Skip to main content
SpringerLink
Log in

Design of Backpropagated Intelligent Networks for Nonlinear Second-Order Lane–Emden Pantograph Delay Differential Systems

  • Research Article-Computer Engineering and Computer Science
  • Published:
Arabian Journal for Science and Engineering Aims and scope Submit manuscript

Abstract

In physical science, nonlinear singular Lane–Emden and pantograph delay differential equations (LE–PDDEs) have abundant applications and thus are of great interest for the researchers. The presented investigation is related to the development of a new application of intelligent computing for the solution of the LE–PDDEs-based system introduced recently by merging the essence of delay differential equation of Pantograph type and standard second-order Lane–Emden equation. Intelligent computing is exploited through Levenberg–Marquardt backpropagation networks (LMBNs) and Bayesian regularization backpropagation networks (BRBNs) to provide the solutions to nonlinear second-order LE–PDDEs. The performance of design LMBNs and BRBNs is substantiated on three different case studies through comparative analysis from known exact/explicit solutions. The correctness of the designed solvers for LE–PDDEs is further certified by accomplishing through assessment on error histograms, regression measures and index of mean squared error.

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
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Rakhshan, S.A.; Effati, S.: A generalized Legendre–Gauss collocation method for solving nonlinear fractional differential equations with time varying delays. Appl. Numer. Math. 146, 342–360 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Kuang, Y. (ed.): Delay Differential Equations: with Applications in Population Dynamics. Academic Press, Cambridge (1993)

    MATH  Google Scholar 

  3. Li, W.; Chen, B.; Meng, C.; Fang, W.; Xiao, Y.; Li, X.; Hu, Z.; Xu, Y.; Tong, L.; Wang, H.; Liu, W.: Ultrafast all-optical graphene modulator. Nano Lett. 14(2), 955–959 (2014)

    Google Scholar 

  4. Saray, B.N.; Lakestani, M.: On the sparse multi-scale solution of the delay differential equations by an efficient algorithm. Appl. Math. Comput. 381, 125291s (2020)

    MathSciNet  MATH  Google Scholar 

  5. Narasingam, A.; Kwon, J.S.I.: Application of Koopman operator for model-based control of fracture propagation and proppant transport in hydraulic fracturing operation. J. Process Control 91, 25–36 (2020)

    Google Scholar 

  6. Beretta, E.; Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33(5), 1144–1165 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Forde, J.E.: Delay Differential Equation Models in Mathematical Biology. University of Michigan, Ann Arbor, pp. 5436–5436 (2005)

    Google Scholar 

  8. Chapra, S.C.: Applied Numerical Methods. McGraw-Hill, Columbus (2012)

    Google Scholar 

  9. Rangkuti, Y.M.; Noorani, M.S.M.: The exact solution of delay differential equations using coupling variational iteration with Taylor series and small term. Bull. Math. 4(01), 1–15 (2012)

    Google Scholar 

  10. Frazier, M.W.: Background: complex numbers and linear algebra. In: An Introduction to Wavelets Through Linear Algebra. Undergraduate Texts in Mathematics, pp. 7–100. Springer, New York (1999). https://doi.org/10.1007/0-387-22653-2_2

  11. Getto, P.; Waurick, M.: A differential equation with state-dependent delay from cell population biology. J. Differ. Equ. 260(7), 6176–6200 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Wazwaz, A.M., et al.: Reliable treatment for solving boundary value problems of pantograph delay differential equation. Rom. Rep. Phys 69, 102 (2017)

    Google Scholar 

  13. Isah, A.; Phang, C.; Phang, P.: Collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations. Int. J. Difer. Equ. 2017, 2097317 (2017). https://doi.org/10.1155/2017/2097317

    Article  MathSciNet  MATH  Google Scholar 

  14. Saray, B.N.; Manafian, J.: Sparse representation of delay differential equation of Pantograph type using multi-wavelets Galerkin method. Eng. Comput. 35(2), 887–903 (2018)

    Google Scholar 

  15. YÜZBAŞI, Ş; IsmailovSaray, N.: A Taylor operation method for solutions of generalized pantograph type delay differential equations. Turk. J. Math. 42(2), 395–406 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Yang, C.: Modified Chebyshev collocation method for pantograph-type differential equations. Appl. Numer. Math. 134, 132–144 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Katani, R.: Multistep block method for linear and nonlinear pantograph type delay differential equations with neutral term. Int. J. Appl. Comput. Math. 3(1), 1347–1359 (2017)

    MathSciNet  Google Scholar 

  18. Wang, W.: Fully-geometric mesh one-leg methods for the generalized pantograph equation: approximating Lyapunov functional and asymptotic contractivity. Appl. Numer. Math. 117, 50–68 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Zhan, W.; Gao, Y.; Guo, Q.; Yao, X.: The partially truncated Euler–Maruyama method for nonlinear pantograph stochastic differential equations. Appl. Math. Comput. 346, 109–126 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Koroma, M.A.; Zhan, C.; Kamara, A.F.; Sesay, A.B.: Laplace decomposition approximation solution for a system of multi-pantograph equations. Int. J. Math. Comput. Sci. Eng. 7(7), 39–44 (2013)

