This work aims to study the nonlinear ordinary differential equations (ODEs) system of magnetohydrodynamic (MHD) past over an inclined plate using Levenberg-Marquardt backpropagation neural networks (LMBNNs). The stochastic procedures LMBNNs are provided with three categories of sample statistics, testing, training, and verification. The nonlinear MHD system past over an inclined plate is divided into three profiles, dimensionless momentum, species (salinity), and energy (heat) conservations. The data is applied 15%, 10%, and 75% for validation, testing, and training to solve the nonlinear system of MHD past over an inclined plate. A reference data set is designed to compare the obtained and proposed solutions for the MHD system. The plots of the absolute error (AE) are provided to check the accuracy and precision of the considered nonlinear system of MHD. The obtained numerical solutions of the nonlinear magnetohydrodynamic system have been considered to reduce the mean square error (MSE). For the capability, dependability, and aptitude of the stochastic LMBNNs procedure, the numerical performances are provided to authenticate the relative arrangements of MSE, error histograms (EHs), state transitions (STs), correlation, and regression.

Ocean energy is considered one of the significant sources of renewable energy globally. Due to the necessity of long-term hydropower systems, various electrical power systems have existed and been installed based on demand and energy usage. For example, the improved and low-cost system is photovoltaic panels [

In recent years, the mass and heat flows have achieved significant attention with inclined MHD. Such studies of different physical parameters affect the energy production systems and how porous inclined plate affects magnetized nanofluid flow [

The above studies discussed the rate of heat flow and generally ignored mass transfer, which is simultaneous species in the process of diffusion. However, as noted previously, the combined impacts of heat as well as mass transfer is important in systems devices. In the ocean, MHD generators have serious issues with salinity. The diffusion of mass and mixed convection flow investigations show that mass transfer obeys the Frckian-law [

The present literature has not studied the MHD ocean system flows from the inclined plate with heat generation and Soret effects. Therefore, we focused on that and examined the MHD ocean-generator, the transport phenomena of thermo-diffusion, and heat-generation effects from an inclined surface. Saltwater liquid transportation phenomena have a Soret effect. The laminar steady-state edge leading system is studied for MHD double-diffusive flow in ocean water convection besides heat generation with the inclined insulated (non-conducting) plate. The velocity, concentration (salinity), and temperature are also a wide range of self-control parameters along with surface-heat transfer, skin-friction coefficient, and mass transfer.

The ocean-generator MHD approach is one of the feasible capabilities of mankind. Numerous Russian, Japanese, American, and French engineers studied renewable energy. In Japan, the floating installation system represents the construction of MHD floating ocean-energy along with the basic MHD ocean-energy production. Here α is the slope of the inclined plate. The initial assumption is that the same temperature, T for both plate and fluid, and gets the same salinity level, S at every point. Also, assumed that both are at rest, the generator plate (channel wall) and the fluid. When the wall moves forward along X-axis with a constant velocity then the temperature of species salinity and the wall becomes

Continuity equation:

Momentum Conservation:

Species (Salinity) Conservation.

Energy (heat) conservation:

The specified boundary-conditions are given below:

Proceeding for transformation and defining dimensionless variables as follows:

These parameters introduced in the systems (c.f.

Parameters | Description |
---|---|

Thermal Grashof-number | |

Force of local magnetohydrodynamic body, | |

Inclination angle | |

Prandtl number | |

Modified Pr | |

The source of heat generation term | |

The Soret number (thermo-diffusion) |

The energy systems in which the coefficient of skin-friction along with the rate of heat and salinity transfer is computed by the below expressions:

Here

The purpose of this investigation is to design a robust computational artificial neural network (ANN) technique using the novel characteristics of the Levenberg-Marquardt backpropagation (LBMBP), i.e., ANN-LBMBP for solving the MHD generators in the ocean system. The nonlinear mathematical form of the boundary-layer wall in the ocean system is in three groups, heat transfer, mass transfer and momentum. The obtained solutions through the proposed ANN-LBMBP procedure for solving the MHD ocean energy generating system are related to the reference work out solutions. The stochastic-based numerical processes have been applied to solve various evolutionary/swarming schemes [

A novel stochastic ANN-LBMBP procedure is presented for solving the energy generating in the ocean system.

The obtained numerical results through the stochastic ANN-LBMBP procedures have been compared with the reference solutions for boundary-layer approximation in the ocean system.

The results correspond with performance, which indicates the exactness of the designed stochastic NN-LBMBP for solving energy generation in the ocean system.

