On GT-convexity and related integral inequalities

1.   Introduction

Artificial neural networks (ANNs) are a significant technique for artificial intelligence. ANNs have developed applications in a wide range of fields owing to their capacity to refabricate and model nonlinear phenomena. System identification, sequence recognition, process control, sensor data analysis, natural resource management, quantum chemistry, data mining, pattern recognition, medical diagnosis, finance, visualization, machine translation, e-mail spam filtering and social network filtering are just a few examples of its applications based on the input that flows via a network during the learning process, either externally or internally, ANNs are evolutionarily versatile under a variety of conditions. The back propagation helps to reliable back propagation stochastic numerical technique. Backpropagation is a supervised learning method that applies the gradient descent approach to lower the gradient of the error curve and thereby minimize error. Paul Werbos devised the backpropagation method in 1974, which Rumelhart and Parker rediscovered. In feed-forward multilayer neural networks, the backpropagation algorithm is commonly used as a learning method. The Levenberg-Marquardt (LM) backpropagation technique for ANNs is a ground-breaking convergent stability methodology that gives numerical solutions to a large extent of fluid flow issues. Numerous researchers have recently experimented with Newtonian and non-Newtonian fluid systems by using an Levenberg-Marquardt back-propagated ANN (LBM-BN). Ly et al. ^[1] provided a metaheuristic analysis for the parameters and construction of LBM-BN to forecast the shear capacity of foamed concrete accurately and quickly. Zhao et al. ^[2] used the LBM-BN method to estimate the defection of reinforced concrete beams. Nguyen et al. ^[3] used ANN-based LM to improve the reliability of robot placement. The ANN approach was adopted by Ali et al. ^[4] to estimate the expulsion over a sharp-crested weir, and the training technique is based on LM. Ye and Kim ^[5] used an LBM-BN in China to evaluate electricity usage in a building. Bharati et al. ^[6] created a neuro-fuzzy system framework and self-organizing maps for superconductor prediction.

Nanofluids are usually utilized as coolants in heat transfer devices such as heat exchangers, electronic cooling systems and radiators because of their improved thermal properties. Nanofluids have unique characteristics that could make them fruitful in a variety of heat transfer applications, including pharmaceutical practices, microelectronics and hybrid-powered engines, as well as in engine cooling, heat exchangers, domestic refrigerators, grinding, chillers and boiler flue gas temperature reduction. Numerous studies based on the nanofluid concept have been published in the previous few decades. Choi and Eastman ^[7] proposed nanoparticle diffusion in a base fluid for engineering disciplines for the first time two decades ago. In an analysis by Uddin et al. ^[8], in the presence of a stretching and contracting sheet, steady two-dimensional laminar mixed convective boundary slip nanofluid flow arises in a permeable Darcian medium. A Din et al. ^[9] investigation found that the temperature dispersion, productivity and temperature of the fin's tip are significantly influenced by thermal and thermo-geometric aspects. According to their analysis, the fading exponential fin is more efficient than the expanding exponential fin, and this offers important benefits for mechanical engineering. The results on a magnetohydrodynamics (MHD) nanofluid in a circular permeable material, as presented by Jalili et al. ^[10], demonstrates that the convection mechanism weakens with an increase in the volume fraction of solid nanoparticles. A stronger and more unified core vortex will result from a substantial rise in the Rayleigh number. Additionally, a favorable Nuave occurs when a magnetic force is applied horizontally. A ferrofluid over a shrinking sheet with effective thermal conductivity was another subject of study by Jalili et al ^[11]. The results indicate that the magnetic and boundary parameters have a similar effect on velocity as the micro-rotation parameter. By examining natural convection in a cavity equipped with a nanofluid, Geridonmez reported in ^[12] that the fluid velocity and heat transmission are boosted in the presence of nanoparticles, and that convective heat transfer is decreased in a rectangular cavity. The squeezing nanofluid flow that was developed by the authors of ^[13,14] investigated the Casson-type hybrid nanofluid with slip and sinusoidal heat conditions. Tawade et al. ^[15] observed increments in the temperature profile for increasing values of the Brownian motion parameter, and that the energy distribution increases with increment in the value of the thermophoresis parameter. Hamid et al. ^[16] investigated the fractional-order unsteady natural convective radiating flow of a nanofluid; they observed that the velocity field decreases with enhancing magnetic field effects. A heat and mass transmission study of radiative and chemical reactive effects on an MHD nanofluid over an infinite moving vertical plate was performed by Arulmozh et al. ^[17]. The behavior of an ionized nano-liquid motion with reference to heat transmission between two parallel discs was studied by Khan et al. ^[18]. Tuz Zohra et al. ^[19] explored the MHD bio-nano convective slip stream with Stefan blowing phenomena across a spinning disc, demonstrating that, compared to blowing, suction offers a better medium for enhancing the amount of heat, mass and microbial transmission. Various studies pertaining to nanofluids can be found in ^{[20,21,22,23]}.

