Skip to main content

Advertisement

SpringerLink
Log in

Neuron analysis through the swarming procedures for the singular two-point boundary value problems arising in the theory of thermal explosion

  • Review
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

The motive of this work is to provide the neural investigations using the artificial neural networks (ANNs) through the particle swarm optimization for the singular two-point (STP) boundary value problems (BVPs), i.e., STP-BVPs arising in the theory of thermal explosion. The main purpose of this work is to perform the neural studies based on the large and small (45, 15, 3) neurons together with the complexity cost. The neuron performances have been designated in the form of absolute error. The best results have been achieved in case of large neurons as compared to small neurons, but the complexity cost gets high. The optimization measures of an error function are performed by using the swarming computational global search scheme along with the local search interior-point algorithms (IPA) for the STP-BVPs arising in the theory of thermal explosion. The exactness of the proposed scheme is approved by using the comparison of the obtained and reference solutions. Moreover, the generalization of the data is proposed in terms of statistical analysis to substantiate the capability and trustworthiness of the designed approach for the STP-BVPs.

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

Similar content being viewed by others

Data availability

All data generated or analyzed during this study are included in this article.

References

  1. Z. Liu, W. Cai, Z.Q.J Xu, Multi-scale deep neural network (MscaleDNN) for solving Poisson-Boltzmann equation in complex domains. 2020 arXiv preprint https://arxiv.org/abs/2007.11207

  2. P. Roul, K. Thula, A new high-order numerical method for solving singular two-point boundary value problems. J. Comput. Appl. Math. 343, 556–574 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. F.S. Guzmán, L.A. Ureña-López, Gravitational atoms: General framework for the construction of multistate axially symmetric solutions of the Schrödinger-Poisson system. Phys. Rev. D 101(8), 081302 (2020)

    Article  ADS  Google Scholar 

  4. A. Selvam, J. Alzabut, R. Dhineshbabu, S. Rashid, M. Rehman, Discrete fractional order two-point boundary value problem with some relevant physical applications. J. Inequal. Appl. 2020(1), 1–19 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. P.K. Pandey, Solution of two-point boundary value problems, a numerical approach: parametric difference method. Appl. Math. Nonlinear Sci. 3(2), 649–658 (2018)

    Article  MathSciNet  Google Scholar 

  6. K. Marynets, On a two-point boundary-value problem in geophysics. Appl. Anal. 98(3), 553–560 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. A.T. Assanova, Z.M. Kadirbayeva, On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations. Comput. Appl. Math. 37(4), 4966–4976 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Makin, Two-point boundary-value problems with nonclassical asymptotics on the spectrum. Electron. J. Differential Equations 2018(95), 1–7 (2018)

    MathSciNet  MATH  Google Scholar 

  9. K. Marynets, A nonlinear two-point boundary-value problem in geophysics. Monatshefte für Mathematik 188(2), 287–295 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Nachman, A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids. SIAM J. Appl. Math. 38(2), 275–281 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973. MR0404788 (53: 8588)

  12. S. Liao, Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14(4), 983–997 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. G. Adomian, Solving frontier problems of physics: the decomposition method (Fundamental Theories of Physics, Kluwer Academic Publishers Group, Dordrecht, With a preface by Yves Cherruault, 1994), p. 1

    MATH  Google Scholar 

  14. ΖΜ Odibat et al., Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7(1), 27–34 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. J.H. He, Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135(1), 73–79 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Z. Sabir, M.R. Ali, R. Sadat, Gudermannian neural networks using the optimization procedures of genetic algorithm and active set approach for the three-species food chain nonlinear model. J. Ambient Intell. Human. Comput. (2022). https://doi.org/10.1007/s12652-021-03638-3

    Article  Google Scholar 

  17. Z. Sabir, Stochastic numerical investigations for nonlinear three-species food chain system. Int. J. Biomath. (2021). https://doi.org/10.1142/S179352452250005X

