System Of Partial Differential Equations Research Articles

Investigation of convective peristaltic flow of non-Newtonian fluids through a non-uniform tapered porous conduit with Ohmic heating and viscous dissipation

ABSTRACT The study of convective magnetohydrodynamic (MHD) biofluid transport through a non-uniform duct is of interest due to its applications to biomedical devices and equipment. Because of this, a comparative study of free convective peristaltic flow of non-Newtonian (Casson and Jeffrey) fluids in a tapered porous symmetric duct with Hall current and viscous dissipation effects is carried out. The proposed dimensional governing equations are converted into a system of non-dimensional partial differential equations by invoking adequate non-dimensional parameters. Furthermore, a long-wavelength approximation is used to reduce the complexity of the system. Then the reduced system of partial differential equations is solved analytically using the regular perturbation method. The fascinating features of the influential parameters on the flow characteristics are demonstrated and analyzed in depth through graphs and tables. The results demonstrate that the velocity and temperature in the Jeffrey fluid flow are larger compared to those in the Casson fluid flow. The thermal Grashof number and the Hall current parameter enhanced the pressure rise profile. This research has significant implications in biomedical devices (MRI, vitamins A and D injections, heart-lung machines, etc.) and pharmaceutical fluid preparation in biomedical sciences.

Large time existence and asymptotic stability of the generalized solution to flow and thermal explosion model of reactive real micropolar gas

We study the long time behavior of the generalized solution of the flow and thermal explosion model of the reactive real micropolar gas. The dynamics of the chemical reaction involved and the usual laws of conservation of mass, momentum, angular momentum, and energy generate a complex governing system of partial differential equations. The fluid is nonideal and non‐Newtonian. In this work, we prove that the problem can be solved in an infinite time domain and establish the asymptotic properties of the solution. Namely, we conclude that for certain parameter values, the solution stabilizes exponentially to a steady‐state solution, while for others the stabilization occurs but at power decay rate. At the end, we conducted a few numerical tests whereby we experimentally confirmed theoretical findings about long‐term behavior of the solution.

A lemma on C 0-semigroups on time scales and approximate controllability of the heat dynamic equation

In this paper, we present a lemma that allows us to characterize a broad class of C 0-semigroups on time scales, which can be applied to prove existence and uniqueness of solutions of systems of partial differential equations where the time domain is a time scale. The result obtained is applied to study the controllability of the heat equation on time scales. The lemma presented in this paper can be seen as a unification of the one proved by H. Leiva in [28], for semigroups in [0,∞).

Convex hull property for elliptic and parabolic systems of PDE

We study the convex hull property for systems of partial differential equations. This is a generalization of the maximum principle for a single equation. We show that the convex hull property holds for a class of elliptic and parabolic systems of non-linear partial differential equations. In particular, this includes the case of the parabolic p-Laplace system. The coupling conditions for coefficients are demonstrated to be optimal by means of respective counterexamples.

Effect of traps on carrier transport in CdSe quantum dot thin films

The influence of trap effects on carrier transport characteristics in quantum dot (QD) thin films is the subject of study, aiming to provide a theoretical basis for the structural design and performance improvement of QD thin film optoelectronic devices. This study presents a specific mathematical description of capturing and releasing charges by traps, which includes the time-varying equation for captured charges. Utilizing the carrier hopping transport model, a system of partial differential equations is employed as the physical field, establishing hopping transport models that account for both shallow traps and a combination of shallow and deep traps. Simulations based on specific experimental samples reveal that the presence of traps introduces asymmetry in the diffusion motion of charge carriers, extending the duration of the photocurrent signal and resulting in the capture of charges, along with a reduction in the peak value of the current signal. The model also simulates carrier transport characteristics under the influence of repetitive light pulses, demonstrating distinct patterns in capturing and releasing charges for both shallow and deep traps.