The activation energy (AE) in a fluid stream is significant in a broad range of industrial applications. Arrhenius proposed the concept of activation energy in 1889. It is the absolute basic minimum of energy required for the organisms to transform the reaction mixture into products. Oil emulsions, geothermal energy, fluid mechanics and chemical engineering all benefit from this phenomenon. Given a Carreau nanofluid with magnetic influence, Irfan et al. ^[24] applied mass flux theory and Arrhenius activation energy. Waqas et al. ^[25] proposed a bio-convection flow of a tangent hyperbolic nanofluid over a Riga plate with activation energy, demonstrating that increasing the chances of mixed convection parameters increases the velocity of the tangent hyperbolic fluid. Bhatti and Michaelides ^[26] also used activation energy to transmit a thermo-bioconvection nanofluid across a Riga plate. The nanofluid and bacteria existing in the base fluid were filled into the Riga plate. The study of modulated heat plus mass fluxes in 3D Eyring-Powell nanofluid nonlinear thermal radiation was done by Muhammad et al. ^[27]. For preventing boundary-layer separation and reducing submarine friction and pressure drag, a Riga plate has been used. In the presence of electromagnetic fields and gyrotactic microorganisms, slip effects on an MHD nanofluid were explored by Habib et al. ^[28], who used a mathematical approach. The MHD peristaltic transport of a Sutterby nanofluid with mixed convection and a Hall current was described by Hayat et al. ^[29]. In contrast to higher activation energies and radiation factors, concentration was seen to increase. Entropy continues to decrease in response to greater diffusion parameters.

In real-life situations, many complex problems demand an ANN due to their complex structure. Due to their extensive applicability in a wide range of domains, analytical and numerical actions that address computational fluid dynamic problems with a deterministic computing model are garnering increasing interest from technical groups, specifically, mathematicians and physicists. When compared to traditional numerical techniques, artificial intelligence algorithms based on stochastic numerical computing have produced accurate and validated results, inspiring authors to work in this paradigm. Researchers in ANNs have recently accomplished some valuable work, like entropy-generated systems ^[30], porous fins ^[31], COVID-19 ^[32], hydromagnetic Williamson fluid flow ^[33], carbon nanotubes ^[34], Emden-Fowler equation ^[35], second-order singular functional differential models ^[36], Darcy-Forchheimer models ^[37], dissipative fluid flow systems ^[38], mosquito dispersal models ^[39] and many others ^{[40,41,42,43]}.

The authors' main goal of this paper is to exploit the strength of artificial back-propagated neural networks with an LBM-BN algorithm that augment the computing power and level of accuracy of the solver to scrutinize the hydromagnetic velocity slip nanofluid flow (VSN) of a viscous material with nonlinear mixed convection over a stretching and rotating disc. No one has used this technique for the proposed problem. It is a novel approach to solve the problem of intelligent computing technique-based supervised learning for VSN model dynamics by using a modern stochastic solution approach based on the artificial intelligence algorithm. Artificial intelligence-based stochastic solution approaches are a more effective and practical substitute for the implementation of numerous linear and nonlinear mathematical frameworks. These problem-solving methods were developed by using a contemporary computing paradigm to address a system of extremely nonlinear ordinary differential equations (ODEs) that describe the mathematical representations of such fluid problems. The main objectives of this research are as follows:

(i). Represent the mathematical modeling for hydromagnetic VSN.

(ii). Explain the solution for designed hydromagnetic VSN via testing, validation and training processes.

(iii). Examine the reference data samples arbitrarily preferred for testing, training and validation and analyze the approximated solutions of the proposed LBM-BN in comparison with the reference results.

