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Neuro-swarm intelligent computing to solve the second-order singular functional differential model

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Abstract

The aim of the present study is to solve the singular second-order functional differential model with the development of neuro-swarm intelligent computing solver ANN–PSO–SQP based on mathematical modeling of artificial neural networks (ANNs) optimized globally search efficacy of particle swarm optimization (PSO) aided with local search efficiency of sequential quadratic programming (SQP). In the scheme ANN–PSO–SQP, an error-based objective function is assembled with the help of continuous mapping of ANN for second-order singular functional differential model and optimized with combination strength of PSO with SQP. The inspiration for the design of ANN–PSO–SQP comes with an objective to present a precise, reliable and feasible frameworks to handle with stiff singular functional models involving the delayed, pantograph and prediction terms. The designed scheme is tested for three different variants of the singular second-order functional differential models. The obtained outcomes on both single as well as multiple runs of the proposed ANN–PSO–SQP are compared with the exact solutions to validate the efficacy, correctness and viability.

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This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There are no external data associated with the manuscript].

References

  1. X. Liu, G. Ballinger, Boundedness for impulsive delay differential equations and applications to population growth models. Nonlinear Anal. Theory Methods Appl. 53(7–8), 1041–1062 (2003)

    MathSciNet  Google Scholar 

  2. M. Dehghan, F. Shakeri, The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics. Phys. Scr. 78(6), 065004 (2008)

    ADS  Google Scholar 

  3. P.W. Nelson, A.S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 179(1), 73–94 (2002)

    MathSciNet  Google Scholar 

  4. M. Villasana, A. Radunskaya, A delay differential equation model for tumor growth. J. Math. Biol. 47(3), 270–294 (2003)

    MathSciNet  Google Scholar 

  5. M.R. Roussel, The use of delay differential equations in chemical kinetics. J. Phys. Chem. 100(20), 8323–8330 (1996)

    Google Scholar 

  6. S.A. Gourley, Y. Kuang, J.D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection. J. Biol. Dyn. 2(2), 140–153 (2008)

    MathSciNet  Google Scholar 

  7. D. Bratsun, D. Volfson, L.S. Tsimring, J. Hasty, Delay-induced stochastic oscillations in gene regulation. Proc. Natl. Acad. Sci. 102(41), 14593–14598 (2005)

    ADS  Google Scholar 

  8. G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model of viral infections. SIAM J. Appl. Math. 70(7), 2693–2708 (2010)

    MathSciNet  Google Scholar 

  9. P.L. Chambre, On the solution of the Poisson–Boltzmann equation with application to the theory of thermal explosions. J. Chem. Phys. 20(11), 1795–1797 (1952)

    ADS  Google Scholar 

  10. A.M. Wazwaz, A new method for solving singular initial value problems in the second-order ordinary differential equations. Appl. Math. Comput. 128(1), 45–57 (2002)

    MathSciNet  Google Scholar 

  11. K. Boubaker, R.A. Van Gorder, Application of the BPES to Lane–Emden equations governing polytropic and isothermal gas spheres. New Astron. 17(6), 565–569 (2012)

    ADS  Google Scholar 

  12. M. Dehghan, F. Shakeri, Solution of an integro-differential equation arising in oscillating magnetic fields using He's homotopy perturbation method. Prog. Electromagn. Res. 78, 361–376 (2008)

    Google Scholar 

  13. O. Abu Arqub, A. El-Ajou, A.S. Bataineh, I. Hashim, A representation of the exact solution of generalized Lane–Emden equations using a new analytical method, in Abstract and Applied Analysis, vol. 2013. Hindawi (2013)

  14. F. Mirzaee, S.F. Hoseini, Solving singularly perturbed differential-difference equations arising in science and engineering with Fibonacci polynomials. Results Phys. 3, 134–141 (2013)

    ADS  Google Scholar 

  15. M.K. Kadalbajoo, K.K. Sharma, Numerical analysis of boundary-value problems for singularly perturbed differential-difference equations with small shifts of mixed type. J. Optim. Theory Appl. 115(1), 145–163 (2002)

    MathSciNet  Google Scholar 

  16. M.K. Kadalbajoo, K.K. Sharma, Numerical treatment of a mathematical model arising from a model of neuronal variability. J. Math. Anal. Appl. 307(2), 606–627 (2005)

    MathSciNet  Google Scholar 

  17. H. Xu, Y. Jin, The asymptotic solutions for a class of nonlinear singular perturbed differential systems with time delays. Sci. World J. (2014). https://doi.org/10.1155/2014/965376

