System Of PDEs Research Articles

Dynamics of magnetized viscous dissipative material of hybrid nanofluid with irregular thermal generation/absorption

The intriguing thermophysical properties of nanoparticles have considerable potential in thermal transportation for engineering applications. The inclusion of solid nanoparticles in the base fluid raises the fluid's thermal conductivity, which improves thermal transfer characteristics. Thus, an analysis is performed by assuming the Tiwari and Das model, for the flow and energy transportation aspects of hybrid nanoparticles in a shrinkable porous medium. The current study focuses on the thermal heat transfer performance on the dynamic of ethylene glycol-based titania and alumina nanoparticles. The flow-controlling system of PDEs is transformed into a system of non-linear ODEs by using similarity transformations, and the resulting equations are then solved numerically through the solver bvp4c in Matlab function with a limit of tolerance 10−6. The multiple branches of solutions are noted for distinct values of suction and shrinking parameters. The outcomes reveal that the inclusion of titania and alumina nanocomposites improves the thermophysical properties of the base fluid significantly. The findings indicate that hybrid nanofluids are more effective for increasing the thermal transport rate. Further, in the upper branch solution, the friction drags, and energy transportation rate raises for the higher influence of suction strength while it declines for the lower branch.

Polynomial decay of a linear system of PDEs via Caputo fractional‐time derivative

An in‐depth study and analysis of the stability of one‐dimensional Linear System of PDEs via Caputo time fractional derivative (LSCFD) was presented. We proved some stability results for the LSCFD in different Hilbert spaces. Indeed, by using Fourier analysis method and the properties of the Mittag–Leffler Function (MLF), some polynomial stability results for LSCFD have been established. Finally, as an application, we used finite difference methods well suited to integer and fractional order derivatives, and performed some numerical experiments to confirm the theoretical results.

Lorentz force and solar energy case study on CNTs and pollytetrafluoroethylene (PTFE) paraffin oil-based hybrid nanofluid flow through a porous divergent/convergent channel

The concept of thermal transfer has fundamental implications in many areas, from the most basic physiological processes to the most complex technological systems. Kerosene oil, often known as paraffin oil, is frequently used as a fuel for jet lighting, engines and heating processes in many technologies. The rate of heat transmission is the fundamental requirement of all phenomena. But this cannot be achieved from base fluid like paraffin oil due to its less efficiency to rise the rate of heat transmission and save energy lost due to high temperatures. Therefore, current analysis is to explore effect of the Lorentz force, viscous dissipation and solar radiation on kerosene oil with the combination of single-walled carbon nanotubes (SWCNTs) and Polytetrafluoroethylene (PTFE) in a porous convergent/divergent and stretching/shrinking channel. SWCNTs and PTFE are added to kerosene oil to produce the hybrid nanofluid. The designed problem is modeled in terms of partial differential equations PDEs. The nonlinear PDEs system is then reduced into ODEs using the transformations framework. The ND Solve technique is applied to numerically simulate the reduced boundary value problem, and the results are sketched and discussed. Obtained results demonstrate that velocity profile increase for the divergent channel and a fall in the convergence channel against higher value of porosity parameter and as well as Hartmann number.

Dynamic–quadratic balancing: A computational approach to balancing and model reduction for affine nonlinear systems

AbstractIn this paper, we discuss a new approach to balancing (known as dynamic–quadratic balancing) and model reduction for affine nonlinear system. We give a fresh look to balancing in terms of the dynamics of the system, rather than simply a structural property. With this perspective, we also develop a new approach to obtaining the balancing transformation in one step, instead of a three‐step process as proposed in earlier methods. Further, we explore the relationship between quadratic balancing and input–output stability. In addition, we also develop a computational approach for obtaining the balancing transformation by solving a coupled system of PDEs or inequalities (Lyapunov/Hamilton–Jacobi type). After that, model reduction can be carried out in the conventional way using Hankel singular‐value functions or using a new criterion. Finally, we present some examples to clarify the results.

