Open Access

ARTICLE

Fractional Order Environmental and Economic Model Investigations Using Artificial Neural Network

Wajaree Weera¹, Chantapish Zamart¹, Zulqurnain Sabir^2,3, Muhammad Asif Zahoor Raja⁴, Afaf S. Alwabli⁵, S. R. Mahmoud⁶, Supreecha Wongaree⁷, Thongchai Botmart^1,*
1 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand
2 Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
3 Department of Mathematical Sciences, UAE University, P. O. Box 15551, Al Ain, UAE
4 Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin, 64002, Taiwan
5 Department of Biological Sciences, Rabigh College of Science and Arts, King Abdulaziz University, Rabigh, Saudi Arabia
6 GRC Department, Faculty of Applied Studies, Jeddah, King Abdulaziz University, Rabigh, Saudi Arabia
7 Institute of General Education, Udon Thani Rajabhat University, Udon Thani, 41000, Thailand
* Corresponding Author: Thongchai Botmart. Email:

Computers, Materials & Continua 2023, 74(1), 1735-1748. https://doi.org/10.32604/cmc.2023.032950

Received 02 June 2022; Accepted 12 July 2022; Issue published 22 September 2022

Abstract

The motive of these investigations is to provide the importance and significance of the fractional order (FO) derivatives in the nonlinear environmental and economic (NEE) model, i.e., FO-NEE model. The dynamics of the NEE model achieves more precise by using the form of the FO derivative. The investigations through the non-integer and nonlinear mathematical form to define the FO-NEE model are also provided in this study. The composition of the FO-NEE model is classified into three classes, execution cost of control, system competence of industrial elements and a new diagnostics technical exclusion cost. The mathematical FO-NEE system is numerically studied by using the artificial neural networks (ANNs) along with the Levenberg-Marquardt backpropagation method (ANNs-LMBM). Three different cases using the FO derivative have been examined to present the numerical performances of the FO-NEE model. The data is selected to solve the mathematical FO-NEE system is executed as 70% for training and 15% for both testing and certification. The exactness of the proposed ANNs-LMBM is observed through the comparison of the obtained and the Adams-Bashforth-Moulton database results. To ratify the aptitude, validity, constancy, exactness, and competence of the ANNs-LMBM, the numerical replications using the state transitions, regression, correlation, error histograms and mean square error are also described.

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Keywords

Environmental and economic model; artificial neural networks; fractional order; nonlinear; Levenberg-Marquardt backpropagation

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