Search Results (124)

This paper investigates the dispersion of atmospheric pollutants in urban environments using a fractional-order convection–diffusion-reaction model with dynamic line sources associated with vehicle traffic. The model includes Caputo fractional time derivatives and Riesz fractional space derivatives to account for memory effects and non-local transport phenomena characteristic of complex urban air flows. Vehicle trajectories are generated stochastically on the road network graph using Dijkstra’s algorithm, and each moving vehicle acts as a mobile line source of pollutant emissions. To reflect the daily variability of emissions, a time-dependent modulation function determined by unknown parameters is included in the source composition. These parameters are inferred by solving an inverse problem using synthetic concentration measurements from several fixed observation points throughout the area. The study presents two main contributions. Firstly, a detailed numerical analysis of how fractional derivatives affect pollutant dispersion under realistic time-varying mobile source conditions, and secondly, an evaluation of the performance of the proposed parameter estimation method for reconstructing time-varying emission rates. The results show that fractional-order models provide increased flexibility for representing anomalous transport and retention effects, and the proposed method allows for reliable recovery of emission dynamics from sparse measurements. Full article

In this paper, we present a new space-time Galerkin-like method, where we treat the discretization of spatial and temporal domains simultaneously. This method utilizes a mixed formulation of the tensor-train (TT) and quantized tensor-train (QTT) (please see Section Tensor-Train Decomposition), designed for the finite element discretization (Q1-FEM) of the time-dependent convection–diffusion–reaction (CDR) equation. We reformulate the assembly process of the finite element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the finite element operators. Recognizing the banded structure inherent in the finite element framework’s discrete operators, we further exploit the QTT format of the CDR to achieve greater speed and compression. Additionally, we present a comprehensive approach for integrating variable coefficients of CDR into the global discrete operators within the TT/QTT framework. The effectiveness of the proposed method, in terms of memory efficiency and computational complexity, is demonstrated through a series of numerical experiments, including a semi-linear example. Full article

In this paper, we propose a high-order compact difference scheme for a class of time-fractional convection–diffusion–reaction equations (CDREs) with variable coefficients. Using the Lagrange polynomial interpolation formula for the time-fractional derivative and a compact finite difference approximation for the spatial derivative, we establish an unconditionally stable compact difference method. The stability and convergence properties of the method are rigorously analyzed using the Fourier method. The convergence order of our discrete scheme is $\mathcal{O}({\tau }^{4-\alpha }+h^8)$, where $\tau$ and h represent the time step size and space step size, respectively. This work contributes to providing a better understanding of the dependability of the method by thoroughly examining convergence and conducting an error analysis. Numerical examples demonstrate the applicability, accuracy, and efficiency of the suggested technique, supplemented by comparisons with previous research. Full article

Nitric oxide (NO) is a well-known member of the reactive oxygen species (ROS) family. The extent of its concentration influences whether it produces beneficial physiological effects or harmful toxic reactions. In a blood system, NO is generally produced by nitric oxide synthase (NOS) in the endothelium. Then, it diffuses into the smooth muscle wall causing a vasodilatation, and it can also be diluted in a lumen blood stream. In the present study, we analyzed a convectional reaction–diffusion of NO in a 3D digital phantom of a short segment of small arteries. NO concentrations were analyzed by applying numerical solutions to the boundary problems, which included the Navier–Stokes equation, Darcy’s law, varying consumption of NO, and the dependence of NOS activity on shear stress. All the boundary problems were evaluated using COMSOL Multiphysics software ver. 5.5. The role of two diffusive barriers surrounding the endothelium producing NO was theoretically proven. When the eNOS rate remains unchanged, an increase in the fenestration of the internal elastic lamina (IEL) and a decrease in the diffusive permeability of a thin layer of endothelial surface glycocalyx (ESG) lead to a notable rise in the NO concentration in the vascular wall. The alterations in pore count in IEL and the viscosity of ESG are considered to be involved in the physiological and pathological regulation of NO concentrations. Full article