Stationary analysis of a constrained Markov fluid model with two buffers

. We consider Markov fluid models with two infinite buffers, whose fluid rates ensure that the fluid level of buffer 1 is never larger than the one of buffer 2. For the model, we derive the system of PDEs describing the transient behavior and expressions for the stationary analysis in the Laplace domain. Examples for the application of the model in the context of network access control are presented. Numerical results are compared with detailed simulation results to show that the numerical approach computes the required measures with sufficient accuracy.

Induced magnetic force and curvature effect of ternary hybrid nanofluid (jeffrey model) in ciliary peristaltic channels

ABSTRACT Ternary hybrid nanoparticles require infected blood cell base fluid flow. Blood vessels feature ciliated walls and peristaltic waves. Anemia comes from cancer treatment-induced red blood cell destruction. This study reproduces Jeffrey model blood flow through a ciliary peristaltic artery using ternary hybrid nanoparticles of gold, iron oxide, and copper. This work examines ternary hybrid nanofluid MHD ciliary peristaltic motion using Jeffrey curves. A nonlinear PDE system has heat and induced magnetic force equations. The Adomian Decomposition Method solved this system without long-wavelength approximation. Visualize how essential biological and physical elements affect velocity, temperature, trapping, magnetic force contours, induced magnetic field components, axial pressure, and normal pressure. Flow, temperature, magnetic features, and other important properties are plotted and discussed against biological and biomedical parameters. We discovered full consistency with modest scenario efforts. The increase in nanoparticles gives ternary hybrid nanofluid a larger friction force than hybrid, so its axial pressure gradient at the wall is stronger, decays at hybrid, decays further from nanofluid to base fluid, and reverses at the intermediate area. Peristaltic ciliary activity alters magnetic force contours, shrinks loops, and forms four contour groups with different implications.

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

A nonlinear crime model is generalized by introducing self- and cross-diffusion terms. The effect of diffusion on the stability of non-negative constant steady states is applied. In particular, the cross-diffusion-driven instability, called Turing instability, is analyzed by linear stability analysis, and several Turing patterns driven by the cross-diffusion are studied through numerical investigations. When the Turing–Hopf conditions are satisfied, the type of instability highlighted in the ODE model persists in the PDE system, still showing an oscillatory behavior.

Derivation and well-posedness for asymptotic models of cold plasmas

In this paper we derive three new asymptotic models for a hyperbolic–hyperbolic–elliptic system of PDEs describing the motion of a collision-free plasma in a magnetic field. The first of these models takes the form of a non-linear and non-local Boussinesq system (for the ionic density and velocity) while the second is a non-local wave equation (for the ionic density). Moreover, we derive a unidirectional asymptotic model of the latter which is closely related to the well-known Fornberg–Whitham equation. We also provide the well-posedness of these asymptotic models in Sobolev spaces. To conclude, we demonstrate the existence of a class of initial data which exhibit wave breaking for the unidirectional model.

Generalized second order vectorial ∞-eigenvalue problems

We consider the problem of minimizing the $L^\infty$ norm of a function of the hessian over a class of maps, subject to a mass constraint involving the $L^\infty$ norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the “hinged” and the “clamped” cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^\infty$ minimizer, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem. Furthermore, we establish upper and lower bounds for the eigenvalue.

Numerical Simulation of MHD Radiative Casson Nanofluid Flow over a Nonlinearly Heated Stretching Sheet Embedded in Porous Medium

This study intends to assess the heat transmission of a steady two-dimensional MHD Casson nanofluid flow across a non-linearly stretching sheet concerning viscous dissipation, heat generation, thermal radiation, and chemical radiate parameters. Mathematically, the flow is outlined as PDE systems, converted to ODE systems by a well-posed transformation, which are afterwards solved numerically in Matlab employing the fourth-order precision program (Bvp4c). The comparison indicates adequate agreement, demonstrating the correctness of the numerical strategy. It is observed that in the existence of a chemical reaction, the temperature is elevated, while the concentration profile is markedly reduced, in contrast to a scenario where no chemical reaction is taking place. In consideration of viscosity, heat radiation, thermophoresis impact, and suction or injection parameter, the Nusselt number reveals a growing tendency. In contrast, it exhibits opposing behavior with regard to the other addressed factors. The findings of this study have applications in the engineering, biomedical and industrial sectors, including food processing, polymer making, glass and fiber production, material processing, and enhanced oil recovery.

