System Of Partial Differential Equations Research Articles

Computational study of the role of reduced gravity on convective heat transfer in the presence of heat source and sink along a moving surface

This study highlights the thermal radiation’s impacts on heat transmission in the presence of reduced effects of gravity using the numerical method. For smooth algorithm and integration, the similarity form of stream functions is employed to transform the system of nonlinear partial differential equations into the system of ordinary differential equations. The shooting approach (BVP4C) is employed to acquire the numerical solution of the current model. The numerical findings are acquired using the MATLAB program and are then displayed in tabular and graph forms. The energy equation in the mathematical model includes the thermal radiation impacts along with the expressions for reduced gravity effects. The physical characteristics of the flow profile and thermal distribution for varying values of reduced gravity parameters ([Formula: see text], radiation parameter [Formula: see text], positive number ([Formula: see text] and Prandtl numbers [Formula: see text] are shown graphically along with the results for skin friction and thermal transmission influenced by various emerging parameters are displayed in tables. The aspect of reduced gravity provides new insight into thermal management in diverse applications such as aerospace, microgravity, environments and high-altitude operations. Further, the incorporation of both heat source and sink into the prescribed mathematical model gives a more comprehensive understanding of heat transfer dynamics. Moreover, this study focuses on a moving surface introducing a dynamic component to the analysis. By combining principles from fluid dynamics, thermodynamics and applied physics, this paper promotes an interdisciplinary approach, which could pave the way for further research in related fields.

Optimal Control of Sediment Deposition in Shallow Waters

ABSTRACTThis article presents an algebraic method for designing an initial bed profile to ensure minimal sediment deposition in open channels. We address the hyperbolic system of partial differential equations, known as the Saint‐Venant‐Exner system, which describes the behavior of water flow and sediment transport. By applying a priori estimation techniques and reformulating the problem as an optimal control problem, we construct a minimizing control law, via the adjoint method. This control law ensures the stability of the system and minimizes sediment deposition. In addition, a finite volume scheme is implemented to numerically solve both the direct system (SVE) and its adjoint system. Numerical simulations are provided for illustration. In theses simulations, an experimental rectangular channel with a unit width and a length 100m was considered. First, based on the control result, a bed profile that minimizes sediment deposition while simultaneously stabilizing the system's state was identified. Then, additional simulations were conducted to illustrate the sensitivity of the control solution to variations in sediment parameters and to the initial conditions.

Numerical simulation of magnetised nanofluid flow with improved finite difference method for drug delivery application

AbstractThe numerical modelling and simulation of blood‐based nanofluids provide significant insights into medical applications. The integration of nanotechnology with applied mathematics enables researchers to investigate and develop devices that analyse the impact of various substances on blood, facilitating progress in drug development and targeted drug delivery systems. However, simulating the mechanisms that regulate the physical phenomena related to blood‐based nanotechnology is both crucial and complex. The current investigation models and numerically simulates the impacts of two nanoparticles, gold and titanium oxide, using pure blood (animal blood) as the base fluid over a radially extended rotating disk. The effects of radiation, the magnetic field, and Cattaneo‐Christov heat flux are also considered to enrich the mathematical model with actual physical phenomena. We also discuss the shape effect of various nanoparticles, including cylinders, bricks, spheres, blades, and platelets. We convert the governing system of partial differential equations into a system of ordinary differential equations using similarity transformation. We propose an enhanced finite difference method, the Keller Box technique, to numerically estimate the governing system in nanotechnology. Increasing Rd elevates the temperature profile, indicating an increase in the temperature of the boundary layer. The sphere‐shaped nanoparticles are the best at transferring heat, while the platelet‐shaped nanoparticles have the best local Nusselt number and skin friction coefficient. The outcomes of this research may facilitate progress in biology, materials science, and drug delivery system management.