Results on solutions of several systems of the product type complex partial differential difference equations

Abstract This article is devoted to exploring the solutions of several systems of the first-order partial differential difference equations (PDDEs) with product type u ( z + c ) [ α 1 u ( z ) + β 1 u z 1 + γ 1 u z 2 + α 2 v ( z ) + β 2 v z 1 + γ 2 v z 2 ] = 1 , v ( z + c ) [ α 1 v ( z ) + β 1 v z 1 + γ 1 v z 2 + α 2 u ( z ) + β 2 u z 1 + γ 2 u z 2 ] = 1 , \left\{\begin{array}{l}u\left(z+c){[}{\alpha }_{1}u\left(z)+{\beta }_{1}{u}_{{z}_{1}}+{\gamma }_{1}{u}_{{z}_{2}}+{\alpha }_{2}v\left(z)+{\beta }_{2}{v}_{{z}_{1}}+{\gamma }_{2}{v}_{{z}_{2}}]=1,\\ v\left(z+c){[}{\alpha }_{1}v\left(z)+{\beta }_{1}{v}_{{z}_{1}}+{\gamma }_{1}{v}_{{z}_{2}}+{\alpha }_{2}u\left(z)+{\beta }_{2}{u}_{{z}_{1}}+{\gamma }_{2}{u}_{{z}_{2}}]=1,\end{array}\right. where c = ( c 1 , c 2 ) ∈ C 2 c=\left({c}_{1},{c}_{2})\in {{\mathbb{C}}}^{2} , α j , β j , γ j ∈ C , j = 1 , 2 {\alpha }_{j},{\beta }_{j},{\gamma }_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2 . Our theorems about the forms of the transcendental solutions for these systems of PDDEs are some improvements and generalization of the previous results given by Xu, Cao and Liu. Moreover, we give some examples to explain that the forms of solutions of our theorems are precise to some extent.

On wavefront patterns in a fractional reaction–diffusion model for predator–prey system with anti-predator behavior

AbstractIn recent studies, much attention has been paid to reaction–diffusion systems with anomalous diffusion. In this paper, we investigate the formation of wavefront patterns in a predator–prey model with anti-predator behavior under the influence of anomalous subdiffusion. We use methods of traveling wave analysis and numerical integration to establish the existence of traveling wavefront solutions. Further, obtained traveling wavefront solutions are validated through direct computer simulations of time-dependent solutions for fractional partial differential equation system. It is found that wavefronts exist in a range of system parameters, which travel faster in the subdiffusive system than in the normal diffusive one.

An immuno-epidemiological model with waning immunity after infection or vaccination

In epidemics, waning immunity is common after infection or vaccination of individuals. Immunity levels are highly heterogeneous and dynamic. This work presents an immuno-epidemiological model that captures the fundamental dynamic features of immunity acquisition and wane after infection or vaccination and analyzes mathematically its dynamical properties. The model consists of a system of first order partial differential equations, involving nonlinear integral terms and different transfer velocities. Structurally, the equation may be interpreted as a Fokker-Planck equation for a piecewise deterministic process. However, unlike the usual models, our equation involves nonlocal effects, representing the infectivity of the whole environment. This, together with the presence of different transfer velocities, makes the proved existence of a solution novel and nontrivial. In addition, the asymptotic behavior of the model is analyzed based on the obtained qualitative properties of the solution. An optimal control problem with objective function including the total number of deaths and costs of vaccination is explored. Numerical results describe the dynamic relationship between contact rates and optimal solutions. The approach can contribute to the understanding of the dynamics of immune responses at population level and may guide public health policies.

Modelling Plasmid-Mediated Horizontal Gene Transfer in Biofilms

In this study, we present a mathematical model for plasmid spread in a growing biofilm, formulated as a nonlocal system of partial differential equations in a 1-D free boundary domain. Plasmids are mobile genetic elements able to transfer to different phylotypes, posing a global health problem when they carry antibiotic resistance factors. We model gene transfer regulation influenced by nearby potential receptors to account for recipient-sensing. We also introduce a promotion function to account for trace metal effects on conjugation, based on literature data. The model qualitatively matches experimental results, showing that contaminants like toxic metals and antibiotics promote plasmid persistence by favoring plasmid carriers and stimulating conjugation. Even at higher contaminant concentrations inhibiting conjugation, plasmid spread persists by strongly inhibiting plasmid-free cells. The model also replicates higher plasmid density in biofilm’s most active regions.

