System Of PDEs Research Articles

A class of symmetric-hyperbolic PDEs modelling fluid and solid continua

We generalize a new symmetric-hyperbolic system of PDEs proposed in [ESAIM:M2AN 55 (2021) 807-831] for Maxwell fluids to a class of systems that define unequivocally multi-dimensional visco-elastic flows. Precisely, within a general setting for continuum mechanics, we specify constitutive assumptions i) that ensure the unequivocal definition of motions satisfying widely-admitted physical principles, and ii) that contain [ESAIM:M2AN 55 (2021) 807-831] as one particular realization of those assumptions. The new class can capture the mechanics of various materials, from solids to viscous fluids, possibly with temperature dependence and heat conduction.

Output-feedback stabilization of an underactuated network of interconnected hyperbolic PDE systems

Output-feedback stabilization of an underactuated network of interconnected hyperbolic PDE systems

Composite Learning Based Adaptive Control of Linear 2×2 Hyperbolic PDE Systems.

This article considers the adaptive stability control of a class of 2×2 linear hyperbolic PDE systems. The PDE model is subject to constant but in-domain and boundary unknown parameters. A novel adaptive controller is developed by leveraging the swapping design technique and composite parameter learning law. With swapping design, several linear and static combinations, including carefully designed filters, unknown parameters, and error terms, are constructed to express the system states. From the static combinations, a composite learning based forgetting-factor least squares law is introduced to guarantee exponential parameter convergence without the persistent excitation (PE). Although inaccurate parameter estimation in the adaptive backstepping control results in asymptotic stability of the system, accurate parameter estimation ensures the exponential convergence of closed-loop system and concomitantly improves the transient performance. Finally, a comparative numerical simulation is performed to validate the effectiveness and advantage of the developed adaptive control scheme.

Backstepping Control of a Hyperbolic PDE System With Zero Characteristic Speed States

Backstepping Control of a Hyperbolic PDE System With Zero Characteristic Speed States

Stabilization and Optimal Control of Interconnected SDE -Scalar PDE System

Stabilization and Optimal Control of Interconnected SDE -Scalar PDE System

Aspects of chemical reaction and mixed convection in ternary hybrid nanofluid with Marangoni convection and heat source

The proposed study examines the effect of inclined magnetic field on a ternary hybrid nanofluid flow that is axisymmetric thermo-solutal Marangoni convective over an infinite disc. Some well-known uses of Marangoni convection include semiconductor production, atomic reactors, crystal growth, fine art mechanisms, melting, thin-film stretching and welding processes. The non-uniform heat generation and viscous dissipation are taken into account. The thermal conductivity and diffusivity coefficient are presumed to vary inversely with linear function of temperature and concentration. The ternary hybrid nanofluid, which consists of silicon dioxide ([Formula: see text]), iron oxide ([Formula: see text]), molybdenum disulfide ([Formula: see text]) and ethylene glycol as base liquid, undergoes an energy transition to improve heat transfer. The system of PDEs is transformed into nonlinear ordinary differential equations (ODEs) by using the appropriate transformations. Using the BVP4C method, this problem is numerically solved. The heat and mass phenomena rates on flow behavior are investigated using tables and graphs to address the impact of several physical and flow parameters on velocity, concentration, and thermal profiles. By increasing the Marangoni convection parameter, the surface tension gradient gets stronger, leading to more efficient heat and mass transfer inside the liquid as well as stronger induced flows. As the temperature and concentration profiles decrease, the outcome is a more consistent dispersion of these properties throughout the liquid.

Significance of Melting Heat Transfer, Inclined Magnetic Field, and Thermal Radiation on the Dynamics of Williamson Nanofluid Above a Stretching Sheet

This study investigates the melting heat transfer in MHD flow that occurs when the flow is past a stretched sheet. With the use of the boundary layer concept, a set of non-linear ODEs is derived from a system of PDEs of motion and energy. After transforming the non-linear ODEs and their boundary conditions into dimensionless form with the use of appropriate similarity conversions, the equations are then numerically solved by employing the MATLAB inbuilt solver bvp4c method in conjunction with the shooting methodology. Through the use of a variety of charts and tables, the authors analyze and describe in depth the effects that relevant factors have on a variety of flow fields. As melting factor (Me) increases, it is discovered that the effect of Me on fluid temperature and velocity distributions decreases. Williamson parameter and inclined magnetic profile are shown to have decreasing impact on velocity and momentum parameter. After that, the findings are compared with the earlier published findings in limited situations of the issue, and it is discovered that they are in exceptional specify with those findings.

Flux vector splitting schemes applied to a conservative 1D blood flow model with transport for arteries and veins

We present novel flux splitting-based numerical schemes for the 1D blood flow equations with an advection equation for a passive scalar, considering tube laws that allow to model blood flow in arteries and veins. Our schemes are inspired by the original flux vector splitting approach of Toro and Vázquez-Cendón (2012) and represent an extension of the work proposed by Toro et al. (2024), which addressed tube laws suitable for describing blood flow in arteries. Our schemes separate advection terms and pressure terms, generating two different systems of PDEs: the advection system and the pressure system, both of which have a very simple eigenstructure compared to that of the full system. We propose discretization schemes of the Godunov type that are simple and efficient. These qualities are evaluated on a suite of test problems with exact solution. A detailed efficiency analysis is performed in order to illustrate situations in which the proposed methodology results advantageous with respect to standard approaches.