MHD peristaltic flow of nanofluid through curved channel with Joule heating and variable viscosity

AbstractOur goal is to examine the peristaltic flow of a nanofluid (Cu/blood) in a curved channel using magnetohydrodynamics (MHD). We are interested in the effects of adding Joule heating and varying viscosity. When creating the governing equations for the system, it is expected that it will have a long wavelength and a low Reynolds number. The system is analysed taking into account the effects of changing viscosity, MHD and Joule heating. Suitable similarity transformations are applied to turn the original system of partial differential equations into a set of nonlinear ordinary differential equations. These transformed equations are then solved in Mathematica by utilizing the built in function NDSolve, which provides the numerical results. The study employs this technique to gain a better understanding of how the variables involved impact a number of flow characteristics, including temperature, velocity, pressure gradient and stream function. In particular, the research demonstrates that for higher magnetic parameter values, the velocity on the two walls of the channel shows an opposite tendency. Additionally, a rise in the Brinkman number is linked to an increase in temperature. The pressure gradient rises in tandem with the volume % of nanoparticles in the nanofluid. Similarly, the streamline graphs demonstrate that the trapped bolus increases with the viscosity coefficient. Overall, this study provides valuable insight into the complex relationships among MHD, changing viscosity and Joule heating in peristaltic nanofluid flow along a curved channel. The study contributes to our understanding of these mechanisms and possibly has ramifications for several technical and biomedical fields.

Structural form finding using the Stress Density Method: Well-posedness and convergence of numerical solutions

Structural form finding using the Stress Density Method: Well-posedness and convergence of numerical solutions

Hybrid nanofluids and bioconvection: insights into bacterial behavior and particle interaction

The collective movement or flow of microorganisms, such as bacteria or algae, brought on by their biological activity is called bioconvection. This study investigates a hybrid nanofluid’s unstable boundary layer stagnation point flow through a permeable sheet containing nanoparticles and gyrotactic bacteria. The potential applications of this finding in bioconvection, which is critical to ecological and biotechnological processes, make it significant. The study takes a mathematical modeling approach, using group-theoretical methods to convert a system of partial differential equations (PDEs) into a system of ordinary differential equations (ODEs). This allows for a thorough examination of the hybrid nanofluid’s flow characteristics. The latest investigation was spurred by the need to examine several parameters, such as the Prandtl number Pr, Peclet number Pe, Brownian motion coefficient DB, microorganism diffusion coefficient DN, thermophoresis diffusion coefficient DT, temperature ratio δT, concentration difference ΔC, and Schmidt number SC. As the value of ΔC increases, the velocity also rises. On the other hand, when Pr and δT increase, the velocity decreases. The temperature rises in proportion to the levels of DB. The values of DT, δT, and DB increase along with the nanoparticles. However, it decreases when ΔC and SC increase. Elevations in DN are associated with increased densities of microorganisms. However, it drops when Pr, DT, and Pe increase. This study adds to the body of knowledge on bioconvection by addressing the interactions between nanoparticles and gyrotactic microorganisms in hybrid nanofluids and their previously unstudied consequences in a stagnation point flow setting, which make it novel and not discussed before.

Construction of Normal‐Regular Solutions of Inhomogeneous System of Partial Differential Equations of Second Order

ABSTRACTIn this paper, we consider an inhomogeneous system of second‐order partial differential equations with regular singularities at the point (0, 0). The Frobenius‐Latysheva method is used to find normally regular solutions. A universal approach to constructing solutions in the vicinity of regular singularities has been developed, the number of linearly independent partial solutions has been determined, and compatibility and integrability conditions have been investigated. Concrete examples demonstrate the use of the method of uncertain coefficients to obtain partial solutions of inhomogeneous systems.

On a Cross-Diffusion Model in Ecohydrology: Theory and Numerics

In this paper, we consider a version of the mathematical model introduced in (Wang et al. in Commun. Nonlinear Sci. Numer. Simul. 42:571–584, 2017) to describe the interaction between vegetation and soil water in arid environments. The model corresponds to a nonlinear parabolic coupled system of partial differential equations, with non-flux boundary conditions, which incorporates, in addition to the natural diffusion of water and plants, a cross-diffusion term given by the hydraulic diffusivity due to the suction of water by the roots. The model also considers a monotonously decreasing vegetation death rate capturing the infiltration feedback between plants and ground water. We first prove the existence and uniqueness of global solutions in a large class of initial data, allowing non-regular ones. These solutions are in a mild setting and under additional regularity assumptions on the initial data and the domain, they are classical. Second, we propose a fully discrete numerical scheme, based on a semi-implicit Euler discretization in time and finite element discretization (with “mass-lumping”) in space, for approximating the solutions of the continuous model. We prove the well-posedness of the numerical scheme and some qualitative properties of the discrete solutions including, positivity, uniform weak and strong estimates, convergence towards strong solutions and optimal error estimates. Finally, we present some numerical experiments in order to showcase the good behavior of the numerical scheme including the formation of Turing patterns, as well as to validate the convergence order in the error estimates obtained in the theoretical analysis.