  18. F.Z. Geng, S.P. Qian, M.G. Cui, Improved reproducing kernel method for singularly perturbed differential-difference equations with boundary layer behavior. Appl. Math. Comput. 252, 58–63 (2015)

    MathSciNet  Google Scholar 

  19. Z. Masood et al., Design of Mexican Hat Wavelet neural networks for solving Bratu type nonlinear systems. Neurocomputing 221, 1–14 (2017)

    Google Scholar 

  20. M.A.Z. Raja, Numerical treatment for boundary value problems of pantograph functional differential equation using computational intelligence algorithms. Appl. Soft Comput. 24, 806–821 (2014)

    Google Scholar 

  21. J. Berg, K. Nyström, A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317, 28–41 (2018)

    Google Scholar 

  22. M.A.Z. Raja, M.A. Manzar, S.M. Shah, Y. Chen, Integrated intelligence of fractional neural networks and sequential quadratic programming for Bagley–Torvik systems arising in fluid mechanics. J. Comput. Nonlinear Dyn. 15(5), 051003-1–051003-12 (2020)

  23. A.H. Bukhari et al., Neuro-fuzzy modeling and prediction of summer precipitation with application to different meteorological stations. Alex. Eng. J. 59(1), 101–116 (2020)

    Google Scholar 

  24. Y. Sagna, Multidimensional BSDE with Poisson jumps of Osgood type. Appl. Math. Nonlinear Sci. 4(2), 387–394 (2019)

    MathSciNet  Google Scholar 

  25. 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(12), 9041–9059 (2019)

    Google Scholar 

  26. A. Mehmood, A Zameer, S.H. Ling et al., Integrated computational intelligent paradigm for nonlinear electric circuit models using neural networks, genetic algorithms and sequential quadratic programming. Neural Comput. Appl (2019). https://doi.org/10.1007/s00521-019-04573-3

  27. W. Waseem et al., A study of changes in temperature profile of porous fin model using cuckoo search algorithm. Alex. Eng. J. 59(1), 11–24 (2020)

    MathSciNet  Google Scholar 

  28. I. Ahmad et al., Design of computational intelligent procedure for thermal analysis of porous fin model. Chin. J. Phys. 59, 641–655 (2019)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  30. M.F. Fateh et al., Differential evolution based computation intelligence solver for elliptic partial differential equations. Front. Inf. Technol. Electron. Eng. 20(10), 1445–1456 (2019)

    Google Scholar 

  31. M.A.Z. Raja, A. Mehmood, A.A.Khan et al. Integrated intelligent computing for heat transfer and thermal radiation-based two-phase MHD nanofluid flow model. Neural Comput. Appl. 32, 2845–2877 (2020). https://doi.org/10.1007/s00521-019-04157-1

  32. H. Chen, J. Jiang, D. Cao, X. Fan, Numerical investigation on global dynamics for nonlinear stochastic heat conduction via global random attractors theory. Appl. Math. Nonlinear Sci. 3(1), 175–186 (2018)

    MathSciNet  Google Scholar 

  33. M.A.Z. Raja, J. Mehmood, Z. Sabir, A.K. Nasab, M.A. Manzar, Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput. Appl. 31(3), 793–812 (2019)

    Google Scholar 

  34. A. Mehmood et al., Integrated intelligent computing paradigm for the dynamics of micropolar fluid flow with heat transfer in a permeable walled channel. Appl. Soft Comput. 79, 139–162 (2019)

    Google Scholar 

  35. A. Hassan et al., Design of cascade artificial neural networks optimized with the memetic computing paradigm for solving the nonlinear Bratu system. Eur. Phys. J. Plus 134(3), 122 (2019)

    ADS  Google Scholar 

  36. Z. Sabir et al., Neuro-heuristics for nonlinear singular Thomas-Fermi systems. Appl. Soft Comput. 65, 152–169 (2018)

    Google Scholar 

  37. Z. Sabir et al., Novel design of Morlet wavelet neural network for solving second order Lane–Emden equation. Math. Comput. Simul. 172, 1–14 (2020)

    MathSciNet  Google Scholar 

  38. M.A.Z. Raja, F.H. Shah, E.S. Alaidarous, M.I. Syam, Design of bio-inspired heuristic technique integrated with interior-point algorithm to analyze the dynamics of heartbeat model. Appl. Soft Comput. 52, 605–629 (2017)

    Google Scholar 

  39. M.A.Z. Raja, M. Umar, Z. Sabir, J.A. Khan, D. Baleanu, A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head. Eur. Phys. J. Plus 133(9), 364 (2018)

    Google Scholar 

  40. M. Umar et al., Unsupervised constrained neural network modeling of boundary value corneal model for eye surgery. Appl. Soft Comput. 85, 105826 (2019)

