholar
  33. 33.
    Kumar M, Yadav N (2015) Numerical solution of Bratu’s problem using multilayer perceptron neural network method. Natl Acad Sci Lett 38(5):425–428MathSciNetCrossRefGoogle Scholar
  34. 34.
    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):377–396CrossRefGoogle Scholar
  35. 35.
    Yadav N, Yadav A, Kumar M, Kim JH (2015) An efficient algorithm based on artificial neural networks and particle swarm optimization for solution of nonlinear Troesch’s problem. Neural Comput Appl. doi: 10.1007/s00521-015-2046-1 Google Scholar
  36. 36.
    Raja MAZ (2014) Stochastic numerical treatment for solving Troesch’s problem. Inf Sci 279:860–873MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    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 60:59–75CrossRefGoogle Scholar
  38. 38.
    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 48:26–39CrossRefGoogle Scholar
  39. 39.
    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):3075–3093MathSciNetCrossRefGoogle Scholar
  40. 40.
    Raja MAZ, Samar R, Manzar MA, Shah SM (2017) Design of unsupervised fractional neural network model optimized with interior point algorithm for solving Bagley-Torvik equation. Math Comput Simul 132:139–158. doi: 10.1016/j.matcom.2016.08.002 MathSciNetCrossRefGoogle Scholar
  41. 41.
    Abu Arqub O, Abo-Hammour Z, Momani S (2014) Application of continuous genetic algorithm for nonlinear system of second-order boundary value problems. Appl Math 8(1):235–248zbMATHGoogle Scholar
  42. 42.
    Arqub OA, Abo-Hammour Z (2014) Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inf Sci 279:396–415MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Arqub OA, Al-Smadi M, Momani S, Hayat T (2016) Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems. Soft Comput 1–16. doi: 10.1007/s00500-016-2262-3
  44. 44.
    Arqub OA (2015) Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations. Neural Comput Appl 1–20. doi: 10.1007/s00521-015-2110-x
  45. 45.
    Abo-Hammour Z, Arqub OA, 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 2014:401696. doi: 10.1155/2014/401696 MathSciNetCrossRefGoogle Scholar
  46. 46.
    Ahmad I, Ahmad F, Raja MAZ, Ilyas H, Anwar N, Azad Z (2016) Intelligent computing to solve fifth-order boundary value problem arising in induction motor models. Neural Comput Appl. doi: 10.1007/s00521-016-2547-6 Google Scholar
  47. 47.
    Zonouz PR, Niaei A, Tarjomannejad A (2016) Modeling and optimization of toluene oxidation over perovskite-type nanocatalysts using a hybrid artificial neural network-genetic algorithm method. J Taiwan Inst Chem Eng 65:276–285CrossRefGoogle Scholar
  48. 48.
    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):1611–1638MathSciNetCrossRefGoogle Scholar
  49. 49.
    Hosseini SA, Alvarez-Galvan MC (2016) Study of physical–chemical properties and catalytic activities of ZnCr2O4 spinel nano oxides obtained from different methods—modeling the synthesis process by response surface methodology and optimization by genetic algorithm. J Taiwan Inst Chem Eng 61:261–269CrossRefGoogle Scholar
  50. 50.
    Pasandideh SHR, Niaki STA, Gharaei A (2015) Optimization of a multiproduct economic production quantity problem with stochastic constraints using sequential quadratic programming. Knowl-Based Syst 84:98–107CrossRefGoogle Scholar
  51. 51.
    Han X, Quan L, Xiong X (2015) A modified gravitational search algorithm based on sequential quadratic programming and chaotic map for ELD optimization. Knowl Inf Syst 42(3):689–708CrossRefGoogle Scholar
  52. 52.
    Raja MAZ, Shah FH, Tariq M, Ahmad I (2016) Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch’s problem arising in plasma physics. Neural Comput Appl 1–27. doi: 10.1007/s00521-016-2530-2

Copyright information

© The Natural Computing Applications Forum 2017

Authors and Affiliations

  1. 1.Department of Electrical EngineeringCOMSATS Institute of Information Technology, Attock CampusAttockPakistan
  2. 2.Department of MathematicsPreston University Kohat, Islamabad CampusKohatPakistan
  3. 3.Department of MathematicsCapital University of Science and TechnologyIslamabadPakistan
  4. 4.School of Mathematical SciencesUniversiti Sains MalaysiaGelugorMalaysia
  5. 5.Hamdard Institute of Engineering and TechnologyHamdard UniversityIslamabadPakistan

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