    Article  Google Scholar 

  18. Z. Sabir et al., Design of Morlet wavelet neural network for solving the higher order singular nonlinear differential equations. Alex. Eng. J. 60(6), 5935–5947 (2021)

    Article  Google Scholar 

  19. Z. Sabir et al., Integrated intelligent computing with neuro-swarming solver for multi-singular fourth-order nonlinear Emden-Fowler equation. Comput. Appl. Math. 39(4), 1–18 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Z. Sabir et al., Heuristic computing technique for numerical solutions of nonlinear fourth order Emden-Fowler equation. Math. Comput. Simul. 178, 534–548 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Umar et al., A stochastic computational intelligent solver for numerical treatment of mosquito dispersal model in a heterogeneous environment. Eur. Phys. J. Plus 135(7), 1–23 (2020)

    Article  Google Scholar 

  22. M. Umar et al., Neuro-swarm intelligent computing paradigm for nonlinear HIV infection model with CD4+ T-cells. Math. Comput. Simul. 188, 241–253 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Umar et al., Stochastic numerical technique for solving HIV infection model of CD4+ T cells. Eur. Phys. J. Plus 135(5), 403 (2020)

    Article  MathSciNet  Google Scholar 

  24. Y. Guerrero-Sánchez et al., Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discret. Contin. Dyn. Syst. -S (2020). https://doi.org/10.3934/dcdss.2020431

    Article  MATH  Google Scholar 

  25. M. Umar et al., A novel study of Morlet neural networks to solve the nonlinear HIV infection system of latently infected cells. Results Phys. 25, 104235 (2021)

    Article  Google Scholar 

  26. Z. Sabir, M. Umar, M.A.Z. Raja, I. Fathurrochman, H. Hasan, Design of Morlet wavelet neural network to solve the non-linear influenza disease system. Appl. Math. Nonlinear Sci. (2022). https://doi.org/10.2478/amns.2021.2.00120

    Article  Google Scholar 

  27. Z. Sabir, T. Botmart, M.A.Z. Raja, R. Sadat, M.R. Ali, A.A. Alsulami, A. Alghamdi, Artificial neural network scheme to solve the nonlinear influenza disease model. Biomed. Signal Process. Control 75, 103594 (2022)

    Article  Google Scholar 

  28. M.A.Z. Raja et al., Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput. Appl. 31(3), 793–812 (2019)

    Article  MathSciNet  Google Scholar 

  29. Z. Sabir et al., FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane-Emden system. Comput. Appl. Math. 39(4), 1–18 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. T. Saeed et al., An advanced heuristic approach for a nonlinear mathematical based medical smoking model. Results Phys. 32, 105137 (2022)

    Article  Google Scholar 

  31. Z. Sabir, M.A.Z. Raja, A.S. Alnahdi, M.B. Jeelani, M.A. Abdelkawy, Numerical investigations of the nonlinear smoke model using the Gudermannian neural networks. Math. Biosci. Eng. 19(1), 351–370 (2022)

    Article  MATH  Google Scholar 

  32. P. Roul et al., A new numerical approach for solving a class of singular two-point boundary value problems. Numer. Algorithms 75(3), 531–552 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Z. Sabir, H.A. Wahab, M.R. Ali, R. Sadat, Neuron analysis of the two-point singular boundary value problems arising in the thermal explosion’s theory. Neural Process. Lett. (2022). https://doi.org/10.1007/s11063-022-10809-6

    Article  Google Scholar 

  34. R.K. Pandey et al., Existence-uniqueness results for a class of singular boundary value problems arising in physiology. Nonlinear Anal. Real World Appl. 9(1), 40–52 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. H. Çağlar et al., B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos, Solitons Fractals 39(3), 1232–1237 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. A.R. Kanth, Cubic spline polynomial for non-linear singular two-point boundary value problems. Appl. Math. Comput. 189(2), 2017–2022 (2007)

