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The complex nonlinear Fokas–Lenells (FL) equation to obtain the explicit travelling wave solutions under different values parameters such as solitary wave solutions, kink solitary wave solutions and periodic wave solutions and others solutions are studied using the dynamical system approach. Also, the optical solitons for the FL equation, as well as the plane-wave, complex dark-singular, and complex periodic-singular solutions are obtained via the e x p (- Φ (ζ)) expansion method. In conclusion, graphical representations of these solutions are provided so that the dynamics of these waves can be viewed. [ABSTRACT FROM AUTHOR]

In this study, we build multi-wave solutions of the KdV-5 model through Hirota's bilinear method. Taking complex conjugate values of the free parameters, various colliding exact solutions in the form of rogue wave, symmetric bell soliton and rogue waves form; breather waves, the interaction of a bell and rogue wave, and two colliding rogue wave solutions are constructed. To explore the characteristics of the breather waves, localized in any direction, the higher-order KdV-5 model, which describes the promulgation of weakly nonlinear elongated waves in a narrow channel, and ion-acoustic, and acoustic emission in harmonic crystals symmetrically is analyzed. With the appropriate parameters that affect and manage phase shifts, transmission routes, as well as energies of waves, a mixed solution relating to hyperbolic and sinusoidal expression are derived and illustrated by figures. All the single and multi-soliton appeared symmetric about an axis of the wave propagation. The analyzed outcomes are functional in achieving an understanding of the nonlinear situations in the mentioned fields. [ABSTRACT FROM AUTHOR]

Rotating-disc electrodes (RDEs) are favored technologies for analyzing electrochemical processes in electrically charged cells and other revolving machines, such as engines, compressors, gearboxes, and generators. The model is based on the concept of the nonlinear entropy convection-diffusion equations, which are constructed using semi-boundaries as an infinite notion. In this model, the surrogate solutions with different parameter values for the mathematical characterization of non-dimensional O H − and H + ion concentrations at a rotating-disc electrode (RDE) are investigated using an intelligent hybrid technique by utilizing neural networks (NN) and the Levenberg–Marquardt algorithm (LMA). Reference solutions were calculated using the RK-4 numerical method. Through the training, validation, and testing sampling of reference solutions, the NN-BLMA approximations were recorded. Error histograms, absolute error, curve fitting graphs, and regression graphs validated the NN-BLMA's resilience and accuracy for the problem. Additionally, the comparison graphs between the reference solution and the NN-BLMA procedure established that our paradigm is reliable and accurate. [ABSTRACT FROM AUTHOR]

H β -Hausdorff functions for β ∈ 0 , 1 are introduced, and common fixed-point theorems for a pair of multivalued operators satisfying generalized contraction conditions are proven in a b -metric space. Our results are proper extensions and new variants of many contraction conditions existing in literature. In order to demonstrate applications of our result, we have proven an existence theorem for a unique common multivalued fractal of a pair of iterated multifunction systems and also an existence theorem for a common solution of a pair of Volterra-type integral equations. [ABSTRACT FROM AUTHOR]

Self-driving vehicles are increasingly vital for various environmental tasks, such as data gathering, remote sensing, and mapping. They are particularly important in transportation, where advanced robotic vehicles can operate effectively without human oversight in challenging conditions. One exciting application is using collaborative autonomous robotic vehicles to locate specific points of interest like accidents, congestion, rain, fog, or icy roads. This article explores multi-robot systems for transportation tasks, considering interference that may impact the robot group's performance. A mathematical model is analysed to quantify the influence of interference and transformed into a dimensionless form for analysis. The system is studied numerically, and a neural soft computing technique, LMA-NN, is proposed. The technique is established on a neural network. For performance evaluation of the LMA-NN technique, different performance benchmarks such as mean square error and absolute error with reference solution are employed. The results are also presented graphically. [ABSTRACT FROM AUTHOR]

