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.

The researcher’s community is taking keen interest for solving the nonlinear environmental and economic (NEE) model as it has several applications in industrial organizations. Conceptual investigations based on the association of the supply chain are implemented to the strategic relationships to increase the operational and financial interest in the industrial presentations and to decrease the cost catalogues in the supply chain management (SCM) [

The mathematical system represents the numerous differential models, e.g., SITR covid model [

The system _{1}, _{2},…., _{8} are the constant coefficients. The initial conditions are represented as _{1}, _{2} and _{3.} The aim of this work is to provide the importance and significance of the FO-NEE model by applying the artificial neural networks (ANNs) and the Levenberg-Marquardt backpropagation method (ANNs-LMBM). The extended form of the FO-NEE model is given as:

The computing stochastic solvers based on the ANNs-LMBM are derived to solve the FO form of the derivative using the NEE model. The stochastic solvers have been tested through the local and global operator performances to solve a few nonlinear, singular, and complicated differential models. Few applications of the stochastic applications are COVID-19 systems [

The aim of the current study is to present the numerical investigations of the FO mathematical system, which is based on the NEE model by using the stochastic performances of the ANNs-LMB. The time-fractional form of the derivative has a number of submissions in various networks. The integer order shows remembrance, whereas, the memory function represents the form of FO. The FO form is applied to the application of the real-world systems [

A design of a fractional order economic and environmental nonlinear mathematical model is presented.

The stochastic performances have never been applied before to solve the mathematical nonlinear FO-NEE model.

The solutions of the mathematical nonlinear FO-NEE model have been successfully presented by using the stochastic ANNs-LMBM.

Three different cases using the FO derivative have been examined to present the numerical performances of the mathematical FO-NEE model.

The brilliance of the stochastic ANNs-LMBM procedures is authenticated by using the comparison of the reference (Adams-Bashforth-Moulton) and calculated results.

The performance through the absolute error (AE) is validated with the matching of order 4 to 6 to solve the mathematical FO-NEE model.

The reliability and constancy of the proposed ANNs-LMBM is tested using the EHs, correlation, STs, MSE and regression to solve the mathematical FO-NEE model.

The other parts are provided as: The proposed ANNs-LMBM is described in Section 2. The simulations of the FO-NEE model are narrated in Section 3. Conclusions are provided in the last Section.

In this section, the proposed artificial neural networks, and the Levenberg-Marquardt backpropagation method (ANNs-LMBM) structure is presented to solve the mathematical fractional order nonlinear environmental and economic (FO-NEE) model. The designed ANNs-LMBM methodology is divided in two phases as: the essential steps using the ANNs-LMBM are provided, while the execution method via designed procedures is described to solve the mathematical FO-NEE model.

This section shows three different variations based on the fractional order (FO) derivative to solve the mathematical fractional order nonlinear environmental and economic (FO-NEE) model using the artificial neural networks and the Levenberg-Marquardt backpropagation method (ANNs-LMBM) structure. The mathematical form of each case is presented as:

The numerical simulations of the FO-NEE model are presented using the ANNs-LMBM with 15 numbers of neurons together with the selection of data to solve the FO-NEE system is executed as 70% for training and 15% for both testing and certification. The structure of the input, hidden and output layers with 15 neurons are plotted in

^{−04}, 1.3409 × 10^{−10} and 2.2393 × 10^{−10}, respectively. The gradient values have been calculated through the ANNs-LMBM for the mathematical FO-NEE system. These values calculated around 4.1547 × 10^{−03}, 9.9851 × 10^{−08} and 9.8482 × 10^{−08}. These illustrations show the convergence of the proposed ANNs-LMBM for the mathematical FO-NEE model. The values of the fitting curves are provided through the comparison of the achieved and reference outcomes in ^{−04}, 9.21 × 10^{−06} and 8.04 × 10^{−06} for case 1–3. The correlation is given in

Case | MSE | Gradient | Performance | Epoch | Mu | Time | ||
---|---|---|---|---|---|---|---|---|

Training | Testing | Authentication | ||||||

1 | 8.56 × 10^{−05} |
1.67 × 10^{−04} |
1.11 × 10^{−04} |
0.001 × 10^{−05} |
7.52 × 10^{−05} |
81 | 1 × 10^{−05} |
2 s |

2 | 7.62 × 10^{−10} |
8.44 × 10^{−11} |
1.34 × 10^{−10} |
9.99 × 10^{−08} |
7.63 × 10^{−10} |
813 | 1 × 10^{−09} |
5 s |

3 | 7.65 × 10^{−10} |
7.45 × 10^{−11} |
2.23 × 10^{−10} |
9.85 × 10^{−08} |
7.66 × 10^{−10} |
150 | 1 × 10^{−09} |
3 s |

The comparison performances through the overlapping of the results along with the AE values have been provided in ^{−02} to 10^{−04}, 10^{−04} to 10^{−06} and 10^{−04} to 10^{−08} for case 1 to 3 based on the mathematical FO-NEE model. The AE values for ^{−02} to 10^{−04} for case 1, 10^{−04} to 10^{−06} for case 2 and 3 for the mathematical FO-NEE model. The AE for ^{−03} to 10^{−05} for case 1, 10^{−05} to 10^{−07} for case 2 and 3 based on the mathematical FO-NEE model. These good AE performances show the accuracy of the ANNs-LMB-NNs for the mathematical FO-NEE model.

The purpose of this work is to introduce a stochastic numerical procedure for the fractional order derivatives using the mathematical form of the environmental and economic system. The dynamics of the nonlinear environmental and economic model become more accurate and precise with the involvement of the fractional order derivative. 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 dynamics of the fractional order NEE model has never been tested before through the stochastic solvers. The ANNs-LMBM method have been presented for the numerical solutions of the mathematical FO-NEE system. Three different cases using the FO derivative have been examined to indicate the performances of the FO-NEE model. The selection of the data to solve the mathematical FO-NEE system is executed as 70% for training and 15% for both testing and certification. Fifteen neurons have been applied for the mathematical NEE system throughout this study and the obtained measures have been compared with the reference Adams-Bashforth-Moulton. The absolute error is calculated in good measures to solve the fractional order derivatives using the mathematical form of the environmental and economic system. To reduce the MSE performances that the numerical results have been performed through ANNs-LMBM. To authorize the aptitude, reliability of the ANNs-LMBM, the simulations have been performed through EHs, STs, MSE, correlation and regression. The accuracy of ANNs-LMBM is observed for the mathematical FO-NEE system through the overlapping of the proposed and reference outcomes. Moreover, the performance is indicated to authenticate the consistency of the ANNs-LMBM scheme.

In future studies, the ANNs-LMBM approaches have been used to solve the numerical performance of the nonlinear models [

_{2}gas with an efficient method