Significance of radial magnetic field and Hall current on magnetic nanoparticle-assisted drug delivery and therapeutics in cylindrical vessels with bioconvection effects

Blood flow patterns are crucial for diagnosing circulatory disorders such as arteriosclerotic disease, promoting bioengineers and medical scientists to focus on analyzing blood flow dynamics within the circulatory system. In this study, we examined the influence of a radial magnetic field on blood flow mediated by magnetic nanoparticles in the presence of Hall current and ion-slip effects. Also, we explored the impact of critical phenomena, which include viscous dissipation, Brownian motion, thermophoresis, and bioconvection caused by micro-organisms. A system of partial differential equations mathematically represents the considered model, and the resulting dimensionless equations were solved using numerical techniques. The physical quantities, including velocity, temperature, and concentration distributions of blood, density profiles of organisms, surface drag force, volume flow rate, heat transfer coefficient, mass transfer, and micro-organism mass transfer coefficients, are calculated and presented by graphical and tabular perspectives. It is remarked that the magnetite nanoparticles are more effective than zinc oxide for controlling blood flow under a radial magnetic field. In addition, the blood velocity at the center of the vessel increased by 14.65%, and 11.74% as the Hall parameter rose from 0.2 to 1.0 for magnetite-infused and zinc oxide-infused blood, respectively. Moreover, the blood concentration and mass transfer rates along the vessel were enhanced by advancing Brownian motion parameters. The results of this research hold substantial potential for applications in targeted drug delivery, magnetic therapy, and therapeutic hyperthermia treatments.

Resummation of multistress tensors in higher dimensions

In the context of holographic conformal field theories (CFTs), a system of linear partial differential equations was recently proposed to be the higher-dimensional analog of the null-state equations in d=2 CFTs at large central charge. Solving these equations in a near-light-cone expansion yields solutions that match the minimal-twist multistress tensor contributions to a heavy-light four-point correlator (or a thermal two-point correlator) computed using holography, the conformal bootstrap, and other methods. This paper explores the exact solutions to these equations. We begin by observing that, in an expansion in terms of the ratio between the heavy operator’s dimension and the central charge, the d=2 correlator involving the level-two degenerate scalars at each order can be represented as a Bessel function; the resummation yields the Virasoro vacuum block. We next observe a relation between the d=2 correlator and the d=4 near-light-cone correlator involving light scalars with the same conformal dimension. The resummed d=4 correlator takes a simple form in the complex frequency domain. Unlike the Virasoro vacuum block, the resummation in d=4 leads to essential singularities. Similar expressions are also obtained when the light scalar’s dimension takes other finite values. These CFT results correspond to a holographic computation with a spherical black hole. In addition, using the differential equations, we demonstrate that the correlators can be reconstructed via certain modes. In d=2, these modes are related to the Virasoro algebra. Published by the American Physical Society 2025

Hyperbolic extensions of constrained PDEs

Systems of partial differential equations (PDEs) comprising a combination of constraints and evolution equations are ubiquitous in physics. For both theoretical and practical reasons, such as numerical integration, it is desirable to have a systematic understanding of the well-posedness of the Cauchy problem for these systems. In this article, we first review the use of hyperbolic reductions, where the evolution equations are singled out for consideration. We then examine in greater detail the extensions, namely, systems in which constraints are evolved as auxiliary variables alongside the original variables, resulting in evolution systems with no constraints. Assuming a particular structure of the original system, we provide sufficient conditions for the strong hyperbolicity of an extension. Finally, this theory is applied to the examples of electromagnetism and a toy model of magnetohydrodynamics.