    Google Scholar 

  41. Z. Sabir, H.A. Wahab, M. Umar, F. Erdoğan, Stochastic numerical approach for solving second order nonlinear singular functional differential equation. Appl. Math. Comput. 363, 124605 (2019)

    MathSciNet  Google Scholar 

  42. S. Lodhi et al., Fractional neural network models for nonlinear Riccati systems. Neural Comput. Appl. 31(1), 359–378 (2019)

    Google Scholar 

  43. M.A.Z. Raja, R. Samar, M.A. Manzar, S.M. Shah, Design of unsupervised fractional neural network model optimized with interior point algorithm for solving Bagley–Torvik equation. Math. Comput. Simul. 132, 139–158 (2017)

    MathSciNet  Google Scholar 

  44. Y. Shi, R.C. Eberhart, Empirical study of particle swarm optimization, in Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406), vol. 3, pp. 1945–1950. IEEE (1999).

  45. A.P. Engelbrecht, Computational Intelligence: An Introduction (Wiley, Hoboken, 2007)

    Google Scholar 

  46. D. Wang, D. Tan, L. Liu, Particle swarm optimization algorithm: an overview. Soft. Comput. 22(2), 387–408 (2018)

    Google Scholar 

  47. A. Mehmood et al., Nature-inspired heuristic paradigms for parameter estimation of control autoregressive moving average systems. Neural Comput. Appl. 31(10), 5819–5842 (2019)

    Google Scholar 

  48. A. Mehmood, A. Zameer, M.S. Aslam et al. Design of nature-inspired heuristic paradigm for systems in nonlinear electrical circuits. Neural Comput. Appl. (2019). https://doi.org/10.1007/s00521-019-04197-7

  49. J. Sánchez-García, D.G. Reina, S.L. Toral, A distributed PSO-based exploration algorithm for a UAV network assisting a disaster scenario. Future Gener. Comput. Syst. 90, 129–148 (2019)

    Google Scholar 

  50. H. Mesloub, M.T. Benchouia, R. Boumaaraf, A. Goléa, N. Goléa, M. Becherif, Design and implementation of DTC based on AFLC and PSO of a PMSM. Math. Comput. Simul. 167, 340–355 (2020)

    MathSciNet  Google Scholar 

  51. D. Bouhadjra, A. Kheldoun, A. Zemouche, Performance analysis of stand-alone six-phase induction generator using heuristic algorithms. Math. Comput. Simul. 167, 231–249 (2020)

    MathSciNet  Google Scholar 

  52. M.A.Z. Raja, A. Zameer, A.K. Kiani, A. Shehzad, M.A.R. Khan, Nature-inspired computational intelligence integration with Nelder-Mead method to solve nonlinear benchmark models. Neural Comput. Appl. 29(4), 1169–1193 (2018)

    Google Scholar 

  53. H. Ghomeshi, M.M. Gaber, Y. Kovalchuk, A non-canonical hybrid metaheuristic approach to adaptive data stream classification. Future Gener. Comput. Syst. 102, 127–139 (2020)

    Google Scholar 

  54. X. Xu, H. Rong, E. Pereira, M. Trovati, Predatory search-based chaos turbo particle swarm optimisation (PS-CTPSO): a new particle swarm optimisation algorithm for web service combination problems. Future Gener. Comput. Syst. 89, 375–386 (2018)

    Google Scholar 

  55. M.A.Z. Raja, U. Ahmed, A. Zameer, A.K. Kiani, N.I. Chaudhary, Bio-inspired heuristics hybrid with sequential quadratic programming and interior-point methods for reliable treatment of economic load dispatch problem. Neural Comput. Appl. 31(1), 447–475 (2019)

    Google Scholar 

  56. S. Akbar et al., Design of bio-inspired heuristic techniques hybridized with sequential quadratic programming for joint parameters estimation of electromagnetic plane waves. Wirel. Pers. Commun. 96(1), 1475–1494 (2017)

    Google Scholar 

  57. F.A. Chaudhry, M. Amin, M. Iqbal, R.D. Khan, J.A. Khan, A novel chaotic differential evolution hybridized with quadratic programming for short-term hydrothermal coordination. Neural Comput. Appl. 30(11), 3533–3544 (2018)

    Google Scholar 

  58. C.L. Xiao, H.X. Huang, Optimal design of heating system in rapid thermal cycling blow mold by a two-step method based on sequential quadratic programming. Int. Commun. Heat Mass Transf. 96, 114–121 (2018)

