A numerical performance of the novel fractional water pollution model through the Levenberg-Marquardt backpropagation method

The aim of the current work is to provide the importance and significance of the fractional order (FO) derivatives for solving the nonlinear water pollution model (FWP) model. The FO derivative to solve the water pollution model is provided to get more precise results. The investigations through the non-integer and nonlinear mathematical form to define the fractional water pollution model are also provided in this study. The composition of the fractional water pollution model is classified into three classes, execution cost of control, system competence of industrial elements and a new diagnostics technical exclusion cost. The mathematical FWP 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 FWP model. The data is selected to solve the mathematical FWP system is 70% for training and 15% for both certification and testing. The exactness of solver is observed through the comparison of the results. To ratify the aptitude, validity, constancy, and exactness of the ANNs-LMBM, the replications using the regression/correlation, state transitions, and error histograms are also described.