Researchers have made significant strides in understanding car-following behaviour and traffic flow, especially with the advent of intelligent and networked technologies. Diverse mathematical models analyse traffic flow, each with pros and cons. This study focuses on a sensitivity-based mathematical model for uphill and downhill highways, examining position, velocity, and acceleration profiles to predict traffic jam occurrence. The model employs an ordinary differential equation and a machine learning-based approach (machine learning procedure neural network) for numerical solutions, exhibiting high accuracy (1 0 − 8 − 1 0 − 10) compared to the reference Runge–Kutta method For accuracy, reliability and stability of the results are evaluated by various performance indicators and statistical terms. For multiple independent executions, mean absolute deviation, root mean square error and error in Nash–Sutcliffe efficiency are calculated. Their values are lies in range 1 0 − 8 − 1 0 − 14 . Moreover, graphical analysis is established for better visualization of traffic flow and congestion. [ABSTRACT FROM AUTHOR]

The current study delivers a numerical investigation on the performance of heat transfer and flow of micropolar fluid in porous Darcy structures with isothermal and isoflux walls (boundary conditions) of a stretching sheet. The dynamics and mechanism of such fluid flows are modelled by nonlinear partial differential equations that are reduced to a system of nonlinear ordinary differential equations by utilizing the porosity of medium and similarity functions. Generally, the explicit or analytical solutions for such nonlinear problems are hard to calculate. Therefore, we have designed a computer or artificial intelligence-based numerical technique. The reliability of neural networks using the machine learning (ML) approach is used with a local optimization technique to investigate the behaviours of different material parameters such as the Prandtl number, micropolar parameters, Reynolds number, heat index parameter, injection/suction parameter on the temperature profile, fluid speed, and spin/rotational behaviour of the microstructures. The approximate solutions determined by the efficient machine learning approach are compared with the classical Runge–Kutta fourth-order method and generalized finite difference approximation on a quasi-uniform mesh. The accuracy of the errors lies around 10−8 to 10−10 between the traditional analytical solutions and machine learning strategy. ML-based techniques solve different problems without discretization or computational work, and are not subject to the continuity or differentiability of the governing model. Moreover, the results are illustrated briefly to help implement microfluids in drug administering, elegans immobilization, and pH controlling processes. [ABSTRACT FROM AUTHOR]

In this paper, a novel soft computing technique is designed to analyze the mathematical model of the steady thin film flow of Johnson–Segalman fluid on the surface of an infinitely long vertical cylinder used in the drainage system by using artificial neural networks (ANNs). The approximate series solutions are constructed by Legendre polynomials and a Legendre polynomial-based artificial neural networks architecture (LNN) to approximate solutions for drainage problems. The training of designed neurons in an LNN structure is carried out by a hybridizing generalized normal distribution optimization (GNDO) algorithm and sequential quadratic programming (SQP). To investigate the capabilities of the proposed LNN-GNDO-SQP algorithm, the effect of variations in various non-Newtonian parameters like Stokes number ( S t ), Weissenberg number ( W e ), slip parameters (a), and the ratio of viscosities (ϕ) on velocity profiles of the of steady thin film flow of non-Newtonian Johnson–Segalman fluid are investigated. The results establish that the velocity profile is directly affected by increasing Stokes and Weissenberg numbers while the ratio of viscosities and slip parameter inversely affects the fluid's velocity profile. To validate the proposed technique's efficiency, solutions and absolute errors are compared with reference solutions calculated by RK-4 (ode45) and the Genetic algorithm-Active set algorithm (GA-ASA). To study the stability, efficiency and accuracy of the LNN-GNDO-SQP algorithm, extensive graphical and statistical analyses are conducted based on absolute errors, mean, median, standard deviation, mean absolute deviation, Theil's inequality coefficient (TIC), and error in Nash Sutcliffe efficiency (ENSE). Statistics of the performance indicators are approaching zero, which dictates the proposed algorithm's worth and reliability. [ABSTRACT FROM AUTHOR]

In this study, we have investigated the mathematical model of an immobilized enzyme system that follows the Michaelis–Menten (MM) kinetics for a micro-disk biosensor. The film reaction model under steady state conditions is transformed into a couple differential equations which are based on dimensionless concentration of hydrogen peroxide with enzyme reaction (H) and substrate (S) within the biosensor. The model is based on a reaction–diffusion equation which contains highly non-linear terms related to MM kinetics of the enzymatic reaction. Further, to calculate the effect of variations in parameters on the dimensionless concentration of substrate and hydrogen peroxide, we have strengthened the computational ability of neural network (NN) architecture by using a backpropagated Levenberg–Marquardt training (LMT) algorithm. NNs–LMT algorithm is a supervised machine learning for which the initial data set is generated by using MATLAB built in function known as "pdex4". Furthermore, the data set is validated by the processing of the NNs–LMT algorithm to find the approximate solutions for different scenarios and cases of mathematical model of micro-disk biosensors. Absolute errors, curve fitting, error histograms, regression and complexity analysis further validate the accuracy and robustness of the technique. [ABSTRACT FROM AUTHOR]