Event-triggered boundary control of an unstable reaction diffusion PDE with input delay

In chemical, biological, or population (epidemiological) processes the feedback action may be considerably delayed by time-consuming chemical measurements or biological tests. With such large delays on the control action in mind, and motivated by the fact that in some of these systems only piecewise-constant inputs can be applied between time instants at which measurements trigger changes in control, we consider the problem of event-triggered stabilization of 1-D reaction–diffusion PDE systems with input delay. The approach relies on reformulating the delay problem as an actuated transport PDE which cascades into the reaction–diffusion PDE, and on the emulation of backstepping control. The paper proposes a static (state-dependent) triggering condition which establishes the time instants at which the control value needs to be updated. It is shown that under the proposed event-triggered boundary control, there exists a minimal dwell-time (independent of the initial conditions) between two triggering times which allows to guarantee the well-posedness of the closed-loop system, and the exponential stability. The stability analysis is based on Input-to-State stability theory for PDEs and small-gain arguments. A simulation example is presented to validate the theoretical results.

Irreversibility Analysis for Magnetized and Nonlinear Convection Flow of Carreau Fluid Past a Nonlinear Surface with Higher-Order Adherence Factor

Studying entropy generation allows scientists and engineers to optimize processes, design more efficient systems, and develop sustainable technologies. These studies aim towards minimizing energy losses, reducing environmental impacts, and advancing the frontiers of thermodynamics and energy science. As such, the current investigation expunges the dynamics of heat, flow, entropy and irreversibility analysis for a magnetized and dissipative Carreau fluid flow over a nonlinearly stretching sheet. The model is subject to thermal and electrical conductivity variations, higher-order adherence factors and nonlinear convective motion over an inclined nonlinear surface. Herein, the heat source in the flow system is considered space and temperature-dependent. The mathematical model presented in PDE systems is transformed to ODEs via similarity transformation. Numerically, the solutions to the distribution profiles are approximated via Chebyshev Collocation Method (CCM) with the pseudospectra (differentiation matrix) technique. In the limited case, the results here benchmark existing literature. Our findings identified that thermal irreversibility dominated the entire flow system, further entropy generation and irreversibility enhancement requires more thermal energy. We as well elucidate that alteration of both heat source modulation affects the heat transfer coefficient while Joule heating effect generates inner energy which induces the Carreau fluid energy. Carreau fluid, Entropy generation Momentum slip, Nonlinear Convection, Pseudo spectral method

Role of nanomaterial on irreversibility and heat transport due to stretching surface driven blood flow in the view of Buongiorno and Tiwari-Das models

Several strategies are used to increase the heat transmission capabilities of ordinary fluid. Using metallic nanomaterials thethermal conductivity is modified whichcan improve the energy transport mechanism. Our primary goal is to observe the dual diffusion and volumetric friction of injected nanoparticles in blood flow simultaneously. TheEyring–Powell fluid is speculated as blood on the basis of shear thinner behavior. The entropy formation in the region of stagnationin MHD Eyring–Powell nanofluid flow is scrutinized. The flow is produced by stretchable surface. The impacts of slip mechanism viscous dissipation and melting heating are included in the current analysis. The single and two phase model are used as coupled model to inquire the effects of nanomaterials. System of PDEs are achieved and reduced to ODEs using appropriate variables. The reduced ODEs are solved using RK-4 methodology. It is evident that blood flow is greatly influenced by Lorentz forces and fluid primary parameter. On the other hand, the secondary and increased volumetric rate encourage the flow of blood. The blood temperature is rising function of thermophoresis, Eckert and magnetic parameter. Similar trend for solutal profile was noted against Brownian and Lewis number. The system irreversibility are rising with Brinkman, magnetic and diffusion variable. The coefficient of energy transport rate and frictional drag are rising function of material parameters and volumetric concentration of injected nanomaterial. It is interesting to see that nearly all profile show comparable trend with higher concentration of nanomaterials.