Electrocoagulation (EC) systems are regaining attention as a promising wastewater treatment technology due to their numerous advantages, including low system and operational costs and environmental friendliness. However, the widespread adoption and further development of EC systems have been hindered by a lack of fundamental understanding, necessitating systematic research to provide essential insights for system developers. In this study, a continuous EC system with a realistic setup is analyzed using an unsteady, two-dimensional physics-based model that incorporates multiphysics. The model captures key mechanisms, such as arsenic adsorption onto flocs, electrochemical reactions at the electrodes, chemical reactions in the bulk solution, and ionic species transport via diffusion and convection. Additionally, it accounts for bulk wastewater flow circulating between the EC cell and an external storage tank. This comprehensive modeling approach enables a fundamental analysis of how operating conditions influence arsenic removal efficiency, providing crucial insights for optimizing system utilization. Furthermore, the developed model is used to generate data under various operating conditions. Seven machine learning models are trained on this data after hyperparameter optimization. These high-accuracy models are then employed to develop processing maps that identify the conditions necessary to achieve acceptable removal efficiency. This study is the first to generate processing maps by synergistically integrating physics-based and data-driven models. These maps provide clear design and operational guidelines, helping researchers and engineers optimize EC systems. This research establishes a framework for combining physics-based and data-driven modeling approaches to generate processing maps that serve as essential guidelines for wastewater treatment applications. Full article

This article describes an effective computing method for singularly perturbed parabolic problems with small negative shifts in convection and reaction terms. To handle the small negative shifts, the Taylor series expansion is used. The asymptotically equivalent singularly perturbed parabolic convection–diffusion–reaction problem is then discretized with the Crank–Nicolson method on a uniform mesh for the time derivative and a hybrid method on Shishkin-type meshes for the space derivative. The method’s stability and parameter-uniform convergence are established. To substantiate the theoretical findings, the numerical results are presented in tables and graphs are plotted. The present results improve the existing methods in the literature. Due to the effect of the small negative shifts in Examples 1 and 2, the numerical results using Shishkin and Bakhvalov–Shishkin meshes are almost the same. Since there are no small shifts in Examples 3 and 4, the numerical results using the Bakhvalov–Shishkin mesh are more efficient than using the Shishkin mesh. We conclude that the present method using the Bakhvalov–Shishkin mesh performs well for singularly perturbed problems without small negative shifts. Full article

This research investigates the impact of second-order slip conditions, Stefan flow, and convective boundary constraints on the stagnation-point flow of couple stress nanofluids over a solid sphere. The nanofluid density is expressed as a nonlinear function of temperature, while the diffusion-thermo effect, chemical reaction, and thermal radiation are incorporated through linear models. The governing equations are transformed using appropriate non-similar transformations and solved numerically via the finite difference method (FDM). Key physical parameters, including the heat transfer rate, are analyzed in relation to the Dufour number, velocity, and slip parameters using an artificial neural network (ANN) framework. Furthermore, response surface methodology (RSM) is employed to optimize skin friction, heat transfer, and mass transfer by considering the influence of radiation, thermal slip, and chemical reaction rate. Results indicate that velocity slip enhances flow behavior while reducing temperature and concentration distributions. Additionally, an increase in the Dufour number leads to higher temperature profiles, ultimately lowering the overall heat transfer rate. The ANN-based predictive model exhibits high accuracy with minimal errors, offering a robust tool for analyzing and optimizing the thermal and transport characteristics of couple stress nanofluids. Full article

This paper presents a robust fitted mesh finite difference method for solving a system of n singularly perturbed two parameter convection–reaction–diffusion delay differential equations defined on the interval $[0,2]$. Leveraging a piecewise uniform Shishkin mesh, the method adeptly captures the solution’s behavior arising from delay term and small perturbation parameters. The proposed numerical scheme is rigorously analyzed and proven to be parameter-robust, exhibiting nearly first-order convergence. A numerical illustration is included to validate the method’s efficiency and to confirm the theoretical predictions. Full article