A hierarchically normalized physics-informed neural network for solving differential equations: Application for solid mechanics problems

Physics-informed neural networks (PINNs) usually confront significant difficulties to accurately solve partial differential equations (PDEs) due to many pathologies caused by gradient failures during training process. In this paper, a gradient disease regarding the ill-conditioned loss function of the PINN methodology is intensively investigated in both theoretical and experimental aspects. Particularly, the regular PINN approaches can obtain good predictions for trivial problems but may fail to learn solutions when their PDE coefficients and/or physical domain sizes vary. Besides, the influences of neural network structure to the PINN training quality are inferred from the theoretical analyses and also confirmed by relating experiments, in which using large-width multi-layer perceptrons (MLPs) is proved to be beneficial in stabilizing PINN training process. To overcome the abovementioned restrictions, a novel PINN methodology using hierarchically normalized technique (hnPINN) motivated from the developed theory is devised. The key idea of the proposed hnPINN is, on the one hand, to transform the original PDE system into one of two proposed dimensionless forms to alleviate the negative effects of PDE coefficients and domain size; on the other hand, to use secondary output scalers to flexibly calibrate the gradient flow for training effectively and improving the solution preciseness. The determination of the secondary output scaler is formulated by a heuristic framework inspired from theoretical analyses on the hnPINN gradient flow. The obtained results from some typical PDEs and common problems in solid mechanics strongly confirm the high effectiveness of the hnPINN in terms of solution accuracy, superior convergence and performance stability in comparison to those of the vanilla PINN and the non-dimensionalized PINN (ndPINN). As a preprocessing procedure, the hnPINN method is independent on network architectures and also greatly potential to combine with other state-of-the-art PINN models for effectively solving challenged issues in practice.

Manufacture of Microstructured Optical Fibers: Problem of Optimal Control of Silica Capillary Drawing Process

The effective control of any technological process is essential in ensuring high-quality finished products. This is particularly true in manufacturing knowledge-intensive and high-tech products, including microstructured photonic crystal fibers (PCF). This paper addresses the issues of stabilizing the optimal control of the silica capillary drawing process. The silica capillaries are the main components of PCF. A modified mathematical model proposed by the authors is used as the basic model of capillary drawing. The uniqueness of this model is that it takes into account the main forces acting during drawing (gravity, inertia, viscosity, surface tension, pressure inside the drawn capillary), as well as all types of heat transfer (heat conduction, convection, radiation). In the first stage, the system of partial differential equations describing heat and mass transfer was linearized. Then, the problem of the optimal control of the drawing process was formulated, and optimization systems for the isothermal and non-isothermal cases were obtained. In the isothermal case, optimal adjustments of the drawing speed were obtained for different objective functionals. Thus, the proposed approach allows for the constant monitoring and adjustment of the observed state parameters (for example, the outer radius of the capillary). This is possible due to the optimal control of the drawing speed to obtain high-quality preforms. The ability to control and promptly eliminate geometric defects in the capillary was confirmed by the analysis of the numerical calculations, according to which even 15% deviations in the outer radius of the capillary during the drawing process can be reduced to 4–5% by controlling only the capillary drawing speed.

Outlier-resistant non-fragile observer-based fuzzy boundary control for interconnected nonlinear PDE systems

This paper investigates an observer-based boundary controller design for interconnected nonlinear partial differential equation (PDE) systems. First, the Takagi–Sugeno (T–S) fuzzy model is adopted to accurately describe the target systems. Then, boundary measurements are employed to reduce the number of sensors. Next, considering the phenomenon of abnormal interference that may lead to measurement outliers and observer parameters’ uncertainties, an outlier-resistant non-fragile observer expressed by a saturation function is designed to guarantee the desired control objectives. Moreover, the boundary control approach is employed to trade-off the cost of system design and system performance. Furthermore, utilizing the membership function-dependent Lyapunov functions and free-weight matrixes, sufficient conditions ensuring the closed-loop systems’ exponential stability are obtained while decreasing the conservativeness of the system stability analysis. Finally, the proposed method’s feasibility and effectiveness are validated by an example.