Hard congestion limit of the dissipative Aw–Rascle system

In this study, we analyse the famous Aw–Rascle system in which the difference between the actual and the desired velocities (the offset function) is a gradient of a singular function of the density. This leads to a dissipation in the momentum equation which vanishes when the density is zero. The resulting system of PDEs can be used to model traffic or suspension flows in one dimension with the maximal packing constraint taken into account. After proving the global existence of smooth solutions, we study the so-called ‘hard congestion limit’, and show the convergence of a subsequence of solutions towards a weak solution of a hybrid free-congested system. This is also illustrated numerically using a numerical scheme proposed for the model studied. In the context of suspension flows, this limit can be seen as the transition from a suspension regime, driven by lubrication forces, towards a granular regime, driven by the contacts between the grains.

Design of Bayesian stochastic networks for numerical treatment of Williamson fluid stretching flow model with mixed convected heat generation

In the presented article, a stochastic network paradigm through Bayesian Regularization backpropagation neural network (BRB-NN) is designed to interpret the dynamics of the Williamson fluid stretching flow model with mixed convected heat generation (WF-SFM). The governing nonlinear PDEs system of WF-SFM is reduced to a set of ODEs by incorporating appropriate transformations. The reference datasets for the anticipated BRB-NN approach are created with Adams solver for numerical solutions of WF-SFM by varying material variable W e , buoyancy factor λ , temperature characteristic parameter ε , thermal relaxation variable γ , heat source factor δ , and Prandtl numbers P r . The knacks of artificial intelligence (AI) based BRB-NN procedure is then employed on the generated dataset for WF-SFM. The bias training, biased testing, and validation of BRB-NN are conducted to compute the approximate numerical results for WF-SFM for sundry scenarios, and outputs are in good agreement in reference data that validate the worthy performance of the proposed BRB-NN which is further justified by results of absolute error, mean squared error, error histogram illustrations and regression measures. The viable performance in terms of mean square error (MSE) is achieved at levels ranging from E − 11 to E − 13 , consistently for all scenarios of WF-SFM. The accuracy and the justification of performance are effectively proven by the low level of MSE, to optimal regression metric index as well as distribution of instances error on histograms with negligible magnitude.

Liouville type theorems involving fractional order systems

Abstract In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: ( − Δ ) α / 2 u ( x ) = f ( u ( x ) , v ( x ) ) , x ∈ R n , ( − Δ ) α / 2 v ( x ) = g ( u ( x ) , v ( x ) ) , x ∈ R n . $\begin{cases}{\left(-{\Delta}\right)}^{\alpha /2}u\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n},\quad \hfill \\ {\left(-{\Delta}\right)}^{\alpha /2}v\left(x\right)=g\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n}.\quad \hfill \end{cases}$ Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang (“A direct method of moving spheres on fractional order equations,” J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu (“Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space,” Potential Anal., vol. 46, pp. 569–588, 2017).

The existence of a weak solution for a compressible multicomponent fluid structure interaction problem

We analyze a system of PDEs governing the interaction between two compressible mutually noninteracting fluids and a shell of Koiter type encompassing a time dependent 3D domain filled by the fluids. The dynamics of the fluids is modeled by a system resembling compressible Navier-Stokes equations with a physically realistic pressure depending on densities of both the fluids. The shell possesses a non-linear, non-convex Koiter energy. Considering that the densities are comparable initially we prove the existence of a weak solution until the degeneracy of the energy or the self-intersection of the structure occurs for two cases. In the first case the adiabatic exponents are assumed to satisfy max⁡{γ,β}>2, min⁡{γ,β}>0, and the structure involved is assumed to be non-dissipative. For the second case we assume the critical case max⁡{γ,β}≥2 and min⁡{γ,β}>0 and the dissipativity of the structure. The result is achieved in several steps involving, extension of the physical domain, penalization of the interface condition, artificial regularization of the shell energy and the pressure, the almost compactness argument, added structural dissipation and suitable limit passages depending on uniform estimates.