The two-dimensional semiconductor material MoS₂, grown via chemical vapor deposition, has shown significant potential to surpass silicon in advanced electronic technologies. However, the mass transfer and chemical reaction processes critical to the nucleation and growth of MoS₂ grains remain poorly understood. In this study, we conducted an in-depth investigation into the mass transfer and chemical reaction processes during the chemical vapor deposition of MoS₂, employing a novel multi-physics coupling model that integrates flow fields, temperature fields, mass transfer, and chemical reactions. Our findings reveal that the intermediate product Mo₃O₉S₄ not only fails to participate directly in MoS₂ film growth but also hinders the diffusion of MoS₆, limiting the growth process. We demonstrate that increasing the growth temperature accelerates the diffusion rate of MoS₆, mitigates the adverse effects of Mo₃O₉S₄, and promotes the layered growth of MoS₂ films. Additionally, lowering the growth pressure enhances the convective diffusion of reactants, accelerating grain growth. This research significantly advances our understanding of the mass transport and reaction processes in MoS₂ film growth and provides critical insights for optimizing chemical vapor deposition systems. Full article

This paper proposes a robust finite difference method on a fitted Shishkin mesh to solve a system of n singularly perturbed convection–reaction–diffusion differential equations with two small parameters. Defined on the interval $[0,1]$, this system exhibits boundary layers due to the presence of small parameters, making accurate numerical approximations challenging. The method employs a piecewise uniform Shishkin mesh that adapts to layer regions and efficiently captures the solution’s behavior. The scheme is proven to be uniformly convergent with respect to the perturbation parameters, achieving nearly first-order accuracy. Comprehensive numerical experiments validate the theoretical results, illustrating the method’s robustness and efficiency in handling parameter-sensitive boundary layers. Full article

Nanofluids, with their enhanced thermal properties, provide innovative solutions for improving heat transfer efficiency in renewable energy systems. This study investigates a numerical simulation of bioconvective flow and heat transfer in a Williamson nanofluid over a stretching wedge, incorporating the effects of chemical reactions and hydrogen diffusion. The system also includes motile microorganisms, which induce bioconvection, a phenomenon where microorganisms’ collective motion creates a convective flow that enhances mass and heat transport processes. This mechanism is crucial for improving the distribution of nanoparticles and maintaining the stability of the nanofluid. The unique rheological behavior of Williamson fluid, extensively utilized in hydrometallurgical and chemical processing industries, significantly influences thermal and mass transport characteristics. The governing nonlinear partial differential equations (PDEs), derived from conservation laws and boundary conditions, are converted into dimensionless ordinary differential equations (ODEs) using similarity transformations. MATLAB’s bvp4c solver is employed to numerically analyze these equations. The outcomes highlight the complex interplay between fluid parameters and flow characteristics. An increase in the Williamson nanofluid parameters leads to a reduction in fluid velocity, with solutions observed for the skin friction coefficient. Higher thermophoresis and Williamson nanofluid parameters elevate the fluid temperature, enhancing heat transfer efficiency. Conversely, a larger Schmidt number boosts fluid concentration, while stronger chemical reaction effects reduce it. These results are generated by fixing parametric values as $0.1<\varpi <1.5, 0.1<\mathit{Nr}<3.0, 0.2<\mathrm{Pr}<0.5, 0.1<\mathit{Sc}<0.4, \mathrm{and} 0.1<\mathit{Pe}<1.5.$ This work provides valuable insights into the dynamics of Williamson nanofluids and their potential for thermal management in renewable energy systems. The combined impact of bioconvection, chemical reactions, and advanced rheological properties underscores the suitability of these nanofluids for applications in solar thermal, geothermal, and other energy technologies requiring precise heat and mass transfer control. This paper is also focused on their applications in solar thermal collectors, geothermal systems, and thermal energy storage, highlighting advanced experimental and computational approaches to address key challenges in renewable energy technologies. Full article