In this paper, we have analyzed the mathematical model of various nonlinear oscillators arising in different fields of engineering. Further, approximate solutions for different variations in oscillators are studied by using feedforward neural networks (NNs) based on the backpropagated Levenberg–Marquardt algorithm (BLMA). A data set for different problem scenarios for the supervised learning of BLMA has been generated by the Runge–Kutta method of order 4 (RK-4) with the "NDSolve" package in Mathematica. The worth of the approximate solution by NN-BLMA is attained by employing the processing of testing, training, and validation of the reference data set. For each model, convergence analysis, error histograms, regression analysis, and curve fitting are considered to study the robustness and accuracy of the design scheme. [ABSTRACT FROM AUTHOR]

Savanna fire has many types: Savanna woody, Savanna vegetation, and grassland. In this paper, Savanna vegetation is studied, characterized by low trees and high grass. It grows in hot and seasonally dry conditions. The Savanna vegetation is described by relating to the environment and climate. Savanna vegetation is considered a metastable mixture of trees and grass and is advanced to explain stability. The Savanna vegetation is modeled with first-order linear differential equations having grass, trees, and sapling (young trees) as components. Furthermore, the model is evaluated numerically by integrating the global search technique Sine-Cosine algorithm and local search technique Interior point algorithm. Comprehensive numerical experiments are conducted to analyze numerical results. To validate solution of proposed technique, Runge-Kutta order four method isolution is taken as a reference solution. The solutions are compared graphically with the results of the reference technique. Performance indicators Mean Absolute Deviation, Root Mean Squared Error, and Error in Nash-Sutcliffe Efficiency are implemented to verify consistency, and multiple independent runs are drawn. Furthermore, the scheme is evaluated through convergence graphs as well. [ABSTRACT FROM AUTHOR]

The present study analyzes the thermal attribute of conductive, convective, and radiative moving fin with thermal conductivity and constant velocity. The basic Darcy’s model is utilized to formulate the governing equation for the problem, which is further nondimensionalized using certain variables. Moreover, an effective soft computing paradigm based on the approximating ability of the feedforword artificial neural networks (FANN’s) and meta-heuristic approach of global and local search optimization techniques is developed to quantify the effect of variations in significant parameters such as ambient temperature, radiation-conduction number, Peclet number, nonconstant thermal conductivity, and initial temperature parameter on the temperature gradient of the rod. The results by the proposed FANN-AOA-SQP algorithm are compared with radial basis function approximation, Runge–Kutta–Fehlberg method and machine-learning algorithms. An extensive graphical and statistical analysis based on solution curves and errors such as absolute errors, mean square error, standard deviations in Nash–Sutcliffe efficiency, mean absolute deviations, and Theil’s inequality coefficient are performed to show the accuracy, ease of implementation, and robustness of the design scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is not only to propose a new extreme distribution, but also to show that the new extreme model can be used as an alternative to well-known distributions in the literature to model various kinds of datasets in different fields. Several of its statistical properties are explored. It is found that the new extreme model can be utilized for modeling both asymmetric and symmetric datasets, which suffer from over- and under-dispersed phenomena. Moreover, the hazard rate function can be constant, increasing, increasing–constant, or unimodal shaped. The maximum likelihood method is used to estimate the model parameters based on complete and censored samples. Finally, a significant amount of simulations was conducted along with real data applications to illustrate the use of the new extreme distribution. [ABSTRACT FROM AUTHOR]