Numerical simulations of third-grade fluid flow with exothermic reactions in porous media-saturated vertical microchannels

The aim of this article is to analyze the electro-osmotic flow behavior of a third-grade (TG) incompressible fluid flowing in a vertical microchannel saturated with porous material. The electro-osmotic flow is regarded as an essential part of many scientific and engineering applications used in microfluidic devices. The flow is subjected to joule heating, exothermic reactions, and asymmetrical convective cooling. The viscosity which is assumed temperature dependent via a Naham-type law and the convective heat transfer rates governing the asymmetric convective cooling at the microchannel walls are modeled via Newton’s law of cooling. The governing system of nonlinear partial differential equations is solved computationally using an efficient and reliable finite difference method. The spatial and temporal convergence of the numerical scheme is showcased graphically. Our study findings reveal that the combination of exothermic reactions and Joule heating significantly influences the electroosmotic flow. Higher values of variable viscosity parameter (α), Brinkman number (Br), and Buoyancy parameter (Gr) result an enhancement in the flow variables. Conversely, a reverse trend is observed when augmenting the Prandtl number (Pr), non-Newtonian parameter (γ), and porous media parameters (Ω2). Numerical results for temperature and velocity are also discussed qualitatively and validated with the data previously published.

Nonlinear thermal radiation analysis for bioconvection peristaltic transport of fourth-grade nanofluid

Nanoparticles mixed with base liquids result in the formation of nanofluids, known for their exceptional thermal conductivity and their ability to enhance the performance of electrical and mechanical devices. Therefore, this study investigates the peristaltic activity of fourth-grade nanomaterial by taking the consideration of magnetic aspects. Analysis is carried out in the presence of motile gyrotactic microorganisms, Ohmic heating, and viscous dissipation. Thermal transport incorporated effects of nonlinear thermal radiation along with Brownian motion and thermophoresis. Thermal and mass convective conditions are imposed on elastic symmetric channels. The assumptions of low Reynolds number and long wavelength approximations have led to the simplification of the nonlinear partial differential equations system into ordinary differential equations. The governing system is solved numerically by NDSolve technique via Mathematica. Graphs are plotted and examined for velocity, thermal field, density of motile gyrotactic microorganisms, and heat transfer rate.

Improving the rheological behavior of magnetiz couple stress Buongiorno's nanofluid through resilient wavy channel with curvature effect: Nonlinear analysis

In biological systems, fluids with complex rheological behavior often encounter complex microenvironments. Couple stress fluids can be applied to model the flow of biological fluids in micro-vessels, capillaries, and other intricate structures. Couple stress fluids can be employed to model drug transport in microfluidic devices and capillaries, providing insights into the behavior of pharmaceuticals at small scales. Studying the couple stress fluids is crucial, particularly when considering the impact of particle viscosity in the existence of induced magnetic fields and nanoparticles that might enhance the characteristic motion. So, we studied the moderate and magnetic Reynold numbers and curvature effects of magnetized couple stress Buongiorno's nanofluid model through resilient wavy channel. The governing equations of this model are formed in non-dimensional form without any approximation, i.e., in the companionship of moderate Reynolds number and curvature effect, which aren't studied in this form. Using the Adomian decomposition approach, it is possible to obtain a solution of nonlinear system of partial differential equations analytically. The semi-analytical results are obtained and represented graphically for characteristic motion and bio-thermal properties for temperature and concentration, as well as magnetic features. Physically, by rising the magnetic Reynolds number, the magnetic permeability raises, so the ability of the couple stress fluid to form an internal magnetic field within itself with the impact of an applied magnetic field increases, which increases the magnetic force.

Dynamics of Open States and Promoter Functioning in the appY_red and appY_green Genetic Constructions Based on the pPF1 Plasmid.

Although the role of dynamic factors in DNA function still remains unclear, research in this direction is a rapidly developing area of molecular biology. In this work, the genetic constructions appY_red and appY_green, based on the plasmid pPF1 and containing a fragment of Escherichia coli (E. coli) DNA with predicted promoter-like regions, are considered complex dynamic systems in which local sites of double helix unwinding, called open states, can arise and propagate. The purpose of the article is to show the existence of a connection between the dynamics of open states and the functioning of predicted promoters. We experimentally verified the functionality of the predicted promoters using a reporter vector. Using a reverse transcription reaction, transcription start sites were identified indicating the presence of two divergent promoters, one on each strand. In mathematical studies, a dynamic model was used that described open states as one-soliton solutions (kinks) of a system of nonlinear partial differential equations, and the influence of the torque Mτ on the dynamics of kinks was taken into account. Fluorescence analysis of colonies of E. coli cells transformed with plasmid constructions showed that one of the two promoters is stronger than the other and that the strength of the promoters depends on the orientation of the fragment under study in the plasmid. On the other hand, using mathematical modeling, the energy profiles of genetic constructions were calculated and the kink trajectories were constructed. In addition, by studying the effect of torsion moment in model studies, we found threshold torque values at which the behavior of kinks changes dramatically: from oscillatory to translational motion. The minimum values of torsion moment required to initiate the transcription process were also assessed. A comparative analysis of the results of experimental data and model calculations showed a good correlation between the preferred starting points and the direction of transcription, which in turn confirmed the existence of a relationship between the dynamics of open states and the functioning of promoters.

