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Scientific inquiry into effective numerical methods for modelling complex physical processes has led to the investigation of fluid dynamics, mainly when non-Newtonian properties and complex heat sources are involved. This paper presents an enhanced exponential time integrator approach to dynamically simulate non-Newtonian boundary layer flow with spatially and temporally varying heat sources. We propose an explicit scheme with second-order accuracy in time, demonstrated to be stable through Fourier series analysis, for solving time-dependent partial differential equations (PDEs). Utilizing this scheme, we construct and solve dimensionless PDEs representing the flow of Williamson fluid under the influence of space- and temperature-dependent heat sources. The scheme discretizes the continuity equation of incompressible fluid and Navier–Stokes, energy, and concentration equations using the central difference in space. Our analysis illuminates how factors affect velocity, temperature, and concentration profiles. Specifically, we observe a rise in temperature profile with enhanced coefficients of space and temperature terms in the heat source. Non-Newtonian behaviours and geographical/temporal variations in heat sources are critical factors influencing overall dynamics. The novelty of our work lies in developing an explicit exponential integrator approach, offering stability and second-order accuracy, for solving time-dependent PDEs in non-Newtonian boundary layer flow with variable heat sources. Our results provide valuable quantitative insights for understanding and controlling complex fluid dynamics phenomena. By addressing these challenges, our study advances numerical techniques for modelling real-world systems with implications for various engineering and scientific applications.

Fractal time-dependent issues in fluid dynamics provide a distinct difficulty in numerical analysis due to their complex characteristics, necessitating specialized computing techniques for precise and economical solutions. This study presents an innovative computational approach to tackle these difficulties. The main focus is applying the Fractal Runge-Kutta Method to model the time-dependent magnetohydrodynamic (MHD) Newtonian fluid with rescaled viscosity flow on Riga plates. An efficient computational scheme is proposed for handling fractal time-dependent problems in flow phenomena. The scheme is comprised of three stages and constructed using three different time levels. The stability of the scheme is shown by employing the Fourier series analysis to solve scalar problems. The scheme's convergence is guaranteed for a time fractal partial differential equations system. The scheme is applied to the dimensionless fractal heat and mass transfer model of incompressible, unsteady, laminar, Newtonian fluid with rescaled viscosity flow over the flat and oscillatory Riga plates under the effects of space- and temperature-dependent heat sources. The first-order back differences discretize the continuity equation. The results show that skin friction local Nusselt number declines by raising the coefficient of the temperature-dependent term of heat source and Eckert number. The numerical simulations provide valuable insights into fluid dynamics, explicitly highlighting the influence of the temperature-dependent coefficient of the heat source and the Eckert number on skin friction and local Nusselt number. [ABSTRACT FROM AUTHOR]

More than 800 million people worldwide suffer from chronic kidney disease (CKD). It stands as one of the primary causes of global mortality, uniquely noted for an increase in death rates over the past twenty years among non-communicable diseases. Machine learning (ML) has promise for forecasting such illnesses, but its opaque nature, difficulty in explaining predictions, and difficulty in recognizing predicted mistakes limit its use in healthcare. Addressing these challenges, our research introduces an explainable ML model designed for the early detection of CKD. Utilizing a multilayer perceptron (MLP) framework, we enhance the model's transparency by integrating Local Interpretable Model-agnostic Explanations (LIME), providing clear insights into the predictive processes. This not only demystifies the model's decision-making but also empowers healthcare professionals to identify and rectify errors, understand the model's limitations, and ascertain its reliability. By improving the model's interpretability, we aim to foster trust and expand the utilization of ML in predicting CKD, ultimately contributing to better healthcare outcomes. [ABSTRACT FROM AUTHOR]