In this paper, the mathematical model of an inclined longitudinal porous fin of trapezoidal, rectangular, and dovetail profiles in the presence of convective and radiative environments is considered to study the heat transfer and heat distribution within the fin. The governing equation for the energy transfer in the porous fin is derived by using the Darcy model that simulates the interaction of fluids and solids. The mathematical model has been analyzed so that a common equation can be used to study the trapezoidal, rectangular, and dovetail profiles. Furthermore, to study the temperature distribution in the fin, a supervised machine learning algorithm is developed using Cascade feedforward backpropagated (CFB) neural networks and Levenberg–Marquardt (LM) algorithm. A reference solution of 1001 points for supervised learning of the design scheme is generated by using a numerical solver (RK-4), which is further utilized by the CFB-LM algorithm with the Log-Sigmoid activation function to train, validate and test the data properly. The design algorithm’s outcomes are compared to the results of the homotopy perturbation method, shooting method, and other machine learning algorithms. Extensive graphical and statistical analyses are conducted to study the influence of variations in inclination angle, tip tapering, wet porous parameter, internal heat generation, porosity, progressive natural convective parameter, and dimensionless radiative parameter on the thermal profile and heat transfer rate of the longitudinal porous fin. The dovetail fin profile achieves the maximum heat transfer rate, followed by rectangular and trapezoidal fin profiles, provided that internal heat production is kept to a minimum. [ABSTRACT FROM AUTHOR]

In this article, we have proposed a new generalization of the odd Weibull-G family by consolidating two notable families of distributions. We have derived various mathematical properties of the proposed family, including quantile function, skewness, kurtosis, moments, incomplete moments, mean deviation, Bonferroni and Lorenz curves, probability weighted moments, moments of (reversed) residual lifetime, entropy and order statistics. After producing the general class, two of the corresponding parametric statistical models are outlined. The hazard rate function of the sub-models can take a variety of shapes such as increasing, decreasing, unimodal, and Bathtub shaped, for different values of the parameters. Furthermore, the sub-models of the introduced family are also capable of modelling symmetric and skewed data. The parameter estimation of the special models are discussed by numerous methods, namely, the maximum likelihood, simple least squares, weighted least squares, Cramér-von Mises, and Bayesian estimation. Under the Bayesian framework, we have used informative and non-informative priors to obtain Bayes estimates of unknown parameters with the squared error and generalized entropy loss functions. An extensive Monte Carlo simulation is conducted to assess the effectiveness of these estimation techniques. The applicability of two sub-models of the proposed family is illustrated by means of two real data sets. [ABSTRACT FROM AUTHOR]

In this paper, a new parametric compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. Relevant mathematical properties are derived. Some new bivariate G families using the theorems of "Farlie-Gumbel-Morgenstern copula", "the modified Farlie-Gumbel-Morgenstern copula", "the Clayton copula", and "the Renyi's entropy copula" are presented. Many special members are derived, and a special attention is devoted to the exponential and the one parameter Pareto type II model. The maximum likelihood method is used to estimate the model parameters. A graphical simulation is performed to assess the finite sample behavior of the estimators of the maximum likelihood method. Two real-life data applications are proposed to illustrate the importance of the new family. [ABSTRACT FROM AUTHOR]

Biofiltration is a method of pollution management that utilizes a bioreactor containing live material to absorb and destroy pollutants biologically. In this paper, we investigate mathematical models of biofiltration for mixing volatile organic compounds (VOCs) for instance hydrophilic (methanol) and hydrophobic (α -pinene). The system of nonlinear diffusion equations describes the Michaelis-Menten kinetics of the enzymic chemical reaction. These models represent the chemical oxidation in the gas phase and mass transmission within the air-biofilm junction. Furthermore, for the numerical study of the saturation of α -pinene and methanol in the biofilm and gas state, we have developed an efficient supervised machine learning algorithm based on the architecture of Elman neural networks (ENN). Moreover, the Levenberg-Marquardt (LM) optimization paradigm is used to find the parameters/ neurons involved in the ENN architecture. The approximation to a solutions found by the ENN-LM technique for methanol saturation and α -pinene under variations in different physical parameters are allegorized with the numerical results computed by state-of-the-art techniques. The graphical and statistical illustration of indications of performance relative to the terms of absolute errors, mean absolute deviations, computational complexity, and mean square error validates that our results perfectly describe the real-life situation and can further be used for problems arising in chemical engineering. [ABSTRACT FROM AUTHOR]

