Evolutionary computing for nonlinear singular boundary value problems using neural network, genetic algorithm and active-set algorithm

Abstract

In this numerical study, a class of nonlinear singular boundary value problem is solved by implementation of a novel meta-heuristic computing tool based on the artificial neural networks (ANNs) modeling of system and the optimization of decision variable of ANNs through the combined strength of global search via genetic algorithms (GA) and local search ability of active-set algorithm (ASA), i.e., ANN–GA–ASA. The proposed intelligent computing solver ANN–GA–ASA exploits the input, hidden, and output layers’ structure of ANNs. This is to represent the differential model in the nonlinear singular second-order periodic boundary value problems, which are connected to form an error-based objective function (OF) and optimize the OF by the integrated heuristics of GA–ASA. The purpose to present this research is to associate the operational legacy of neural networks and to challenge such kinds of inspiring models. Two different examples of the singular periodic model have been investigated to observe the robustness, proficiency and stability of the ANN–GA–ASA. The proposed outcomes of ANN–GA–ASA are compared with reference to true results so as to establish the value of the designed scheme. Exhaustive comparison has been made and presented between the Log-sigmoidal ANNs results and the radial basis ANNs outcomes. The reliability of the results obtained is endorsed by using both types of networks as well as the value of designed schemes.

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References

  1. 1.

    R.K. Sabirov, Solutions of the Lane-emden and thomas-fermi equations. Russ. Phys. J. 45(2), 129–132 (2002)

    Article  Google Scholar 

  2. 2.

    S.H. Lam, D.A. Goussis, Understanding complex chemical kinetics with computational singular perturbation, in Symposium (International) on Combustion, vol. 22, no. 1 (Elsevier, 1989), pp. 931–941

  3. 3.

    M.O. Ogunniran, O.A. Tayo, Y. Haruna, A.F. Adebisi, Linear stability analysis of Runge–Kutta methods for singular lane-emden equations. J. Niger. Soc. Phys. Sci. 2, 134–140 (2020)

    Article  Google Scholar 

  4. 4.

    R.P. Agarwal et al., Degree theoretic methods in the study of nonlinear periodic problems with nonsmooth potentials. Differ. Integr. Equ. 19(3), 279–296 (2006)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    F. Geng et al., Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space. Appl. Math. Comput. 192(2), 389–398 (2007)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    D. Jiang et al., Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces. J. Math. Anal. Appl. 286(2), 563–576 (2003)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Z. Zhang et al., On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations. J. Math. Anal. Appl. 281(1), 99–107 (2003)

    MathSciNet  Article  Google Scholar 

  8. 8.

    F. Li et al., Existence of positive periodic solutions to nonlinear second order differential equations. Appl. Math. Lett. 18(11), 1256–1264 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9.

    V. Šeda et al., Periodic boundary value problems for nonlinear higher order ordinary differential equations. Appl. Math. Comput. 48(1), 71–82 (1992)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    J.L. Guirao, Z. Sabir, A.Z. Raja, M. Shoaib, A neuro-swarming intelligence based computing for second order singular periodic nonlinear boundary value problems. Front. Phys. 8, 224 (2020)

    Article  Google Scholar 

  11. 11.

    M. Sharma et al., A comprehensive analysis of nature-inspired meta-heuristic techniques for feature selection problem. Arch. Comput. Methods Eng. (2020). https://doi.org/10.1007/s11831-020-09412-6

    Article  Google Scholar 

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

    Z. Sabir et al., Intelligence computing approach for solving second order system of Emden–Fowler model. J. Intell. Fuzzy Syst. 38(6), 7391–7406 (2020)

    Article  Google Scholar 

  14. 14.

    V. Yepes et al., Heuristic techniques for the design of steel-concrete composite pedestrian bridges. Appl. Sci. 9(16), 3253 (2019)

    Article  Google Scholar 

  15. 15.

    M.A.Z. Raja et al., 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. 29(6), 83–109 (2018)

    Article  Google Scholar 

  16. 16.

    Z. Sabir et al., Design of neuro-swarming-based heuristics to solve the third-order nonlinear multi-singular Emden–Fowler equation. Eur. Phys. J. Plus 135(6), 1–17 (2020)

    Article  Google Scholar 

  17. 17.

    Z. Sabir et al., Neuro-swarm intelligent computing to solve the second-order singular functional differential model. Eur. Phys. J. Plus 135(6), 474 (2020)

    Article  Google Scholar 

  18. 18.

    M. Umar et al., Stochastic numerical technique for solving HIV infection model of CD4+ T cells. Eur. Phys. J. Plus 135(6), 403 (2020)

    Article  Google Scholar 

  19. 19.

    M. Umar et al., Intelligent computing for numerical treatment of nonlinear prey–predator models. Appl. Soft Comput. 80, 506–524 (2019)

    Article  Google Scholar 

  20. 20.

    M.A.Z. Raja et al., Design of hybrid nature-inspired heuristics with application to active noise control systems. Neural Comput. Appl. 31(7), 2563–2591 (2019)

    Article  Google Scholar 

  21. 21.

    I. Ahmad et al., Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels. Neural Comput. Appl. 31, 1–19 (2019)

    Google Scholar 

  22. 22.

    M.A.Z. Raja et al., Bio-inspired computational heuristics for parameter estimation of nonlinear Hammerstein controlled autoregressive system. Neural Comput. Appl. 29(12), 1455–1474 (2018)

    Article  Google Scholar 

  23. 23.

    M.A. Mohammed et al., A real time computer aided object detection of nasopharyngeal carcinoma using genetic algorithm and artificial neural network based on Haar feature fear. Future Gener. Comput. Syst. 89, 539–547 (2018)

    Article  Google Scholar 

  24. 24.

    H. Kim et al., Reliability–redundancy allocation problem considering optimal redundancy strategy using parallel genetic algorithm. Reliab. Eng. Syst. Saf. 159, 153–160 (2017)

    Article  Google Scholar 

  25. 25.

    I. Ahmad et al., Intelligent computing to solve fifth-order boundary value problem arising in induction motor models. Neural Comput. Appl. 29(7), 449–466 (2018)

    Article  Google Scholar 

  26. 26.

    M. Mönnigmann, On the structure of the set of active sets in constrained linear quadratic regulation. Automatica 106, 61–69 (2019)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Y. Li et al., An active-set algorithm for solving large-scale nonsmooth optimization models with box constraints. PLoS ONE 13(1), e0189290 (2018)

    Article  Google Scholar 

  28. 28.

    C. Buchheim et al., An active set algorithm for robust combinatorial optimization based on separation oracles. Math. Program. Comput. 11(4), 755–789 (2019)

    MathSciNet  Article  Google Scholar 

  29. 29.

    M. Klaučo et al., Machine learning-based warm starting of active set methods in embedded model predictive control. Eng. Appl. Artif. Intell. 77, 1–8 (2019)

    Article  Google Scholar 

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Correspondence to Zulqurnain Sabir.

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Sabir, Z., Khalique, C.M., Raja, M.A.Z. et al. Evolutionary computing for nonlinear singular boundary value problems using neural network, genetic algorithm and active-set algorithm. Eur. Phys. J. Plus 136, 195 (2021). https://doi.org/10.1140/epjp/s13360-021-01171-y

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