(iv). Generate datasets through the use of LM, and in training/validation/testing processes, as a target for identifying the approximated solution of the proposed LBM-BN.

(v). The suggested technique efficiently examines the dynamics of the problem for many scenarios based on the variation of pertinent parameters to depict velocity, concentration and temperature profiles.

(vi). The LBM-BN validity and verification are based on a thorough examination of accuracy assessments, histograms and regression analysis conducted for the VSN, which are given graphically and numerically in sufficient detail.

(vii). Examine the various scenarios of the VSN by varying the stretching parameter, magnetic variable, Eckert number, thermal Biot numbers and second-order slip parameters.

The proposed mathematical model is illustrated in Section II. Section III describes the methodology of the designed LBM-BNs with the interpretation of the result. Section IV discusses the effects of several physical parameters on axial $f\left(\xi \right),$ radial $f{'}\left(\xi \right),$ thermal $\theta \left(\xi \right),$ and tangential velocities $g\left(\xi \right)$. The current study's conclusion is drawn in Section V.

2.   Problem model

Nonlinear mixed convection hydromagnetic slip flow for nanofluid flow is considered. The slip velocity is of second order. The considered fluid is incompressible, electrically conductive and flowing along a stretching and rotating surface. The disk is located at z = 0 and rotating with a frequency $\Omega$ about the z axis. At $z > 0$, the heat transfer is taken out matter to thermal radiation, Joule heating, Brownian motion and thermophoresis diffusion are also incorporated in the fluid flow system. Figure 1 is a physical illustration of the model.

The governing flow model is given in ^{[44,45,46,47]}. The stimulated magnetic field is ignored because of small values of the following:

Here, from ^[46],

The dimensionless equations are as follows:

Here,

3.   Solution methodology

The novel concept of the proposed LBM-BN-based method is applied to examine the steady 3D incompressible hydromagnetic VSN flow for a rotating disc surface with nonlinear mixed convection. Nonlinear PDEs are altered into ODEs through appropriate transformations, and the computational results can be calculated by utilizing the Lobatto IIIA formula in the bvp4c technique in MATLAB software. For boundary-value problems, the Lobatto IIIA approach has been studied to test the stability of attributes. The Lobatto IIIA method is used by many researchers in different fields ^{[48,49,50,51]}. This technique is applied for the solution of ODEs and generates a dataset for the LBM-BN. The datasets are attained from Lobatto IIIA based on the variation of influential parameters. Later on, a solution can be approximated through a training, testing and validation procedure in MATLAB by using the nftool module, and a comparison is made between the standard results and those of the LBM-BN. The designed neural network weights are connected with nodes and optimized via hidden layers and activation functions. If the weights are not optimized through a forward pass, then the backward pass is taken into account to tune the values of weights, which provides the best solution for the governing mathematical systems. The procedure for the ANNs is provided in ^[52,53]. This dataset can be used to assist in the calculation of the approximated solution of the VSN problem in MATLAB. Figure 2 illustrates the ANN for the VSN. Figure 3 illustrates the flowchart for the given problem. Regarding the influence of different parameters of the VSN, the graphs are presented for axial $f\left(\xi \right),$ radial $f{'}\left(\xi \right),$ thermal $\theta \left(\xi \right),$ and tangential velocity $g\left(\xi \right)$ profiles. Also, Table 1 shows all of the scenarios and cases for variants of the VSN. Table 2 reflects the outcomes of the LBM-BN for all scenarios of the VSN. Table 2 contains the epochs, mean square error (MSE) values, performances, mu parameters, gradients and times for all scenarios related to the VSN. For validation of the performance of the LBM-BN, error histogram (EH) regression analysis was performed and the transition state (TS) and MSE results are analyzed.