We present new lemmas, theorem and corollaries to construct interactions among higher-order rogue waves, n-periodic waves and n-solitons solutions (n → ∞) to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation. Several examples for theories are given by choosing definite interactions of the wave solutions for the model. In particular, we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave, a rogue and a cross-bright bell wave, a rogue and a one-, two-, three-, four-periodic wave. In addition, we also present multi-types interactions between a rogue and a periodic cross-bright bell wave, a rogue and a periodic cross-bright-bark bell wave. Finally, we physically explain such interaction solutions of the model in the 3D and density plots. [ABSTRACT FROM AUTHOR]

The design and development of secure nonlinear cryptographic Boolean function plays an unavoidable measure for modern information confidentiality schemes. This ensure the importance and applicability of nonlinear cryptographic Boolean functions. The current communication is about to suggest an innovative and energy efficient lightweight nonlinear multivalued cryptographic Boolean function of modern block ciphers. The proposed nonlinear confusion element is used in image encryption of secret images and information hiding techniques. We have suggested a robust LSB steganography structure for the secret hiding in the cover image. The suggested approach provides an effective and efficient storage security mechanism for digital image protection. The technique is evaluated against various cryptographic analyses which authenticated our proposed mechanism. [ABSTRACT FROM AUTHOR]

Jamming transition in traffic flow refers to the sudden transition from a free-flowing state to a jammed state as the traffic density increases. This transition is of great interest to traffic engineers and physicists, as it can have significant implications for traffic safety, efficiency, traffic management, and urban planning. Homogeneous car following models is a popular framework used to study the jamming transition phenomenon. The mathematical structure of the problem is governed by the classical Lorenz system to consider the fluctuational effects. The analytical solution of such nonlinear oscillatory differential equations does not exist. Therefore, this study aims to utilize the machine learning approach with the optimization technique that could be used to fine-tune the weights/ parameters of a neural network model to predict the accurate and reliable solutions for the jamming transition in traffic flow. The headway deviations have been studied by considering the multiple scenarios based on the acceleration and braking of the vehicle. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model for the system of prey-predator with immigrant prey has been analyzed to find an approximate solution for immigrant prey population density, local prey population density, and predator population density. Furthermore, we present a novel soft computing technique named LeNN-WOA-NM algorithm for solving the mathematical model of the prey-predator system with immigrant prey. The proposed algorithm uses a function approximating ability of Legendre polynomials based on Legendre neural networks (LeNNs), global search ability of the whale optimization algorithm (WOA), and a local search mechanism of the Nelder–Mead algorithm. The LeNN-WOA-NM algorithm is applied to study the effect of variations on the growth rate, the force of interaction, and the catching rate of local prey and immigrant prey. The statistical data obtained by the proposed technique establish the effectiveness of the proposed algorithm when compared with techniques in the latest literature. The efficiency of solutions obtained by LeNN-WOA-NM is validated through performance measures including absolute errors, MAD, TIC, and ENSE. [ABSTRACT FROM AUTHOR]

This article considers Falkner–Skan flow over a dynamic and symmetric wedge under the influence of a magnetic field. The Hall effect on a magnetic field is negligible for small magnetic Reynolds numbers. The magnetic field B (x) is considered over x-axis, which is in line with the wedge i.e., parallel, while the flow is transverse over the y-axis. This study has numerous device-centric applications in engineering, such as power generators, cooling reactor and heat exchanger design, and MHD accelerators. The Third and second-ordered ordinary differential equations characterize the system. A novel hybrid computational technique is designed for the surrogate solutions of the Falkner–Skan flow system. The designed technique is based on the sine–cosine optimization algorithm and sequential quadratic programming. Reference solutions are calculated by using the Runge–Kutta numerical technique. Performance matrices evaluate the accuracy and stability of our surrogate solutions, mean-absolute deviation (MAD), root-mean-square error (RMSE), and error in Nash-–Sutcliffe efficiency (ENSE). Furthermore, graphical representations in terms of convergence graphs, mesh graphs, stem graphs, stairs plots, and boxplots are presented to establish the symmetry, reliability, and validity of our solutions. [ABSTRACT FROM AUTHOR]