Stability analysis and numerical simulations of the infection spread of epidemics as a reaction–diffusion model

This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results.

Dynamics of methanol conveying mono and hybrid nanoparticles to optimization of heat transfer across stretching cylinder

This manuscript investigates the comparative analysis of SiC and SiC − Ag with methanol-base fluid. The nano species are more important in many fields such as medicine, catalysis, energy-based research, imaging, and environmental sciences. The present study includes theoretical and numerical studies of the Prandtl hybrid nanofluid flow model, which takes into account the effects of heat radiation across a stretched cylinder, inclined magnetohydrodynamics (MHD), and Darcy-Forchheimer. The cylinder’s surface is subjected to the convective slip boundary condition as part of the study. The main aim of this study is to enhance heat transformation. The flow problem is formulated in a system of nonlinear partial differential equations. By employing a similarity transformation, a nonlinear system of partial differential equations can be converted into a linear system of ODEs. A feasible numerical technique Bvp4c is used to solve the reduced system of ODEs through MATLAB software. The results of temperature, concentration, and velocity profiles are discussed graphically. In section and injection scenarios, the velocity profile of both fluids decreased with increasing inputs of magnetic and Darcy-Forchheimer parameters. The heat transmission improved for higher inputs of Prandtl and thermal radiation parameters for both section and injection cases. Furthermore, our findings align with the body of current literature. The conclusions are supported by a careful comparison with pertinent research that has been published in earlier publications.

Numerical scrutinization of heat transfer subject to physical quantities through bioconvective nanofluid flow via stretching permeable surfaces

Background: The mechanics of heat and mass transfer via nanofluid flow across many media are currently being discussed. “Nanofluids” are fluids that include highly heat-conductive nanoparticles, and they are essential for resolving engineering problems. Under the effects of activation energy, thermal radiation, and motile microorganisms, the process of heat and mass transfer through steady nanofluid flow crosses over stretched surfaces in this scenario.Methodology: For mathematical evaluation, the system of partial differential equations (PDEs) is used to describe this physical framework. By introducing suitable similarity variables with a set of boundary conditions, this mathematical system of PDEs has become a system of ordinary differential equations (ODEs). To obtain numerical results, the MATLAB built-in program “bvp4c” is used to solve the system of first-order equations.Results: In the findings and discussion section, the resulting outcomes are thoroughly examined and visually shown. The flow rate in these systems increases due to the erratic movement of microorganisms. The graphical representation shows the impacts of involving physical factors on the microorganism, thermal, concentration, and momentum profiles. Variations/changes in these profiles can be observed by adjusting the parametric values, as depicted in the graphs. Consequently, thermal transport is boosted by 25%. Additionally, the skin friction, Nusselt, Sherwood, and microbe density numbers are determined numerically. The findings demonstrate that increasing the magnetic field parameter causes the velocity profile to decrease, increasing the radiation parameter leads to an increase in temperature description, and increasing the Lewis number causes the microorganism profile’s transport rate to decrease.

Thermodynamic survey of a reactive MHD couple stress fluid through saturated porous substances with convective boundary conditions

This study explores the thermodynamic behavior of a reactive hydromagnetic liquid flowing through permeable materials, with convective cooling applied to the walls. This study holds practical significance in optimization of thermal management systems and it is crucial for enhancing the efficiency of controlling thermal runaway. The flow is modeled as a system of partial differential equations which are numerically solved. The modified Adomian Decomposition Method and Pade approximation technique are utilized in solving the equations. The results acquired for velocity and temperature distributions are thereby employed to estimate entropy generation rate with critical values over numerous effects of various boundaries over improvement of thermal runaway utilizing Pade approximation technique to show the significant impact of convective cooling term (Biot number) and other thermophysical parameters on the fluid flow. The outcomes show that fluid velocity continuously increases with rising values of inverse couple stress and fluid temperature increases Biot number.