Software sensors design for a class of non linear coupled PDE systems: The Vlasov–Poisson dynamical system

This paper presents the synthesis of a state observer for the 1D×1D Vlasov–Poisson (VP) equations. To derive the LPV (linear parameter-varying) formulation of the system, the Vlasov equation is approximated using the discontinuous Galerkin method and the Poisson problem is approximated using the Raviart–Thomas mixed finite element approach. The paper demonstrates the asymptotic and exponential stability of the discretized VP system. Furthermore, the synthesis is extended to include H∞ state estimation. A simulation code has been developed to validate the obtained results.

Numerical analysis of Marangoni convected dusty second-grade nanofluid flow in a suspension of chemically reactive microorganisms

The present work investigates the Magneto-Marangoni convective flow of dusty second-grade nanofluid flow over a porous sheet in the presence of stagnation-point. The effects of chemical reaction, gyrotactic microorganisms are taken into account along with variable mass diffusivity, thermal radiation, and non-uniform heat source. The thermal reflection of nanoparticle molybdenum disulfide ([Formula: see text]) when it interacts with kerosene oil (KO) base fluid is determined. Nanoparticles and dust particles are presumptively spherical. The arising system of nonlinear PDEs is transformed into nonlinear ODEs by applying similarity transformations. Conservation of momentum, mass, concentration, energy, and density of microorganisms are used in to model the flow equations. The numeric expressions are computed by using the RKF-45th based shooting scheme. The outcomes revealed that the microorganisms, concentration, and temperature profiles decay in both phases, but velocity profiles of dust and fluid phases rise as the Marangoni ratio parameter grows.

Multiscale modelling of fluid transport in vascular tumours subjected to electrophoresis anticancer therapies

Electrophoresis facilitated cancer treatment has demonstrated experimental efficacy in enhancing drug delivery within vascularised tumours. However, the lack of realistic mathematical models with direct measurements in the context of electrochemotherapy poses a challenge. We investigate the impact of an applied electric potential on the flow of Darcian-type fluid occurring in two distinct phases: the tumour and healthy regions. We employ the asymptotic homogenisation technique, assuming that the macroscale of the tumour domain is larger than the microscale characterised by vessel heterogeneities. We retain information about the microstructure by encoding information in the homogenised coefficients. We take into account both vascularisation and the microscale variations of the leading order and fine scale electric potential. The resulting effective differential problem reads as a Darcy-type system of PDEs, where the flow is driven by an effective source. The novel model can be used to predict the effect of an applied electric field on cancerous biological tissues, paving a new way of improving current electrochemotherapy protocols.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

Energy transmission through radiative ternary nanofluid flow with exponential heat source/sink across an inclined permeable cylinder/plate: numerical computing

The steady two-dimension (2D) ternary nanofluid (TNF) flow across an inclined permeable cylinder/plate is analyzed in the present study. The TNF flow has been examined under the consequences of heat source/sink, permeable medium and mixed convection. For the preparation of TNF, the magnesium oxide (MgO), cobalt ferrite (CoFe2O4) and titanium dioxide (TiO2) are dispersed in water. The rising need for highly efficient cooling mechanisms in several sectors and energy-related processes ultimately inspired the current work. The fluid flow and energy propagation is mathematically described in the form of coupled PDEs. The system of PDEs is reduced into non-dimensional forms of ODEs, which are further numerically handled through the Matlab package (bvp4c). It has been observed that the results display that the porosity factor advances the thermal curve, whereas drops the fluid velocity. The effect of heat source/sink raises the energy field. Furthermore, the plate surface illustrates a leading behavior of energy transport over cylinder geometry versus the variation of ternary nanoparticles (NPs). The energy dissemination rate in the cylinder enhances from 4.73 to 11.421%, whereas for the plate, the energy distribution rate boosts from 6.37 to 13.91% as the porosity factor varies from 0.3 to 0.9.

Flux-corrected transport stabilization of an evolutionary cross-diffusion cancer invasion model

In the present work, we investigate a model of the invasion of healthy tissue by cancer cells which is described by a system of nonlinear PDEs consisting of a cross-diffusion-reaction equation and two additional nonlinear ordinary differential equations. We show that when the convective part of the system, the haptotaxis term, is dominant, then straightforward numerical methods for the studied system may be unstable. We present an implicit finite element method using conforming P1 or Q1 finite elements to discretize the model in space and the θ-method for discretization in time. The discrete problem is stabilized using a nonlinear flux-corrected transport approach. It is proved that both the nonlinear scheme and the linearized problems used in fixed-point iterations are solvable and positivity preserving. Several numerical experiments are presented in 2D to demonstrate the performance of the proposed method.