    MathSciNet  MATH  Google Scholar 

  37. S.A. Khuri et al., A novel approach for the solution of a class of singular boundary value problems arising in physiology. Math. Comput. Model. 52(3–4), 626–636 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. R.K. Pandey et al., On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology. J. Comput. Appl. Math. 166(2), 553–564 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. A.R. Kanth et al., He’s variational iteration method for treating nonlinear singular boundary value problems. Comput. Math. Appl. 60(3), 821–829 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Li, W. Yaoyang, S. Fong, A.J. Tallón-Ballesteros, X.S. Yang, S. Mohammed, W. Feng, A binary PSO-based ensemble under-sampling model for rebalancing imbalanced training data. J. Supercomput. 78(5), 7428–7463 (2021). https://doi.org/10.1007/s11227-021-04177-6

    Article  Google Scholar 

  41. Y.S. Saito, Optimal distribution of the relaxation behavior of linear viscoelastic materials by the particle swarm optimization method applied to the problem of a twisting shaft. 2018

  42. M. Sharif, J. Amin, M. Raza, M. Yasmin, S.C. Satapathy, An integrated design of particle swarm optimization (PSO) with fusion of features for detection of brain tumor. Pattern Recogn. Lett. 129, 150–157 (2020)

    Article  ADS  Google Scholar 

  43. X. He, X. Fu, Y. Yang, Energy-efficient trajectory planning algorithm based on multi-objective PSO for the mobile sink in wireless sensor networks. IEEE Access 7, 176204–176217 (2019)

    Article  Google Scholar 

  44. S. Shahbeig, M.S. Helfroush, A. Rahideh, A fuzzy multi-objective hybrid TLBO–PSO approach to select the associated genes with breast cancer. Signal Process. 131, 58–65 (2017)

    Article  Google Scholar 

  45. M. Zavvar, S. Garavand, E. Sabbagh, M. Rezaei, H. Khalili, M.H. Zavvar, H. Motameni, Measuring service quality in service-oriented architectures using a hybrid particle swarm optimization algorithm and artificial neural network (PSO-ANN). In 2017 3th International Conference on Web Research (ICWR) (pp. 78–83). IEEE. 2017

  46. M. Maitra, A. Chatterjee, A hybrid cooperative–comprehensive learning based PSO algorithm for image segmentation using multilevel thresholding. Expert Syst. Appl. 34(2), 1341–1350 (2008)

    Article  Google Scholar 

  47. M.A. Abdelkareem, H. Rezk, E.T. Sayed, A. Alaswad, A.M. Nassef, A.G. Olabi, Data on fuzzy logic based-modelling and optimization of recovered lipid from microalgae. Data Brief (2020). https://doi.org/10.1016/j.dib.2019.104931

    Article  Google Scholar 

  48. C. Bertocchi, E. Chouzenoux, M.C. Corbineau, J.C. Pesquet, M. Prato, Deep unfolding of a proximal interior point method for image restoration. Inverse Probl. 36(3), 034005 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. J. Wambacq, J. Ulloa, G. Lombaert, S. François, Interior-point methods for the phase-field approach to brittle and ductile fracture. Comput. Methods Appl. Mech. Eng. 375, 113612 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. N.P. Theodorakatos, A nonlinear well-determined model for power system observability using Interior-Point methods. Measurement 152, 107305 (2020)

    Article  Google Scholar 

  51. J. Kardos, D. Kourounis, O. Schenk, R. Zimmerman, R., Complete results for a numerical evaluation of interior point solvers for large-scale optimal power flow problems. (2018) arXiv preprint https://arxiv.org/abs/1807.03964.

