Machine learning-based prediction of heat transfer performance in annular fins with functionally graded materials.

This paper presents a study investigating the performance of functionally graded material (FGM) annular fins in heat transfer applications. An annular fin is a circular or annular structure used to improve heat transfer in various systems such as heat exchangers, electronic cooling systems, and power generation equipment. The main objective of this study is to analyze the efficiency of the ring fin in terms of heat transfer and temperature distribution. The fin surfaces are exposed to convection and radiation to dissipate heat. A supervised machine learning method was used to study the heat transfer characteristics and temperature distribution in the annular fin. In particular, a feedback architecture with the BFGS Quasi-Newton training algorithm (trainbfg) was used to analyze the solutions of the mathematical model governing the problem. This approach allows an in-depth study of the performance of fins, taking into account various physical parameters that affect its performance. To ensure the accuracy of the obtained solutions, a comparative analysis was performed using guided machine learning. The results were compared with those obtained by conventional methods such as the homotopy perturbation method, the finite difference method, and the Runge-Kutta method. In addition, a thorough statistical analysis was performed to confirm the reliability of the solutions. The results of this study provide valuable information on the behavior and performance of annular fins made from functionally graded materials. These findings contribute to the design and optimization of heat transfer systems, enabling better heat management and efficient use of available space.

Bifurcation analysis on ion sound and Langmuir solitary waves solutions to the stochastic models with multiplicative noises.

This article explores on a stochastic couple models of ion sound as well as Langmuir surges propagation involving multiplicative noises. We concentrate on the analytical stochastic solutions including the travelling and solitary waves by using the planner dynamical systematic approach. To apply the method, First effort is to convert the system of equations into the ordinary differential form and present it in form of a dynamic structure. Next analyze the nature of the critical points of the system and obtain the phase portraits on various conditions of the corresponding parameters. The analytic solutions of the system in an account of distinct energy states for each phase orbit are performed. We also show how the results are highly effective and interesting to realize their exciting physical as well as the geometrical phenomena based on the demonstration of the stochastic system involving ion sound as well as Langmuir surges. Descriptions of effectiveness of the multiplicative noise on the obtained solutions of the model, and its corresponding figures are demonstrated numerically.

Quantitative Study of Non-Linear Convection Diffusion Equations for a Rotating-Disc Electrode.

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 OH- 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.

Dynamical interaction of solitary, periodic, rogue type wave solutions and multi-soliton solutions of the nonlinear models.

This study presents a modification form of modified simple equation method, namely new modified simple equation method. Multiple waves and interaction of soliton solutions of the Phi-4 and Klein-Gordon models are investigated via the scheme. Consequently, we derive various novels and more general interaction, and multiple wave solutions in term of exponential, hyperbolic, and trigonometric, rational function solutions combining with some free parameters. Taking special values of the free parameters, interaction of two dark bells, interaction of two bright bells, two kinks, two periodic waves, kink and soliton, kink-rogue wave solutions are obtained which is the key significance of this method. Properties of the achieved solutions have many useful descriptions of physical behavior, correlated to the solutions are attained in this work through plentiful 3D figures, density plot and 2D contour plots. The results derived may increase the prospect of performing significant experimentations and carry out probable applications.

Analysis of heat transmission in convective, radiative and moving rod with thermal conductivity using meta-heuristic-driven soft computing technique.

Abstract: 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.

Heat Transfer Analysis of Nanofluid Flow in a Rotating System with Magnetic Field Using an Intelligent Strength Stochastic-Driven Approach.

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.

Investigation of Three-Dimensional Condensation Film Problem over an Inclined Rotating Disk Using a Nonlinear Autoregressive Exogenous Model.

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.

Analysis of Nanofluid Particles in a Duct with Thermal Radiation by Using an Efficient Metaheuristic-Driven Approach.

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 (Nt), Hartmann number (Ha), Brownian (Nb) and radiation (Rd) motion parameters, Eckert number (Ec), Reynolds number (Re) and Schmidt number (Sc) 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.

Study of Rolling Motion of Ships in Random Beam Seas with Nonlinear Restoring Moment and Damping Effects Using Neuroevolutionary Technique.

In this paper, a mathematical model for the rolling motion of ships in random beam seas has been investigated. The ships' steady-state rolling motion with a nonlinear restoring moment and damping effect is modeled by the nonlinear second-order differential equation. Furthermore, an artificial neural network (NN)-based, backpropagated Levenberg-Marquardt (LM) algorithm is utilized to interpret a numerical solution for the roll angle (x(t)), velocity (x'(t)), and acceleration (x''(t)) of the ship in random beam seas. A reference data set based on numerical examples of the mathematical model for a rolling ship for the LM-NN algorithm is generated by the numerical solver Runge-Kutta method of order 4 (RK-4). The LM-NN algorithm further uses the created data set for the validation, testing, and training of approximate solutions. The outcomes of the design paradigm are compared with those of the homotopy perturbation method (HPM), optimal homotopy analysis method (OHAM), and RK-4. Statistical analyses of the mean square error (MSE), regression, error histograms, proportional performance, and computational complexity further validate the worth of the LM-NN algorithm.

Study of Nonlinear Models of Oscillatory Systems by Applying an Intelligent Computational Technique.

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.

An Optimistic Solver for the Mathematical Model of the Flow of Johnson Segalman Fluid on the Surface of an Infinitely Long Vertical Cylinder.

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 (St), Weissenberg number (We), 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.

Mathematical Analysis of Reaction-Diffusion Equations Modeling the Michaelis-Menten Kinetics in a Micro-Disk Biosensor.

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.

Bayesian and Frequentist Inferences on a Type I Half-Logistic Odd Weibull Generator with Applications in Engineering.

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.

A New Family of Continuous Probability Distributions.

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.