Computational Insights Into Nanoscale Heat Dynamics of Chemically Reactive and Magnetized Carreau Hybrid Bio‐Nanofluid Using a Multilayer Supervised Neural Computing Scheme

ABSTRACTThe use of well‐designed nanoparticles in blood fluid can enhance heat transfer during medical interventions by improving thermophysical characteristics. It enables for targeted heat delivery to specific sites by increasing surface area for better heat exchange, which is crucial in more efficient treatments. The current attempt emphasizes on the enhanced thermal transport mechanism in an aluminium alloy suspended Copper‐based blood nanofluid over an inclined cylindrical surface containing motile gyrotactic microbes. The Carreau fluid viscosity model is implemented to expose the intricate nature of bio‐nanofluid, while the heating source is used to simulate the bio‐convective heat transport mechanism. In addition, the viscosity of hybrid bio‐nanofluids exhibits temperature effects that depend on nanoparticle volume friction dependencies related to the dynamics of spherical and cylindrical shapes with distinct shape factors. The physical generated system of partial differential equations (PDEs) is derived and then transformed into a dimensionless system of ordinary differential equations (ODEs) using similarity functions. The resulting system is reduced into first‐order differential equations and a numerical solution is obtained by using a hybrid computational procedure. The trend of fluid profiles is examined by mean of governing parameters. Results are interpreted via tabular data and MATLAB visualization. It is observed that gravity and surface friction impede the flow direction with inclined magnetic field orientation which causes a decrease in velocity and an increase in the temperature profile. A declining trend is noted in the microbe profile due to higher values of the Peclet number and numeric growth in the value of the motile microbe's factor. Heat transport rate and drag force coefficients for both spherical and cylindrical nanoparticles differ by reasonable amounts. The proposed results build a bridge between traditional computational‐based simulations and advanced ANN‐based approaches, establishing a robust foundation for advanced applications in biomedical engineering.

Study of the Solution of one Nonlinear Integro-differential Equation of Fourth Order.

There is considered a nonlinear integro-differential equation in partial derivatives of the fourth order, the existence and uniqueness of a local solution are proved, and the conditions for the existence of a local solution are determined. The proof is carried out using the method of an additional argument. Recently, this method has been used to reduce nonlinear differential equations in partial derivatives of high order into integral equations and systems of integral equations. This method was developed by Kyrgyz scientists and used to solve nonlinear differential equations in partial derivatives of the first order. Currently, the method of the additional argument is being developed for various new classes of nonlinear partial differential equations of higher order and systems of nonlinear partial differential equations. The nonlinear partial differential equation of the fourth order discussed in the article is presented in operator form, with the method of the additional argument applied consecutively several times. As a result, an integral equation is derived, and the existence and uniqueness of the solution to this integral equation are determined using the contraction mapping principle. The unknown function in the integral equation contains an additional argument, and by equating this additional argument to time, a local solution to the original initial value problem can be obtained. The results presented in the article can be applied in the study of solutions to other nonlinear differential and integro-differential equations of higher order.

Feedback stabilisation of tank-liquid system with robustness to surface tension

We construct a robust stabilising feedback law for the viscous Saint-Venant system of Partial Differential Equations with surface tension and without wall friction. The Saint-Venant system describes the movement of a tank which contains a viscous liquid. We assume constant contact angles between the liquid and the walls of the tank and we achieve a spill-free exponential stabilisation with robustness to surface tension by using a Control Lyapunov Functional. The proposed Control Lyapunov Functional provides a parameterised family of sets which approximate the state space from the interior. Based on the Control Lyapunov Functional, we construct a nonlinear stabilising feedback law which ensures that the closed-loop system converges exponentially to the desired equilibrium point in the sense of an appropriate norm.

Generalized (ψ,φ)-contraction to investigate Volterra integral inclusions and fractal fractional PDEs in super-metric space with numerical experiments

Abstract This article demonstrates the behavior of generalized ( ψ , φ \psi ,\varphi )-type contraction mappings involving expressions of rational-type in the context of super-metric spaces. In this direction, we obtained unique and common fixed point results for a pair of mappings. The obtained results are then utilized to establish some corollaries. Moreover, numerical examples and applications related to the system of integral inclusions and fractal fractional partial differential equations have been presented to validate the established results. The central objective of this research is to provide a more comprehensive framework for generalizing classical results in the context of super-metric space.