Some computational tests for inverse conductivity problems based on vector, variational principles: The 2D case

We focus on several direct vector, variational principles to tackle the practical recovery problem of an unknown conductivity coefficient from boundary measurements. Though the problem is classical and have received a lot of attention because of its practical significance, the variational methods we explore are not so. Despite difficulties associated with the vector nature of the problems, including lack of (quasi, poly)-convexity, experiments show remarkable performance of some of the functionals examined. Beyond the specific meaning of such computations for inverse conductivity problems, the task of approximating the optimal solutions of vector, variational problems, as in hyperelasticity models or non-linear PDE systems at the level of optimality, is also interesting on its own right.

Generalized Bernoulli–Laguerre Polynomials: Applications in Coupled Nonlinear System of Variable-Order Fractional PDEs

Generalized Bernoulli–Laguerre Polynomials: Applications in Coupled Nonlinear System of Variable-Order Fractional PDEs

Investigation of cross-diffusion effect on radiative Jeffery-Hamel flow in convergent/divergent stretchable channel with Lorentz force and Joule heating

This article documents the hydro-magnetic, incompressible Jeffery-Hamel flow in convergent/divergent (CD) channels with stretchable walls. Although there seem to be several works on the subject in the available literature, only a few attempts have been made to work into mass and heat transfer analysis. Consequently, novelty of present study is to investigate the significance of Soret/Dufour. Joule heating, and chemical reaction effects on Jeffery-Hamel flow. The governing coupled system of PDEs showing the mathematical framework of the current physical problems is converted into ODEs through proper transformations. The numerical solution of the obtained couple of ODEs is simulated with the assistance of Mathematica-11 solver NDSolve. The computed outcomes depict that Lorentz force developed due to the magnetic field decays the fluid motion for both convergent and divergent channels. Furthermore, incorporating the heat source and Dufour effects yields a sustainable increase in temperature distribution, whereas the Soret number has reverse features on concentration. Furthermore, the numerical outcomes reflect the significant variation in Nusselt number against magnetic, radiation parameter and Soret number.

Application of machine learning in the investigation of entropy generation incurred during electroosmotic flows of Casson nanofluid

The present research article delves into the application of machine learning techniques in the study of entropy generation arising from the electro‐MHD Casson nanofluid flow in a symmetric microchannel. The inertial effect on fluid flow through a non‐Darcy porous medium is considered where we have taken into account the velocity, thermal, and concentration slips at boundaries. The resulting system of PDEs is converted to system of ODEs using appropriate wave frame transformations. The metamorphosed system is solved using bvp4c routine of MATLAB. The impact of several factors represented by respective parameters on velocity, nanofluid temperature, fraction of nanoparticles, entropy generation, and Bejan number are shown through graphs. This study also employs a machine learning technique in which Particle Swarm Optimization (PSO) algorithm has been used in conjunction with an Artificial Neural Network (ANN) to develop a predictive model capable of optimizing entropy generation within this specific fluidic context. Such an analysis could be useful in determining the optimum entropy generation in pulsatile transport of physiological fluid through intestine. With the use of above mentioned machine learning technique, the considered model offers the minimum entropy generation for specific values of the electroosmotic parameter, dimensionless temperature difference, and velocity slip parameter. It is observed that the flow with minimum electroosmosis effect and maintained at a specific velocity slip gives the optimal entropy generation.

On state feedback control of a class of NonLinear PDE systems in finite dimension

In this note, we propose some new results on state feedback stabilization of a class of non linear PDE systems that are described by Vlasov-Poisson equations. For doing so, thanks to the Galerkin discontinuous discretization approach, we establish first an explicit state space model in finite dimension. A state feedback controller is then proposed. One of the main futures is that the asymptotic stabilization may be guaranteed under an easy tractable LMI condition. Extension to the observer based control case, in the presence of disturbances or not, was also established. We provide an LMI condition to assure convergence in the H∞ sense.

High-Order ADER Discontinuous Galerkin Schemes for a Symmetric Hyperbolic Model of Compressible Barotropic Two-Fluid Flows

This paper presents a high-order discontinuous Galerkin (DG) finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) model of compressible two-phase flow, introduced by Romenski et al. in [59, 62], in multiple space dimensions. In the absence of algebraic source terms, the model is endowed with a curl constraint on the relative velocity field. In this paper, the hyperbolicity of the system is studied for the first time in the multidimensional case, showing that the original model is only weakly hyperbolic in multiple space dimensions. To restore the strong hyperbolicity, two different methodologies are used: (i) the explicit symmetrization of the system, which can be achieved by adding terms that contain linear combinations of the curl involution, similar to the Godunov-Powell terms in the MHD equations; (ii) the use of the hyperbolic generalized Lagrangian multiplier (GLM) curl-cleaning approach forwarded. The PDE system is solved using a high-order ADER-DG method with a posteriori subcell finite-volume limiter to deal with shock waves and the steep gradients in the volume fraction commonly appearing in the solutions of this type of model. To illustrate the performance of the method, several different test cases and benchmark problems have been run, showing the high order of the scheme and the good agreement when compared to reference solutions computed with other well-known methods.