Numerical investigations of the nonlinear smoke model using the Gudermannian neural networks

Zulqurnain Sabir; Muhammad Asif Zahoor Raja; Abeer S. Alnahdi; Mdi Begum Jeelani; M. A. Abdelkawy; Zulqurnain Sabir; Muhammad Asif Zahoor Raja; Abeer S. Alnahdi; Mdi Begum Jeelani; M. A. Abdelkawy

doi:10.3934/mbe.2022018

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 1: 351-370. doi: 10.3934/mbe.2022018

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Research article

Numerical investigations of the nonlinear smoke model using the Gudermannian neural networks

1.
Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
2.
Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan, R.O.C
3.
Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Academic Editor: Zhi-Cheng Wang

Received: 25 August 2021 Accepted: 17 October 2021 Published: 15 November 2021

These investigations are to find the numerical solutions of the nonlinear smoke model to exploit a stochastic framework called gudermannian neural works (GNNs) along with the optimization procedures of global/local search terminologies based genetic algorithm (GA) and interior-point algorithm (IPA), i.e., GNNs-GA-IPA. The nonlinear smoke system depends upon four groups, temporary smokers, potential smokers, permanent smokers and smokers. In order to solve the model, the design of fitness function is presented based on the differential system and the initial conditions of the nonlinear smoke system. To check the correctness of the GNNs-GA-IPA, the obtained results are compared with the Runge-Kutta method. The plots of the weight vectors, absolute error and comparison of the results are provided for each group of the nonlinear smoke model. Furthermore, statistical performances are provided using the single and multiple trial to authenticate the stability and reliability of the GNNs-GA-IPA for solving the nonlinear smoke system.
- Gudermannain neural networks,
- nonlinear smoke model,
- Runge-Kutta,
- active-set algorithm,
- genetic algorithms,
- numerical results
Citation: Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Abeer S. Alnahdi, Mdi Begum Jeelani, M. A. Abdelkawy. Numerical investigations of the nonlinear smoke model using the Gudermannian neural networks[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 351-370. doi: 10.3934/mbe.2022018

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Abstract

These investigations are to find the numerical solutions of the nonlinear smoke model to exploit a stochastic framework called gudermannian neural works (GNNs) along with the optimization procedures of global/local search terminologies based genetic algorithm (GA) and interior-point algorithm (IPA), i.e., GNNs-GA-IPA. The nonlinear smoke system depends upon four groups, temporary smokers, potential smokers, permanent smokers and smokers. In order to solve the model, the design of fitness function is presented based on the differential system and the initial conditions of the nonlinear smoke system. To check the correctness of the GNNs-GA-IPA, the obtained results are compared with the Runge-Kutta method. The plots of the weight vectors, absolute error and comparison of the results are provided for each group of the nonlinear smoke model. Furthermore, statistical performances are provided using the single and multiple trial to authenticate the stability and reliability of the GNNs-GA-IPA for solving the nonlinear smoke system.

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