The blood flow over a stenotic artery is important investigation in mathematical fluid mechanics due to its significance in biomedical sciences. The present investigation aims to examine how nanoparticles affect circulation in a stenotic artery. We examine the significance of magnetized Casson nanofluid flow over a stenotic artery under consideration of the mathematical flow problem. By using the suitable self-similarity variables, the partial differential equation is transformed into ordinary differential equations. Then, the non-dimensional equations are solved using the MATLAB software in the Bvp5c scheme. By increasing the magnetic properties of the circulatory system's cells, which is a scheme that was previously utilized by raising the magnetic field parameter, there was a predictable decrease in the blood flow. Covering the stenosed artery with a greater amount of copper nanoparticles improves its heat transmission efficiency. The present technique may help distribute medications throughout the circulatory system.

Joule dissipation has an important role in the conversion of mechanical energy to heat within a fluid due to the internal friction and viscosity. Moreover, Darcy friction is a measure of the resistance to flow in a porous medium. In response to the efficient heat transfer performance, a robust statistical approach was established to optimize the heat transfer rate in a two-dimensional flow of a nanofluid over a permeable surface embedded with a porous matrix. The electrically conducive fluid affected the flow phenomena to include a carbon nanotube nanoparticle in the conventional liquid water for the enhanced heat transfer properties; additionally, the power-law heat flux model was considered. Appropriate transformation rules were adopted to obtain a non-dimensional system that brought a developed model equipped with several factors. The traditional numerical technique (i.e., shooting based Runge-Kutta) was proposed to handle the coupled nonlinear system. Furthermore, the statistical response surface methodology (RSM) was adopted to obtain an efficient optimized model for the heat transportation rate of the considered factors. An analysis of variance (ANOVA) was utilized to validate the result of the regression analysis. However, it was evident that the nanoparticle concentrations were useful to augment the fluid velocity and the temperature distributions; the statistical approach adopted for the heat transfer rate displayed an optimized effect as compared to a conventional effect.

This study outlines a quantitative and data-driven study of the mixed convection heat transfer processes that concern Cu-water nanofluids in a $\mathrm{\Gamma }$-shaped enclosure with one to five rotating cylinders. The dimensionless equations of mass, momentum, and energy are solved using the finite element method as implemented in the COMSOL Multiphysics 6.3 software in different rotating Reynolds numbers and cylinder geometries. An artificial Neural Network that is trained using Bayesian Regularization on data produced by the COMSOL is utilized to estimate the average Nusselt numbers. The analysis is conducted for a wide range of rotational Reynolds numbers $({\text{Re}}_{\omega }=0-100)$, with the fixed Prandtl number. Results are presented in terms of streamline patterns, isotherm contours, and Nusselt numbers to assess heat transfer behavior. Findings revealed that increasing the number of cylinders and optimizing their placement significantly enhances convective mixing and thermal transport. The ANN model accurately predicts the Nusselt numbers across all configurations with negligible errors. Among all configurations, the third arrangement in Scenario 5 exhibits the highest heat transfer rates, attributed to intensified vortex interaction and reduced thermal resistance. Artificial neural networks and finite element-based models will be of great value to the design of miniature and energy-efficient enclosures and electronics cooling mechanisms that make use of nanofluids.

In this study, we examined the nonlinear dynamics of the Boussinesq equation, a foundational equation in ocean engineering to model and investigate the behavior of waves in shallow water. The novel (G'/G²)-expansion method was employed to obtain different soliton solutions, including periodic, bright, W-type, and bell-shaped soliton solutions. These solutions are illustrated through 2D, 3D, and contour plots. We discovered different dynamical behavior, including periodic, quasi-periodic, and weak chaos, depending on the choice of initial conditions and parameters. The important outcomes included the detection of multistable attractors and the presence of weak chaotic behavior supported by Lyapunov exponents. These understandings have important effects in practical uses such as energy harvesting and wave control in ocean systems, where handling and understanding system transitions and stability is crucial. These findings also give a framework for further examination of stability and control in nonlinear wave systems.

This research is aimed at finding exact soliton solutions of the nonlinear fractional Kairat-X equation, which describes soliton behavior in nonlinear media and has applications in quantum physics, materials science, signal processing, and telecommunications. We use a unified method that generalizes the tanh-function method to find new exact soliton solutions in trigonometric, hyperbolic, and plane wave forms. Computational simulations with fixed parameters are performed to produce two-dimensional and three-dimensional visualizations, e.g., contour and density plots, representing the physical properties of the derived solitons. The simulations result in the identification of several soliton types, namely kink wave solitons, dark solitons, bright solitons, and periodic wave solitons. Our results increase the knowledge of the solution properties of the Kairat-X equation and give a platform to interpret a variety of significant physical phenomena. The systematicity and stability of our methodology prove its usefulness as a device for solving other nonlinear partial differential equations in applied physics and mathematics, always returning different exact solutions.

The present investigation focused on the influence of magnetohydrodynamic Gold-Fe₃O₄ hybrid nanofluid flow over a stretching surface in the presence of a porous medium and linear thermal radiation. This article demonstrates a novel method for implementing an intelligent computational solution by using a multilayer perception (MLP) feed-forward back-propagation artificial neural network (ANN) controlled by the Levenberg-Marquard algorithm. We trained, tested, and validated the ANN model using the obtained data. In this model, we used blood as the base fluid along with Gold-Fe₃O₄ nanoparticles. By using the suitable self-similarity variables, the partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs). After that, the dimensionless equations were solved by using the MATLAB solver in the Fehlberg method, such as those involving velocity, energy, skin friction coefficient, heat transfer rates and other variables. The goals of the ANN model included data selection, network construction, network training, and performance assessment using the mean square error indicator. The influence of key factors on fluid transport properties is presented via tables and graphs. The velocity profile decreased for higher values of the magnetic field parameter and we noticed an increasing tendency in the temperature profile. This type of theoretical investigation is a necessary aspect of the biomedical field and many engineering sectors.

This research paper investigates the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony equation. The new Kudryashov and generalized Arnous methods are employed to obtain the generalized solitary wave solution. The phase plane theory examines the bifurcation analysis and illustrates phase portraits. Finally, the external perturbation terms are considered to reveal its chaotic behavior. These findings contribute to a deeper understanding of the dynamics of the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony wave equation and its applications in real-world phenomena.

This paper addresses the new concatenation model incorporating quintic-order dispersion, incorporating four well-known nonlinear models. The concatenated models are the nonlinear Schrödinger equation, the Hirota equation, the Lakshmanan-Porsezian-Daniel equation, and the nonlinear Schrödinger equation with quintic-order dispersion. The model itself is innovative and serves as an encouragement for investigating and analyzing the extracted optical solitons. These models play a crucial role in nonlinear optics, especially in studying optical fibers. Three integration algorithms are implemented to investigate the optical solitons with the governing model. These techniques are the Weierstrass-type projective Riccati equation expansion method, the addendum to Kudryashov's method, and the new mapping method. The solutions obtained include various solitons, such as bright, dark, and straddled solitons. Additionally, the paper reports hyperbolic solutions and Weierstrass-type doubly periodic solutions. These solutions are novel and have never been reported before. Visual depictions of some selected solitons illustrate these solutions' dynamic behavior and wave structure.