Dark energy and dark matter configurations for wormholes and solitionic hierarchies of nonmetric Ricci flows and F(R,T,Q,Tm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(R,T,Q,T_{m})$$\\end{document} gravity

We extend the anholonomic frame and connection deformation method, AFCDM, for constructing exact and parametric solutions in general relativity, GR, to geometric flow models and modified gravity theories, MGTs, with nontrivial torsion and nonmetricity fields. Following abstract geometric or variational methods, we can derive corresponding systems of nonmetric gravitational and matter field equations which consist of very sophisticate systems of coupled nonlinear PDEs. Using nonholonomic frames with dyadic spacetime splitting and applying the AFCDM, we prove that such systems of PDEs can be decoupled and integrated in general forms for generic off-diagonal metric structures and generalized affine connections. We generate new classes of quasi-stationary solutions (which do not depend on time like coordinates) and study the physical properties of some physically important examples. Such exact or parametric solutions are determined by nonmetric solitonic distributions and/or ellipsoidal deformations of wormhole hole configurations. It is not possible to describe the thermodynamic properties of such solutions in the framework of the Bekenstein–Hawking paradigm because such metrics do not involve, in general, certain horizons, duality, or holographic configurations. Nevertheless, we can always elaborate on associated Grigori Perelman thermodynamic models elaborated for nonmetric geometric flows. In explicit form, applying the AFCDM, we construct and study the physical implications of new classes of traversable wormhole solutions describing solitonic deformation and dissipation of non-Riemannian geometric objects. Such models with nontrivial gravitational off-diagonal vacuum are important for elaborating models of dark energy and dark matter involving wormhole configurations and solitonic-type structure formation.

Simulation of the cancer cell growth and their invasion into healthy tissues using local radial basis function method

Applying mathematical models to simulate dynamic biological processes has been a common practice for a long time. In recent decades, cancer research has also adopted this approach to understand how cancer cell populations grow and spread. This study focuses on a mathematical model that uses a system of PDEs to explain the time-dependent reaction–diffusion interaction among cancer cells, extracellular matrix, and matrix degradation enzymes. We use a computational method that involves the discrete Galerkin technique by employing local radial basis functions (LRBFs) as its basis to approximate the behavior of cancer cells as they grow and invade neighboring healthy tissues. This novel approach employs a smaller set of nodes to approximate the solution, instead of considering all data in the given domain of the cancer growth model. Utilizing locally supported radial basis functions, this method significantly reduces the computational volume required, in contrast to globally supported radial basis functions. Finally, we provide experimental examples to validate and illustrate the effectiveness of this new scheme in modeling the growth and behavior of cancer cells at different stages.

Numerical thermal investigation of radiative magnetohydrodynamics axisymmetric Cu-Al2O3/H2O hybrid nanofluid flow over an unsteady radially stretched surface

The current work focuses on conducting a numerical thermal investigation of nanofluid and hybrid nanofluid flow over unsteady radial stretching surfaces. To perform a comprehensive analysis, lamina and hexahedron-shaped Aluminum Oxide (Al2O3) and Copper (Cu) nanomaterial are immersed in water-based fluid. Furthermore, we take into account the impact of magnetohydrodynamics (MHD) and thermal radiation. The cylindrical coordinate system is utilised for mathematical modeling. The governing system of nonlinear PDEs is converted to a system of ODEs by utilising appropriate similarity transformation and then solved numerically in MATLAB. By using graphs and bar charts, we visually depict the impact of significant physical attributes on velocity, temperature, skin friction, and Nusselt numbers. As per the findings, The nanofluid and hybrid nanofluid flow experienced a notable deceleration due to the influence of the suction parameter (S), unsteadiness parameter (A), and magnetic parameter (M). Furthermore, hybrid nanofluid exhibits the lowest temperature and the slowest velocity. In the case of nanofluid composed of Aluminum Oxide (Al2O3) and water (H2O), the highest Nusselt number is observed along with the lowest skin friction. These recent findings hold the potential to offer advantages to academic research, lubrication processes, and the polymer industry. Abbreviations: PDE’s Partial Differential Equation; ODE’s Ordinary Differential Equation; MHD Magnetohydrodynamic