4.   Results and discussion

The effects of the VSN model were resolved by using an ANN with back-propagated LM. The reference dataset was generated by applying the Lobatto IIIA procedure using MATLAB software. The numerical solution achieved for the allocation of axial $f\left(\xi \right),$ radial $f{'}\left(\xi \right),$ thermal $\theta \left(\xi \right),$ and tangential velocity $g\left(\xi \right)$ profiles has been studied and explained through diagrams. All six different scenarios of each have four different cases that are listed in Table 1. The input ranged between 0 and 7, with a step size of 0.07 for each case of the proposed LBM-BN for the VSN model. After the dataset was generated, the 'nftool' command was employed to evaluate the solution for the VSN. The solutions of the LBM-BN for all scenarios of Case IV, in the form of the MSE, curve fitness plot, error histogram, regression and training state, are demonstrated in Figures 4–8. Figures 4–8 show the outcomes of the stretching parameter $\left({A}_{1}\right)$, magnetic variable ($M$, Eckert number $\left(Ec\right)$, thermal Biot numbers $\left({B}_{1}\right),$ and second-order slip parameters $\left({L}_{2}, {L}_{4}\right)$ for the VSN model. Additionally, convergence through MSE curves for training and testing, as well as the gradient, best performance index, time taken, mu and epochs are presented in Table 2 for each scenario of all four cases.

Figure 4 shows the training, validation and testing MSE curves for Case IV for all six scenarios of the VSN. Figure 4(a) shows the MSE curves for Scenario 1 of Case IV. The MSE curves for the best validation performance were attained at 2.97E-10 with 145 epochs in 9 s. Figure 4(b) shows the MSE curves for Scenario 2 of Case IV. The MSE curves for the best validation performance were attained at 3.42E-10 with 84 epochs in 1 s. Figure 4(c) shows the MSE curves for Scenario 3 of Case IV. The MSE curves for the best validation performance were attained at 6.22E-11 with 58 epochs in 0 s. Figure 4(d) shows the MSE curves for Scenario 4 of Case IV. The MSE curves for the best validation performance were attained at 2.25E-10 with 148 epochs in 0 s. Figure 4(e) shows the MSE curves for Scenario 5 of Case IV. The MSE curves for the best validation performance were attained at 1.20E-09 with 103 epochs in 0 s. Figure 4(f) shows the MSE curves for Scenario 6 of Case IV. The MSE curves for the best validation performance were attained at 2.54E-09 with 193 epochs in 1 s.

Figure 5 illustrates the TS for Case IV for all six scenarios, which was obtained by using the LBM-BN. The impacts of the axial $f\left(\xi \right),$ radial $f{'}\left(\xi \right),$ temperature $\theta \left(\xi \right),$ and tangential velocities $g\left(\xi \right)$ on the stretching parameter $\left({A}_{1}\right)$, magnetic variable ($M$, Eckert number $\left(Ec\right)$, thermal Biot number $\left({B}_{1}\right),$ and second-order slip parameters $\left({L}_{2}, {L}_{4}\right)$ were investigated. The gradient had values of 9.93E-08, 9.95E-08, 9.66E-08, 9.90E-08, 9.67E-08 and 9.71E-08, where the values of the mu parameter were 1.00E-08, 1.00E-09, 1.00E-10, 1.00E-09, 1.00E-09 and 1.00E-09 for the LBM-BN, as shown in the depicted plots. The enhanced networks training and test the enhanced convergence of the results can be obtained for the smallest values of gradient, and Mu provide the best convenience.

Figures 6(a–f) and 7(a–f) represent the fitness curves and EHs for Case IV in terms of ${A}_{1}$, $M, {L}_{2}, {L}_{4, }{B}_{1}$ and $Ec$, as obtained by using the LBM-BN for the VSN model. The scrutiny of the EHs reveals that the maximum values depicting the errors were found to be very close to zero, which validates the worth of the solver.

Figure 8(a–f) reflects the graphical illustration of the regression analysis for Case IV for all scenarios of the LBM-BN VSN. The regression plots show that the solver gave the most optimal solution corresponding to the scenarios in terms of ${A}_{1}$, $M, {L}_{2}, {L}_{4, }{B}_{1}$ and $Ecforf\left(\xi \right), f{'}\left(\xi \right), g\left(\xi \right), \wedge \theta \left(\xi \right)$, as associated with the presented fluid flow system.

Figure 9 demonstrates the behavior of all physical quantities ${A}_{1}$, $M, {L}_{2}, {L}_{4, }{B}_{1}$ and $Ec$ for $f\left(\xi \right), \; f{'}\left(\xi \right), g\left(\xi \right), \wedge \theta \left(\xi \right)$. From Figure 9(a, b) it can be perceived that, for a higher value of ${A}_{1}$, the flow increases in the radial and axial velocity directions. Substantially, for a larger ${A}_{1}$, the stretching rate increases and creates more disturbance in the liquid. Thus, the velocity increases. Figure 9(c, d) shows the effect of ${L}_{2}$ for $f\left(\xi \right), \wedge f{'}\left(\xi \right)$. Clearly, we examined that the velocity components $f\left(\xi \right), \wedge f{'}\left(\xi \right)$decline due to an enrichment in ${L}_{2}.$

Figure 9(e) depicts the impact of the tangential velocity $g\left(\xi \right)$ due to the second-order slip parameter ${L}_{4}.$ It can be seen that the tangential velocity declines with the increase of second order slip parameter.