This paper presents a robust fitted mesh finite difference method for solving a dynamical system of two parameter convection–reaction–diffusion delay differential equations defined on the interval $[0,2]$. The method incorporates a piecewise uniform Shishkin mesh to accurately resolve the solution behavior caused by small perturbation parameters and delay terms. The proposed numerical scheme is proven to be parameter-robust and achieves almost first-order convergence. Numerical illustrations are provided to showcase the method’s effectiveness, highlighting its capability to address boundary and interior layers with improved accuracy. The results, supported by symmetrical considerations in the figures, enhance the precision and serve as validation for the theoretical results. Full article

In this study, we discuss the symmetries underlying Bernstein polynomial differentiation matrices, as they are used in the collocation method approach to approximate solutions of initial and boundary value problems. The symmetries are brought into connection with those of the Chebyshev pseudospectral method (Chebyshev polynomial differentiation matrices). The treatment discussed here enables a faster and more accurate generation of differentiation matrices. The results are applied in numerical solutions of several initial value problems for the partial differential equation of convection–diffusion reaction type. The method described herein demonstrates the combination of advanced numerical techniques like polynomial interpolation, stability-preserving timestepping, and transformation methods to solve a challenging nonlinear PDE efficiently. The use of Bernstein polynomials offers a high degree of accuracy for spatial discretization, and the CGL nodes improve the stability of the polynomial approximation. This analysis shows that exploiting symmetry in the differentiation matrices, combined with the wise choice of collocation nodes (CGL), leads to both accurate and efficient numerical methods for solving PDEs and accuracy that approach pseudospectral methods that use well-known orthogonal polynomials such as Chebyshev polynomials. Full article

The effects of relaxation, convection, and anisotropy on a two-dimensional, two-equation system of nonlinearly coupled, second-order hyperbolic, advection–reaction–diffusion equations are studied numerically by means of a three-time-level linearized finite difference method. The formulation utilizes a frame-indifferent constitutive equation for the heat and mass diffusion fluxes, taking into account the tensorial character of the thermal diffusivity of heat and mass diffusion. This approach results in a large system of linear algebraic equations at each time level. It is shown that the effects of relaxation are small although they may be noticeable initially if the relaxation times are smaller than the characteristic residence, diffusion, and reaction times. It is also shown that the anisotropy associated with one of the dependent variables does not have an important role in the reaction wave dynamics, whereas the anisotropy of the other dependent variable results in transitions from spiral waves to either large or small curvature reaction fronts. Convection is found to play an important role in the reaction front dynamics depending on the vortex circulation and radius and the anisotropy of the two dependent variables. For clockwise-rotating vortices of large diameter, patterns similar to those observed in planar mixing layers have been found for anisotropic diffusion tensors. Full article

One of the most significant applications of mathematical numerical methods in biology is the theoretical description of the convectional reaction–diffusion of chemical compounds. Initial biological objects must be appropriately mimicked by digital domains that are suitable for further use in computational modeling. In the present study, an algorithm for the creation of a digital phantom describing a local part of nervous tissue—namely, a synaptic contact—is established. All essential elements of the synapse are determined using a set of consistent Boolean operations within the COMSOL Multiphysics software 6.1. The formalization of the algorithm involves a sequence of procedures and logical operations applied to a combination of 3D Voronoi diagrams, an experimentally defined inner synapse area, and a simple ellipsoid under different sets of biological parameters. The obtained digital phantom is universal and may be applied to different types of neuronal synapses. The clear separation of the designed domains reveals that the boundary’s conditions and internal flux dysconnectivity functions can be set up explicitly. Digital domains corresponding to the parts of a synapse are appropriate for further application of the derived numeric meshes, with various capacities of the included elements. Thus, the obtained digital phantom can be effectively used for further modeling of the convectional reaction–diffusion of chemical compounds in nervous tissue. Full article