The absolute error (AE) performances are found in good ranges to solve the waste energy in the ocean system. The rest of the paper’s parts are summarized: The procedure for the MHD in the ocean’s nonlinear ODEs system past over an inclined plate is provided in Section 2. The numerical results are given in Section 3. The concluding remarks and future work directions are provided in Section 4.

In this section, the designed stochastic LMBNNs procedures are provided in two steps for solving the MHD nonlinear ODEs system past over an inclined plate. Firstly, the necessary details of the stochastic process for the ODEs are explained, whereas the execution actions of the stochastic procedures are provided in the second phase.

An appropriate optimization LMBNNs procedure is provided in

The numerical results based on stochastic LMBNNs procedures for solving the MHD system past over an inclined plate are provided in the input [0, 1]. The ‘nftool’ command is built-in MATLAB procedure using the statistics, 15%, 10% and 75% for testing, validation, and training to solve the MHD past over an inclined plate. Seven numbers of neurons have been used throughout this numerical study. The obtained numerical representations of the stochastic procedures for the nonlinear MHD past over an inclined plate are illustrated in

The numerical representations to solve the nonlinear MHD system past over an inclined plate are drawn in ^{−10}, 1.13 × 10^{−11} and 6.16 × 10^{−12}, respectively. ^{−08}, 9.98 × 10^{−08} and 9.81× 10^{−08}. These illustrations designate the accuracy and precision of the proposed stochastic procedures for the MHD past over an inclined plate. The fitting curve representations are illustrated in ^{−08}, 4.93 × 10^{−07} and 3.83 × 10^{−07} for cases 1, 2 and 3. The correlation performances are observed in

The comparison plots are drawn in ^{−05} to 10^{−08} for case 1, 10^{−05} to 10^{−07} for case 2 and 10^{−05} to 10^{−06} for case 3. ^{−05} to 10^{−08} for case 1, 10^{−05} to 10^{−07} for case 2 and 10^{−04} to 10^{−06} for case 3, respectively. Likewise, the AE values for ^{−05} to 10^{−07} for case 1, 10^{−04} to 10^{−06} for case 2 and 10^{−05} to 10^{−06} for case 3, respectively. These closely overlapped results of the AE values designate the correctness and exactness of the nonlinear MHTWF past over a stretched surface.

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

[Training] | [Verification] | [Testing] | ||||||

1 | 7.66 × 10^{−12} |
2.96 × 10^{−10} |
2.51 × 10^{−11} |
7.66 × 10^{−12} |
9.79 × 10^{−08} |
1 × 10^{−10} |
116 | 3 |

2 | 6.27 × 10^{−12} |
1.13 × 10^{−11} |
8.70 × 10^{−12} |
6.27 × 10^{−12} |
9.98 × 10^{−08} |
1 × 10^{−10} |
116 | 3 |

3 | 5.49 × 10^{−12} |
6.16 × 10^{−12} |
9.37 × 10^{−12} |
5.49 × 10^{−12} |
9.81 × 10^{−08} |
1 × 10^{−10} |
122 | 3 |

The purpose of this study was to perform the numerical simulations based on the ANNs along with the novel stochastic procedures of the LMB for solving the system of MHD past over an inclined plate. The nonlinear system of MHD past over an inclined plate was categorized into three profiles, Momentum Conservation, Species (Salinity) Conservation, and Energy (heat) conservation. The procedures based on the stochastic paradigms were provided with three different categories of sample statistics, testing, training, and validation. The statistics 10%, 15% and 75% for testing, validation and training were given to solve the MHD past over an inclined plate. Seven numbers of neurons have used in this numerical study for solving the nonlinear transport model. A reference dataset was proposed for the comparison of the obtained and proposed solutions for the MHD system in the ocean. One can observe the overlying of the solutions, which demonstrates the correctness of the LMBNNs. The absolute error plots were provided in good measures that validated the accuracy and precision of the nonlinear ODEs system of MHD past over an inclined plate. The obtained numerical solutions of the nonlinear above mention system using the LMBNNs performances have calculated to reduce the MSE. For the capability, dependability and aptitude of the stochastic procedure, the numerical performances were provided to authenticate the proportional arrangements of MSE, error histograms (EHs), state transitions (STs), correlation and regression. The MSE convergence presentations were applied to the best curve, training, validation and testing for each variation of the nonlinear transfer model. The correlation representations were capable to confirm the regression actions. The gradient measures were also provided for the nonlinear transfer model. Furthermore, the precision was pragmatic using the numerical conformations as well as graphical procedures through the STs, MSE, EHs, regression and convergence measures, respectively.

In future work, the designed LMBNNs procedures can be applied to solve the biological, fluid, singular, fractional, and lonngren-wave models [