    Google Scholar 

  21. Eriqat, T.; El-Ajou, A.; Moa’ath, N.O.; Al-Zhour, Z.; Momani, S.: A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations. Chaos Solitons Fractals 138, 109957 (2020)

    MathSciNet  Google Scholar 

  22. Ezz-Eldien, S.S.; Wang, Y.; Abdelkawy, M.A.; Zaky, M.A.; Aldraiweesh, A.A.; Machado, J.T.: Chebyshev spectral methods for multi-order fractional neutral pantograph equations. Nonlinear Dyn. 100, 1–13 (2020)

    Google Scholar 

  23. Alsuyuti, M.M.; Doha, E.H.; Ezz-Eldien, S.S.; Youssef, I.K.: Spectral Galerkin schemes for a class of multi-order fractional pantograph equations. J. Comput. Appl. Math. 384, 113157 (2020)

    MathSciNet  MATH  Google Scholar 

  24. Wang, L.P.; Chen, Y.M.; Liu, D.Y.; Boutat, D.: Numerical algorithm to solve generalized fractional pantograph equations with variable coefficients based on shifted Chebyshev polynomials. Int. J. Comput. Math. 96(12), 2487–2510 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Dehestani, H.; Ordokhani, Y.; Razzaghi, M.: Numerical technique for solving fractional generalized pantograph-delay differential equations by using fractional-order hybrid bessel functions. Int. J. Appl. Comput. Math. 6(1), 1–27 (2020)

    MathSciNet  MATH  Google Scholar 

  26. Hashemi, M.S.; Atangana, A.; Hajikhah, S.: Solving fractional pantograph delay equations by an effective computational method. Math. Comput. Simul. 177, 295–305 (2020)

    MathSciNet  MATH  Google Scholar 

  27. Rabiei, K.; Ordokhani, Y.: Solving fractional pantograph delay differential equations via fractional-order Boubaker polynomials. Eng. Comput. 35(4), 1431–1441 (2019)

    Google Scholar 

  28. Raja, M.A.Z.; Ahmad, I.; Khan, I.; Syam, M.I.; Wazwaz, A.M.: Neuro-heuristic computational intelligence for solving nonlinear pantograph systems. Front. Inf. Technol. Electron. Eng. 18(4), 464–484 (2017)

    Google Scholar 

  29. Sun, H.; Hou, M.; Yang, Y.; Zhang, T.; Weng, F.; Han, F.: Solving partial differential equation based on Bernstein neural network and extreme learning machine algorithm. Neural Process. Lett. 50(2), 1153–1172 (2019)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  32. Khan, I., et al.: Design of neural network with Levenberg–Marquardt and Bayesian regularization backpropagation for solving pantograph delay differential equations. IEEE Access 8, 137918–137933 (2020)

    Google Scholar 

  33. Mosavi, A.; Shokri, M.; Mansor, Z.; Qasem, S.N.; Band, S.S.; Mohammadzadeh, A.: Machine learning for modeling the singular multi-pantograph equations. Entropy 22(9), 1041 (2020)

    MathSciNet  Google Scholar 

  34. Mandelzweig, V.B.; Tabakin, F.: Quasi linearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput. Phys. Commun. 141(2), 268–281 (2001)

    MATH  Google Scholar 

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

    Google Scholar 

  36. Khan, J.A., et al.: Nature-inspired computing approach for solving non-linear singular Emden–Fowler problem arising in electromagnetic theory. Connect. Sci. 27(4), 377–396 (2015)

    Google Scholar 

  37. Bhrawy, A.H.; Aloi, A.S.; Van Gorder, R.A.: An efficient collocation method for a class of boundary value problems arising in mathematical physics and geometry. Abstr. Appl. Anal. 2014, 425648 (2014). https://doi.org/10.1155/2014/425648

    Article  MathSciNet  MATH  Google Scholar 

  38. Luo, T.; Xin, Z.; Zeng, H.: Nonlinear asymptotic stability of the Lane–Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities. Commun. Math. Phys. 347(3), 657–702 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Rach, R.; Duan, J.S.; Wazwaz, A.M.: Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 52(1), 255–267 (2014)

    MathSciNet  MATH  Google Scholar 

  40. Taghavi, A.; Pearce, S.: A solution to the Lane–Emden equation in the theory of stellar structure utilizing the Tau method. Math. Methods Appl. Sci. 36(10), 1240–1247 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Ramos, J.I.: Linearization methods in classical and quantum mechanics. Comput. Phys. Commun. 153(2), 199–208 (2003)

    MathSciNet  MATH  Google Scholar 

  42. Rădulescu, V.; Repovš, D.: Combined effects in nonlinear problems arising in the study of anisotropic continuous media. Nonlinear Anal. Theory Methods Appl. 75(3), 1524–1530 (2012)