Elementary Wave Interactions in Two‐Layer Blood Flow Model

ABSTRACTIn this work, we consider the Riemann problem for a hyperbolic system of partial differential equations (PDEs) governing two‐layer blood flows through arteries. The characteristics of Riemann invariants lead us to employ a projection of the elementary curves onto the reduced phase plane. Furthermore, exploiting the properties of these elementary curves, we establish the solution of the associated Riemann problem in this phase plane. Moreover, we have provided a necessary and sufficient condition on Riemann data for the existence of a vacuum. Finally, we investigate all possible cases of wave interaction rigorously in the phase plane.

Математическое описание процесса автоклавной обработки изделий из ячеистого бетона как объекта с распределенными параметрами

A mathematical description of the autoclave treatment process for cellular concrete products has been developed as a non-stationary control object under conditions of parameter distribution in the form of a system of partial differential equations. The equations take into account heat and mass transfer both in the autoclave medium and in the cellular concrete masses, as well as the process of steam diffusion through the boundary layer and internal heat release during the synthesis of hydro-silicates in the structure of the autoclaved raw material in the form of tobermorite. The resulting system of equations is linked with boundary conditions according to the scheme proposed in the work. Based on the obtained system of partial differential equations, and considering the developed scheme of boundary conditions, a calculation scheme for the autoclave and its corresponding computational model in the SOLIDWORKS Flow Simulation software environment have been created. Numerical experiments were conducted to assess the temperature dynamics at the center and surface of the masses, as well as in the autoclave medium. A comparison of the results of numerical modeling with known data from field experiments by determining the root mean square deviation of temperature values at the center and surface of cellular concrete masses confirmed the adequacy of the developed model under the accepted assumptions. The data obtained from numerical modeling can be used in the synthesis of automated control systems with the autoclave treatment process to determine the temperature gradient in the volume of autoclaved cellular concrete, which, in turn, will optimize production processes and improve the quality of the final product while minimizing energy consumption for its production.

Nonlinear partial differential equations in neuroscience: From modeling to mathematical theory

Many systems of partial differential equations (PDEs) have been proposed as simplified representations of complex collective behaviors in large networks of neurons. In this survey, we briefly discuss their derivations and then review the mathematical methods developed to handle the unique features of these models, which are often nonlinear and nonlocal. The first part focuses on parabolic Fokker–Planck equations: the Nonlinear Noisy Leaky Integrate and Fire neuron model, stochastic neural fields in PDE form with applications to grid cells and rate-based models for decision-making. The second part concerns hyperbolic transport equations, namely, the model of the Time Elapsed since the last discharge and the jump-based Leaky Integrate and Fire model. The last part covers some kinetic mesoscopic models, with particular attention to the kinetic Voltage-Conductance model and FitzHugh–Nagumo kinetic Fokker–Planck systems.

Alternative Transient and Steady State Analysis of Some Gaver’s Parallel Systems

This paper presents one of the most interesting generalizations of Gaver’s basic two-unit parallel system sustained by a cold standby unit and attended by a repairman with multiple vacations. The system was studied using the supplementary variables technique, like many other similar semi-Markov systems. D.P. Gaver, Jr. was the first to apply this method for constructing and studying reliability models, and since then it has been then widely used by other researchers to study various reliability problems. As a result, non-classical boundary value problem of mathematical physics with nonlocal boundary conditions has been obtained. Until now, a solution to this problem was obtained in terms of Laplace transforms. Naturally, the most significant part of the problem is a system of partial differential equations (Kolmogorov forward equations). In this study, we demonstrate that Kolmogorov equations are redundant and we can solve the problem by avoiding the necessity of using them. We present here a novel, purely probabilistic approach. The results are formulated as rigorous mathematical statements, offering a significant simplification in the reliability analysis of stochastic systems. Our findings show that this novel approach can be applied to study both semi-Markov and some non semi-Markov models where the supplementary variables technique is used.

On Mathematical Analysis of a Micro-Scale Boltzmann-Maxwell’s PDEs Model for a Plasma Flow Influence by Non-linear Sinusoidal External Electric Field: Novel Irreversibility Analysis of Plasma Kinetic Theory

On Mathematical Analysis of a Micro-Scale Boltzmann-Maxwell’s PDEs Model for a Plasma Flow Influence by Non-linear Sinusoidal External Electric Field: Novel Irreversibility Analysis of Plasma Kinetic Theory