Breast cancer is ranked as a significant cause of mortality among females globally. Its complex nature poses principal challenges for physicians and researchers for rapid diagnosis and prognosis. Hence, machine learning algorithms are employed to forecast and identify diseases. This study discusses the comparative analysis of seven machine learning models, e.g., logistic regression (LR), support vector machine (SVM), k-nearest neighbor classifier (KNN), decision tree classifier (DT), random forest classifier (RF), Naïve Bayes (NB), and artificial neural network (ANN) to predict breast cancer using Wisconsin breast cancer and breast cancer datasets. In the Wisconsin breast cancer dataset, KNN depicted 99% accuracy, followed by RF (98%), SVM (96%), NB (96%), LR (96%), ANN (93%), and DT (92%). On the contrary, in the breast cancer (BC) dataset, the highest accuracy was achieved by LR at 83%, and the lowest was achieved by DT (65%), which depicted that the numeric dataset WBC has better accuracy than the breast cancer dataset. [ABSTRACT FROM AUTHOR]

An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized by first-order backward difference formulas, whereas the dimensionless Navier–Stokes equation, energy, and equation for nanoparticle volume fraction are discretized by the proposed scheme in fractal time. The effect of different parameters involved in the velocity, temperature, and nanoparticle volume fraction are displayed graphically. The velocity profile rises as the parameter I grows. We primarily apply this computational approach to analyze a non-Newtonian fluid's fractal time-dependent boundary layer flow over flat and oscillatory sheets. Considering spatial and temperature-dependent heat generation is a crucial factor that introduces additional complexity to the analysis. The continuity equation for the incompressible fluid is discretized using first-order backward difference formulas. On the other hand, the dimensionless Navier–Stokes equation, energy equation, and the equation governing nanoparticle volume fraction are discretized using the proposed fractal time-dependent scheme. [ABSTRACT FROM AUTHOR]

Researchers can incorporate uncertainties in computational fluid dynamics (CFD) that go beyond the inaccuracies caused by numerical discretization thanks to stochastic simulations. This study confirms the validity of current stochastic modeling tools by providing examples of stochastic simulations in conjunction with numerical solutions for incompressible flows. A numerical technique for solving deterministic and stochastic models is developed in this work. Our approach employs the Euler‐Maruyama method for stochastic modeling, representing a stochastic version of the third‐order explicit‐implicit scheme. For the deterministic model, the scheme is third‐order accurate. The consistency and stability of the constructed scheme are provided in the mean square sense. The scheme is the predictor–corrector type that is built on two time levels. Moreover, a mathematical model of the Casson nanofluid flow with variable thermal conductivity is given with the effect of the chemical reaction. The appropriate transformations are used to condense the set of partial differential equations (PDEs) down to one that is dimensionless. The scheme is applied for the deterministic and stochastic models of dimensionless flow problems. The velocity profile's deterministic and stochastic behavior are shown using contour plots. Results show that growing values of the thermal mixed convection parameter enhance the velocity profile. This article presents the progress made in stochastic computational fluid dynamics (SCFD) and highlights the energy‐related aspects of our discoveries. Our computational approach and stochastic modeling techniques provide new insights into the energy properties of Casson nanofluid flow, specifically regarding the variability of thermal conductivity and chemical processes. Our objective is to clarify the complex interaction of these factors on energy dynamics. This article presents a contemporary summary of the latest SCFD advancements. Additionally, it highlights potential directions for future research and unresolved issues that require attention from the members of the field of computational mathematics. [ABSTRACT FROM AUTHOR]

A novel stochastic numerical scheme is introduced to solve stochastic differential equations. The development of the scheme is based on two different parts. One part finds the solution for the deterministic equation, and the second part is the numerical approximation for the integral part of the Wiener process term in the stochastic partial differential equation. The scheme's stability and consistency in the mean square sense are also ensured. Additionally, a respective mathematical model of the boundary layer flow of Casson fluid on a flat and oscillatory plate is formulated. Wiener process terms perturb the model to be studied. This scheme will be solved in contexts including deterministic and stochastic. The influence of different parameters on velocity, temperature, and concentration profiles is demonstrated in various graphical representations. The main objective of this study is to present a reliable numerical approach that surpasses the limitations of traditional numerical methods to analyze non-Newtonian mixed convective fluid flows with varying transport parameters. Our objective is to demonstrate the capabilities of the new stochastic finite difference scheme in enhancing our comprehension of stochastic fluid flow phenomena. This will be achieved by comprehensively examining its mathematical foundations and computer execution. Our objective is to develop a revolutionary method that will serve as a valuable resource for scientists and engineers studying the modeling and understanding of stochastic unstable non-Newtonian mixed convective fluid flow. This method will address the challenges posed by the fluid's changing thermal conductivity and mass diffusivity. [ABSTRACT FROM AUTHOR]