This paper analyzed the model of swinging oscillation of a solid circular sector arising in hydrodynamical machines, electrical engineering, heat transfer applications, and civil engineering. Nonlinear differential equations govern the mathematical model for frequency oscillation of the system. Furthermore, a computational strength of Cascade neural networks (CNNs) is utilized with backpropagated Levenberg–Marquardt (BLM) algorithm to study the oscillations in angular displacement θ , velocity θ ′ , and acceleration θ ″ . A data set for the supervised learning of the CNN-BLM algorithm for different angles α and radius R are generated by Runge–Kutta (RK-4) method. The BLM algorithm further interprets the dataset with log-sigmoid as an activation function for the solutions' validation, testing, and training. The results obtained by the design scheme are compared with Akbari–Ganji's (AG) method. The rapid convergence and quality of the solutions are validated through performance indicators such as mean absolute deviations (MAD), root means square error, and error in Nash–Sutcliffe efficiency (ENSE). The statistics demonstrate the design scheme's applicability and efficiency to highly singular nonlinear problems. [ABSTRACT FROM AUTHOR]

This paper investigates the heat transfer of two-phase nanofluid flow between horizontal plates in a rotating system with a magnetic field and external forces. The basic continuity and momentum equations are considered to formulate the governing mathematical model of the problem. Furthermore, certain similarity transformations are used to reduce a governing system of non-linear partial differential equations (PDEs) into a non-linear system of ordinary differential equations. Moreover, an efficient stochastic technique based on feed-forward neural networks (FFNNs) with a back-propagated Levenberg–Marquardt (BLM) algorithm is developed to examine the effect of variations in various parameters on velocity, gravitational acceleration, temperature, and concentration profiles of the nanofluid. To validate the accuracy, efficiency, and computational complexity of the FFNN–BLM algorithm, different performance functions are defined based on mean absolute deviations (MAD), error in Nash–Sutcliffe efficiency (ENSE), and Theil's inequality coefficient (TIC). The approximate solutions achieved by the proposed technique are validated by comparing with the least square method (LSM), machine learning algorithms such as NARX-LM, and numerical solutions by the Runge–Kutta–Fehlberg method (RKFM). The results demonstrate that the mean percentage error in our solutions and values of ENSE, TIC, and MAD is almost zero, showing the design algorithm's robustness and correctness. [ABSTRACT FROM AUTHOR]

In this study, we provide a discretized system of a continuous dynamical model for enhancing crop production in the presence of insecticides and insects. Crops are assumed to grow logistically but are limited by an insect population that entirely depends on agriculture. To protect crops from insects, farmers use insecticides, and their overmuch use is harmful to human health. We assumed that external efforts are proportional to the gap between actual production and carrying capacity to increase the field's development potential. We use the Levenberg–Marquardt algorithm (LMA) based on artificial neural networks (NNs) to investigate the approximate solutions for different insecticide spraying rates. "NDSolve" tool in Mathematica generated a data collection for supervised LMA. The NN-LMA approximation's value is achieved by the training, validation, and testing reference data sets. Regression, error histograms, and complexity analysis help to validate the technique's robustness and accuracy. [ABSTRACT FROM AUTHOR]

This study investigated the steady two-phase flow of a nanofluid in a permeable duct with thermal radiation, a magnetic field, and external forces. The basic continuity and momentum equations were considered along with the Buongiorno model to formulate the governing mathematical model of the problem. Furthermore, the intelligent computational strength of artificial neural networks (ANNs) was utilized to construct the approximate solution for the problem. The unsupervised objective functions of the governing equations in terms of mean square error were optimized by hybridizing the global search ability of an arithmetic optimization algorithm (AOA) with the local search capability of an interior point algorithm (IPA). The proposed ANN-AOA-IPA technique was implemented to study the effect of variations in the thermophoretic parameter (N t) , Hartmann number (H a) , Brownian (N b) and radiation (R d) motion parameters, Eckert number (E c) , Reynolds number (R e) and Schmidt number (S c) on the velocity profile, thermal profile, Nusselt number and skin friction coefficient of the nanofluid. The results obtained by the designed metaheuristic algorithm were compared with the numerical solutions obtained by the Runge–Kutta method of order 4 (RK-4) and machine learning algorithms based on a nonlinear autoregressive network with exogenous inputs (NARX) and backpropagated Levenberg–Marquardt algorithm. The mean percentage errors in approximate solutions obtained by ANN-AOA-IPA are around 10 − 6 to 10 − 7 . The graphical analysis illustrates that the velocity, temperature, and concentration profiles of the nanofluid increase with an increase in the suction parameter, Eckert number and Schmidt number, respectively. Solutions and the results of performance indicators such as mean absolute deviation, Theil's inequality coefficient and error in Nash–Sutcliffe efficiency further validate the proposed algorithm's utility and efficiency. [ABSTRACT FROM AUTHOR]