  52. J.Y. Lee, N.W. Kim, T.W. Kim, M. Jehanzaib, Feasible ranges of runoff curve numbers for Korean watersheds based on the interior point optimization algorithm. KSCE J. Civ. Eng. 23(12), 5257–5265 (2019)

    Article  Google Scholar 

  53. B.L. Gorissen, Interior point methods can exploit structure of convex piecewise linear functions with application in radiation therapy. SIAM J. Optim. 32(1), 256–275 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  54. A. Mhlanga, A theoretical model for the transmission dynamics of HIV/HSV-2 co-infection in the presence of poor HSV-2 treatment adherence. Appl. Math. Nonlinear Sci. 3(2), 603–626 (2018)

    Article  MathSciNet  Google Scholar 

  55. H.M. Baskonus et al., New complex hyperbolic structures to the lonngren-wave equation by using sine-gordon expansion method. Appl. Math. Nonlinear Sci. 4(1), 141–150 (2019)

    MathSciNet  Google Scholar 

  56. E. Ilhan et al., A generalization of truncated M-fractional derivative and applications to fractional differential equations. Appl. Math. Nonlinear Sci. 5(1), 171–188 (2020)

    Article  MathSciNet  Google Scholar 

  57. T. Sajid et al., Impact of oxytactic microorganisms and variable species diffusivity on blood-gold Reiner-Philippoff nanofluid. Appl. Nanosci. (2020). https://doi.org/10.1007/s13204-020-01581-x

    Article  Google Scholar 

  58. K. Vajravelu et al., Influence of velocity slip and temperature jump conditions on the peristaltic flow of a Jeffrey fluid in contact with a Newtonian fluid. Appl. Math. Nonlinear Sci. 2(2), 429–442 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  59. T. Sajid et al., Impact of activation energy and temperature-dependent heat source/sink on maxwell–sutterby fluid. Math. Probl. Eng. (2020). https://doi.org/10.1155/2020/5251804

    Article  MathSciNet  MATH  Google Scholar 

  60. M.S.M. Selvi et al., Application of modified wavelet and homotopy perturbation methods to nonlinear oscillation problems. Appl. Math. Nonlinear Sci. 4(2), 351–364 (2019)

    Article  MathSciNet  Google Scholar 

  61. M. Umar et al., The 3-D flow of Casson nanofluid over a stretched sheet with chemical reactions, velocity slip, thermal radiation and Brownian motion. Thermal Sci. 24(5), 2929–2939 (2020)

    Article  MathSciNet  Google Scholar 

  62. M.K. Ammar et al., Visibility intervals between two artificial satellites under the action of Earth oblateness. Appl. Math. Nonlinear Sci. 3(2), 353–374 (2018)

    Article  MathSciNet  Google Scholar 

  63. M.K. Ammar et al., Calculation of line of site periods between two artificial satellites under the action air drag. Appl. Math. Nonlinear Sci. 3(2), 339–352 (2018)

    Article  MathSciNet  Google Scholar 

  64. H. Duru et al., New travelling wave solutions for KdV6 equation using sub equation method. Appl. Math. Nonlinear Sci. 5(1), 455–460 (2020)

    Article  MathSciNet  Google Scholar 

  65. T.A. Sulaiman et al., On the exact solutions to some system of complex nonlinear models. Appl. Math. Nonlinear Sci. 6(1), 29–42 (2021)

    Article  MathSciNet  Google Scholar 

  66. W. Zhao, T. Sh, L. Wang, Fault diagnosis and prognosis of bearing based on hidden Markov model with multi-features. Appl. Math. Nonlinear Sci. 5(1), 71–84 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, United Arab Emirates University, P. O. Box 15551, Al Ain, UAE

    Zulqurnain Sabir

  2. Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan

    Zulqurnain Sabir

Authors
  1. Zulqurnain Sabir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zulqurnain Sabir.

Ethics declarations

Conflict of interest

The authors state that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sabir, Z. Neuron analysis through the swarming procedures for the singular two-point boundary value problems arising in the theory of thermal explosion. Eur. Phys. J. Plus 137, 638 (2022). https://doi.org/10.1140/epjp/s13360-022-02869-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-022-02869-3

Search

Navigation