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Computational intelligence approach using Levenberg–Marquardt backpropagation neural networks to solve the fourth-order nonlinear system of Emden–Fowler model

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A Correction to this article was published on 11 April 2022

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Abstract

The present investigations are related to design an integrated computing numerical approach through Levenberg–Marquardt backpropagation (LMB) neural networks (NNs), i.e., LMB-NNs. The designed LMB-NNs approach is presented to solve the fourth-order nonlinear system of Emden–Fowler model (FO-SEFM). The solution of six different examples based on the FO-SEFM using the designed methodology LMB-NNs is numerically treated along with the discussion of singular point and shape factor. The comparison of the obtained results from the LMB-NNs and the exact solutions of each example has been presented. To evaluate the approximate results of the FO-SEFM for different problems, the testing, training, and authentication procedures are accompanied to adapt the NNs by reducing the functions of mean square error (MSE) through the LMB. The proportional investigations and performance studies based on the results of error histograms, MSE, regression, and correlation establish the effectiveness and correctness of the designed LMB-NNs approach.

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Author information

Authors and Affiliations

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

    Zulqurnain Sabir

  2. Faculty of Engineering and Technology, Future University, Cairo, Egypt

    Mohamed R. Ali

  3. Department of Basic Science, Faculty of Engineering at Benha, Benha University, Benha, 13512, Egypt

    Mohamed R. Ali

  4. 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

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

    Muhammad Asif Zahoor Raja

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

    Muhammad Shoaib

  7. Department of General Studies, National Autonomous University of Chota, Chota, Peru

    Rafaél Artidoro Sandoval Núñez

  8. Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, Zagazig, Egypt

    R. Sadat

Authors
  1. Zulqurnain Sabir
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  2. Mohamed R. Ali
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  3. Muhammad Asif Zahoor Raja
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  4. Muhammad Shoaib
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  5. Rafaél Artidoro Sandoval Núñez
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  6. R. Sadat
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Correspondence to R. Sadat.

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Sabir, Z., Ali, M.R., Raja, M.A.Z. et al. Computational intelligence approach using Levenberg–Marquardt backpropagation neural networks to solve the fourth-order nonlinear system of Emden–Fowler model. Engineering with Computers 38 (Suppl 4), 2975–2991 (2022). https://doi.org/10.1007/s00366-021-01427-2

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  • DOI: https://doi.org/10.1007/s00366-021-01427-2

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