The effects of reducing the influence of the tangential velocity and its relationship with larger values of M are depicted in Figure 9(f). Physically, for larger values of M, the Lorentz force increases, which is a force that is resistant to the motion of producing material elements. Hence, the velocity decreases. Figure 9(g) displays the effect of $Ec$ on temperature. The parameter Ec was used to calculate the effect of self-heating on the dissipation properties of a liquid. At extreme flow rates, the thermal field in a fluidic framework is swamped by the temperature gradients appearing in the framework. Also, the effects of dissipation caused by the internal friction temperature are enhanced with the increase of Ec. Figure 9(h) was sketched to discuss the performance of ${B}_{1}$ on temperature function. We can see that $\theta \left(\xi \right)$ increases with increment in the value of ${B}_{1}.$

Figure 10 shows the graphical representation of absolute error (AE) with the variation of $f\left(\xi \right)$, ${f}^{{'}}\left(\xi \right), g\left(\xi \right)$ and $\theta \left(\xi \right)$, which validates the performance of the proposed technique. In Figure 10(a), the AE lies between 10^-4 and 10^-7 with the increment in ${A}_{1}$ for $f\left(\xi \right)$, whereas, in Figure 10(b), the AE ranges between 10^-4 and 10^-6 with the increase in ${A}_{1}$ for ${f}^{{'}}\left(\xi \right).$ Figure 10(c, d) illustrates the AE of ${L}_{2}$for $f\left(\xi \right)$ and ${f}^{{'}}\left(\xi \right)$ in the range of 10^-4 to 10^-7. Figure 10(e, f) depicts the AE of ${L}_{4}$ and M for the tangential velocity $g\left(\xi \right)$in the range of 10^-4 to 10^-7 and 10^-4 to 10^-6, respectively. Figure 10(g, h) exhibits the AE in the range of 10^-4 to 10^-7 for both Ec and ${B}_{1}$ for the temperature function $\theta \left(\xi \right).$

5.   Conclusions

For the present analysis, the author's main aim was exploitation of the concept of the VSN through the application of an LBM-BN to a stretched rotating disk for the inspection of a second-order velocity slip in the presence of activation energy. The PDEs indicating the VSN were converted into ODEs. The designed approach is efficient and accurate based on the MSEs obtained via the training, testing and validation procedures. Moreover, the convergence and stability of LBM-BN mappings have been validated on the basis of achieved accuracy through the application of regression-based statistical analysis and EHs for the proposed model. Lobatto IIIA was utilized to resolve the ODEs and generate the reference dataset for the LBM-BN. The reference dataset was used to facilitate calculation of the approximated solution of the VSN in MATLAB. The flow consequences on the $f\left(\xi \right), f{'}\left(\xi \right), g\left(\xi \right), \wedge \theta \left(\xi \right)$ profiles were analyzed from the perspectives of different physical quantities, i.e., ${A}_{1}$, $M, {L}_{2}, {L}_{4, }{B}_{1}$ and $Ec$. The substantial findings taken from this research are as follows:

● The axial and radial velocity functions grow as values of ${A}_{1}$ increase, but a reverse trend can be seen for ${L}_{2}.$

● The tangential velocity decreases with increment in ${L}_{4}$.

● With larger values of M, the tangential velocity increases.

● The temperature function is increased with increment in Ec and ${B}_{1}.$

Further, the proposed technique can also be applied to a micropolar nanofluid ^[54], a Casson nanofluid ^[55], the waterborne spread and control of diseases ^[56] and computer epidemic viruses ^[57]. As a future work direction, the authors will consider other machine learning techniques to study the activation energy of various chemical processes.

Acknowledgments

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IF2/PSAU/2022/01/21873).

Conflict of interest

The authors disclosed no conflicts of interest in publishing this paper.

Appendix