    Google Scholar 

  59. M.A.Z. Raja, F.H. Shah, M. Tariq, I. Ahmad, 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)

    Google Scholar 

  60. M.A.Z. Raja, A. Zameer, A.U. Khan, A.M. Wazwaz, A new numerical approach to solve Thomas–Fermi model of an atom using bio-inspired heuristics integrated with sequential quadratic programming. SpringerPlus 5(1), 1400 (2016)

    Google Scholar 

  61. I. Ahmad, F. Ahmad, M. Bilal, Neuro-heuristic computational intelligence for nonlinear Thomas–Fermi equation using trigonometric and hyperbolic approximation. Measurement 156, 107549 (2020)

    Google Scholar 

  62. I. Ahmad et al., Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model. Neural Comput. Appl. 28(1), 929–944 (2017)

    Google Scholar 

  63. I. Ahmad et al., Bio-inspired computational heuristics to study Lane–Emden systems arising in astrophysics model. SpringerPlus 5(1), 1866 (2016)

    Google Scholar 

  64. K. Majeed et al., A genetic algorithm optimized Morlet wavelet artificial neural network to study the dynamics of nonlinear Troesch’s system. Appl. Soft Comput. 56, 420–435 (2017)

    Google Scholar 

  65. M.A.Z. Raja, Stochastic numerical treatment for solving Troesch’s problem. Inf. Sci. 279, 860–873 (2014)

    MathSciNet  Google Scholar 

  66. M.A.Z. Raja, A. Mehmood, A.A. Khan, A. Zameer, Integrated intelligent computing for heat transfer and thermal radiation-based two-phase MHD nanofluid flow model. Neural Comput. Appl. 32, 2845–2877 (2020)

    Google Scholar 

  67. A. Mehmood et al., Design of neuro-computing paradigms for nonlinear nanofluidic systems of MHD Jeffery–Hamel flow. J. Taiwan Inst. Chem. Eng. 91, 57–85 (2018)

    Google Scholar 

  68. M.A.Z. Raja, A. Mehmood, A. ur Rehman, A. Khan, A. Zameer, Bio-inspired computational heuristics for Sisko fluid flow and heat transfer models. Appl. Soft Comput. 71, 622–648 (2018)

    Google Scholar 

  69. F. Evirgen et al., System analysis of HIV infection model with CD4+ T under non-singular kernel derivative. Appl. Math. Nonlinear Sci. 5(1), 139–146 (2020)

    MathSciNet  Google Scholar 

  70. M.A.Z. Raja, S.A. Niazi, S.A. Butt, An intelligent computing technique to analyze the vibrational dynamics of rotating electrical machine. Neurocomputing 219, 280–299 (2017)

    Google Scholar 

  71. 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)

    Google Scholar 

  72. Z. Sabir, H. Günerhan, J.L. Guirao, On a new model based on third-order nonlinear multisingular functional differential equations. Math. Probl. Eng. 2020, 9 (2020)

  73. M. Modanli, A. Akgül, On solutions of fractional order telegraph partial differential equation by Crank–Nicholson finite difference method. Appl. Math. Nonlinear Sci. 5(1), 163–170 (2020)

    MathSciNet  Google Scholar 

  74. M.A.Z. Raja, K. Asma, M.S. Aslam, Bio-inspired computational heuristics to study models of hiv infection of CD4+ T-cell. Int. J. Biomath. 11(02), 1850019 (2018)

    MathSciNet  Google Scholar 

  75. M.A.Z. Raja, F.H. Shah, M.I. Syam, Intelligent computing approach to solve the nonlinear Van der Pol system for heartbeat model. Neural Comput. Appl. 30(12), 3651–3675 (2018)

    Google Scholar 

  76. A. Ara et al., Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling. Adv. Differ. Equ. 2018(1), 8 (2018)

    MathSciNet  Google Scholar 

  77. A.H. Bukhari et al., Fractional neuro-sequential ARFIMA-LSTM for financial market forecasting. IEEE Access 8, 71326–71338 (2020)

    Google Scholar 

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Authors and Affiliations

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

    Zulqurnain Sabir & Muhammad Umar

  2. Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin, 64002, Taiwan, ROC

    Muhammad Asif Zahoor Raja

  3. Department of Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, Attock, 43600, Pakistan

    Muhammad Asif Zahoor Raja

  4. Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, 43600, Pakistan

    Muhammad Shoaib

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  1. Zulqurnain Sabir
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  3. Muhammad Umar
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Correspondence to Zulqurnain Sabir.

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Sabir, Z., Raja, M.A.Z., Umar, M. et al. Neuro-swarm intelligent computing to solve the second-order singular functional differential model. Eur. Phys. J. Plus 135, 474 (2020). https://doi.org/10.1140/epjp/s13360-020-00440-6

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