This paper analyzed the three-dimensional (3D) condensation film problem over an inclined rotating disk. The mathematical model of the problem is governed by nonlinear partial differential equations (NPDE's), which are reduced to the system of nonlinear ordinary differential equations (NODE's) using a similarity transformation. Furthermore, the system of NODEs is solved by the supervised machine learning strategy of the nonlinear autoregressive exogenous (NARX) neural network model with the Levenberg–Marquardt algorithm. The dimensionless profiles of velocity, acceleration, and temperature are investigated under the effect of variations in the Prandtl number and normalized thickness of the film. The results demonstrate that increasing the Prandtl number causes an increase in the fluid's temperature profile. The solutions obtained by the proposed algorithm are compared with the state-of-the-art techniques that show the accuracy of the approximate solutions by NARX-BLM. The mean percentage errors in the results by the proposed algorithm for Θ η , Ψ η , k η , − s η , and θ η are 0.0000180 % , 0.000084 % , 0.0000135 % , 0.000075 % , and 0.00026 % , respectively. The values of performance indicators, such as mean square error and absolute errors, are approaching zero. Thus, it validates the worth and efficiency of the design scheme. [ABSTRACT FROM AUTHOR]

In this article, we take a look at an Ordinary Differential Equation model that describes the bacteria’s role in anaerobic biodegradation dynamics of domestic garbage in a landfill. A nonlinear Ordinary Differential Equation system is used to describe biological activities. In the current study, the Levenberg–Marquardt Backpropagation Neural Network is used to locate alternate solutions for the model. The Runge–Kutta order four (RK-4) method is employed to produce reference solutions. Different scenarios were looked at to analyse our surrogate solution models. The reliability to verify the equilibrium of the mathematical model, physical quantities such as the half-saturation constant (KS\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_S$$\end{document}), the maximum growth rate (μm\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu _m$$\end{document}), and the inhibition constant (KI\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_I$$\end{document}), can be modified. We categorise our potential solutions into training, validation and testing groups in order to assess how well our machine learning strategy works. The advantages of the Levenberg-Marquardt Backpropagation Neural Network scheme have been shown by studies that compare statistical data based on Mean Square Error Function, efficacy, regression plots, and error histograms. From the whole process we conclude that Levenberg–Marquardt Backpropagation Neural Network is accurate and authentic. [ABSTRACT FROM AUTHOR]

This study investigated the steady two-phase flow of a nanofluid in a permeable duct with thermal radiation, a magnetic field, and external forces. The basic continuity and momentum equations were considered along with the Buongiorno model to formulate the governing mathematical model of the problem. Furthermore, the intelligent computational strength of artificial neural networks (ANNs) was utilized to construct the approximate solution for the problem. The unsupervised objective functions of the governing equations in terms of mean square error were optimized by hybridizing the global search ability of an arithmetic optimization algorithm (AOA) with the local search capability of an interior point algorithm (IPA). The proposed ANN-AOA-IPA technique was implemented to study the effect of variations in the thermophoretic parameter (N t) , Hartmann number (H a) , Brownian (N b) and radiation (R d) motion parameters, Eckert number (E c) , Reynolds number (R e) and Schmidt number (S c) on the velocity profile, thermal profile, Nusselt number and skin friction coefficient of the nanofluid. The results obtained by the designed metaheuristic algorithm were compared with the numerical solutions obtained by the Runge–Kutta method of order 4 (RK-4) and machine learning algorithms based on a nonlinear autoregressive network with exogenous inputs (NARX) and backpropagated Levenberg–Marquardt algorithm. The mean percentage errors in approximate solutions obtained by ANN-AOA-IPA are around 10 − 6 to 10 − 7 . The graphical analysis illustrates that the velocity, temperature, and concentration profiles of the nanofluid increase with an increase in the suction parameter, Eckert number and Schmidt number, respectively. Solutions and the results of performance indicators such as mean absolute deviation, Theil's inequality coefficient and error in Nash–Sutcliffe efficiency further validate the proposed algorithm's utility and efficiency. [ABSTRACT FROM AUTHOR]