    MathSciNet  MATH  Google Scholar 

  43. Ahmad, I., et al.: Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically. Neural Comput. Appl. (2020). https://doi.org/10.1007/s00521-020-05355-y

    Article  Google Scholar 

  44. Umar, M., 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)

    Google Scholar 

  45. Mehmood, A., et al.: Integrated computational intelligent paradigm for nonlinear electric circuit models using neural networks, genetic algorithms and sequential quadratic programming. Neural Comput. Appl. 32(14), 10337–10357 (2020)

    Google Scholar 

  46. Bukhari, A.H., et al.: Design of a hybrid NAR-RBFs neural network for nonlinear dusty plasma system. Alex. Eng. J. 59, 3325–3345 (2020)

    Google Scholar 

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

    Google Scholar 

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

  49. Ahmad, I., 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 

  50. Siraj-ul-Islam, A., et al.: A new heuristic computational solver for nonlinear singular Thomas–Fermi system using evolutionary optimized cubic splines. Eur. Phys. J. Plus 135(1), 1–29 (2020)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  52. Raja, M.A.Z.; Zameer, A.; Khan, A.U.; Wazwaz, A.M.: 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 

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

    MathSciNet  MATH  Google Scholar 

  54. Sabir, Z., 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)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  57. Raja, M.A.Z.; Umar, M.; Sabir, Z.; Khan, J.A.; Baleanu, D.: 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 

  58. Adel, W.; Sabir, Z.: Solving a new design of nonlinear second-order Lane–Emden pantograph delay differential model via Bernoulli collocation method. Eur. Phys. J. Plus 135(5), 427 (2020)

    Google Scholar 

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

    Google Scholar 

  60. Raja, M.A.Z.; Khan, J.A.; Chaudhary, N.I.; Shivanian, E.: Reliable numerical treatment of nonlinear singular Flierl–Petviashvili equations for unbounded domain using ANN, GAs, and SQP. Appl. Soft Comput. 38, 617–636 (2016)

    Google Scholar 

  61. Raja, M.A.Z.; Samar, R.; Manzar, M.A.; Shah, S.M.: 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  MATH  Google Scholar 

  62. Raja, M.A.Z.; Khan, J.A.; Haroon, T.: Stochastic numerical treatment for thin film flow of third grade fluid using unsupervised neural networks. J. Taiwan Inst. Chem. Eng. 48, 26–39 (2015)

    Google Scholar 

  63. Shamshirband, S.; Fathi, M.; Dehzangi, A.; Chronopoulos, A.T.; Alinejad-Rokny, H.: A review on deep learning approaches in healthcare systems: taxonomies, challenges, and open issues. J. Biomed. Inf. 113, 103627 (2020)

    Google Scholar 

  64. Shamshirband, S.; Rabczuk, T.; Chau, K.W.: A survey of deep learning techniques: application in wind and solar energy resources. IEEE Access 7, 164650–164666 (2019)

    Google Scholar 

  65. Rajaei, P.; Jahanian, K.H.; Beheshti, A.; Band, S.S.; Dehzangi, A.; Alinejad-Rokny, H.: VIRMOTIF: a user-friendly tool for viral sequence analysis. Genes 12(2), 186 (2021)

    Google Scholar 

  66. Ilyas, H., et al.: A novel design of Gaussian WaveNets for rotational hybrid nanofluidic flow over a stretching sheet involving thermal radiation. Int. Commun. Heat Mass Transf. 123, 105196 (2021)

    Google Scholar 

  67. Shoaib, M., et al.: A stochastic numerical analysis based on hybrid NAR-RBFs networks nonlinear SITR model for novel COVID-19 dynamics. Comput. Methods Programs Biomed. 202, 105973 (2021)

    Google Scholar 

  68. Chen, Y.; Yu, H.; Meng, X.; Xie, X.; Hou, M.; Chevallier, J.: Numerical solving of the generalized Black-Scholes differential equation using Laguerre neural network. Digit. Signal Process. 112, 103003 (2021)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, KP, Pakistan

    Imtiaz Khan & Saeed Islam

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

    Muhammad Asif Zahoor Raja

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

    Muhammad Abdul Rehman Khan

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

    Muhammad Shoaib

  5. Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand

    Zahir Shah

Authors
  1. Imtiaz Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad Asif Zahoor Raja
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muhammad Abdul Rehman Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Muhammad Shoaib
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Saeed Islam
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Zahir Shah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Asif Zahoor Raja.

Ethics declarations

Conflict of Interest

All the authors of the manuscript declared that there is 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

Khan, I., Raja, M.A.Z., Khan, M.A.R. et al. Design of Backpropagated Intelligent Networks for Nonlinear Second-Order Lane–Emden Pantograph Delay Differential Systems. Arab J Sci Eng 47, 1197–1210 (2022). https://doi.org/10.1007/s13369-021-05814-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13369-021-05814-1

Keywords

Search

Navigation