Opportunities for magnetohydrodynamic mixed convection include modeling firefighting, combustion engineering, and cooling of electronic components. Nanotechnology has just provided a novel passive technique for improving heat transfer. Structured colloidal suspensions of magnetite nanoparticles in a base fluid are known as magneto nanofluids, and they are intended for use in thermal transfer uses, such as micro device cooling mechanisms. The current model is intended for examination of the nanofluid stagnation point flow using magnetohydrodynamics. A stretched sheet was exposed to magnetic strength and thermal radiation effects with outstanding results on heat transfer enhancements under mixed convection conditions. Furthermore, over the boundary effects of thermal slip and velocity are considered. The persuading system of partial differential equations of the governed fluid model is transformed into a scheme of coupled non-linear ordinary differential equations and expounded using a suitable numerical method. The fluid's velocity, temperature, and concentration of nanoparticles are discussed graphically for a range of newly emerging parametric values. It has been noted that when there is a magnetic field, the fluid's temperature increases, but its velocity decreases. After obtaining the numerical solution, parameters characterizing the flow, such as the local skin friction coefficient, local Sherwood number, and local Nusselt number, are thoroughly investigated. [ABSTRACT FROM AUTHOR]

This paper introduces an exponential time integrator scheme for solving partial differential equations in time, specifically addressing the scalar time-dependent convection-diffusion equation. The proposed second-order accurate scheme is demonstrated to be stable. It is applied to analyze the heat and mass transfer mixed convective flow of power-law nanofluid over flat and oscillatory sheets. The governing equations are transformed into a dimensionless set of partial differential equations, with the continuity equation discretized using a first-order scheme. The proposed time integrator scheme is employed in the time direction, complemented by second-order central discretization in the space direction for the momentum, energy, and nanoparticle volume fraction equations. Quantitative results indicate intriguing trends, indicating that an increase in the Prandtl number and thermophoresis parameter leads to a decrease in the local Nusselt number. This modified time integrator is a valuable tool for exploring the dynamics of unsteady power-law nanofluid flow over moving sheets across various scenarios. Its versatility extends to the examination of unstable fluid flows. This work improves engineering and technological design and operation in nanofluid dynamics. Improving numerical simulations' precision and computational efficiency deepens our comprehension of fundamental physics, yielding helpful information for enhancing systems that rely on nanofluids. [ABSTRACT FROM AUTHOR]

This contribution addresses a fractal numerical scheme that can be employed for handling fractal time-dependent parabolic equations. The numerical scheme presented in this contribution can be used to discretize integer order and fractal derivatives in a given differential equation. Therefore, the scheme and results can be used for both cases. The proposed finite difference scheme is based on two stages. Fractal time derivatives are discretized by employing the proposed approach. For the scalar convection–diffusion equation, we derive the stability condition of the proposed fractal scheme. Using a nonlinear chemical reaction, the approach is also used to solve the Quantum Calculus model of a Williamson nanofluid's unsteady Darcy–Forchheimer flow over flat and oscillatory sheets. The findings indicate a negative correlation between the velocity profile and the porosity parameter and inertia coefficient, with an increase in these factors resulting in a drop in the velocity profile. Additionally, the fractal scheme under consideration is being compared to the fractal Crank–Nicolson method, revealing that the proposed scheme exhibits a superior convergence speed compared to the fractal Crank–Nicolson method. Several problems involving the motion of non-Newtonian nanofluids through magnetic fields and porous media can be investigated with the help of the proposed numerical scheme. This research has implications for developing more efficient heat transfer and energy conversion devices based on nanofluids. [ABSTRACT FROM AUTHOR]