System Of Nonlinear Partial Differential Equations Research Articles

Numerical Study of Unsteady MHD Flow and Entropy Generation in a Rotating Permeable Channel with Slip and Hall Effects**Support of the National Natural Science Foundation of China under Grant Nos. 51709191 and 51706149, and Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education under Grant No. ARES-2018-10

This article investigates an unbiased analysis for the unsteady two-dimensional laminar flow of an incompressible, electrically and thermally conducting fluid across the space separated by two infinite rotating permeable walls. The influence of entropy generation, Hall and slip effects are considered within the flow analysis. The problem is modeled based on valid physical arguments and the unsteady system of dimensionless PDEs (partial differential equations) are solved with the help of Finite Difference Scheme. In the presence of pertinent parameters, the precise movement of the flow in terms of velocity, temperature, entropy generation rate, and Bejan numbers are presented graphically, which are parabolic in nature. Streamline profiles are also presented, which exemplify the accurate movement of the flow. The current study is one of the infrequent contributions to the existing literature as previous studies have not attempted to solve the system of high order non-linear PDEs for the unsteady flow with entropy generation and Hall effects in a permeable rotating channel. It is expected that the current analysis would provide a platform for solving the system of nonlinear PDEs of the other unexplored models that are associated to the two-dimensional unsteady flow in a rotating channel.

Oscillations and asymptotic convergence for a delay differential equation modeling platelet production

We analyze the existence of oscillating solutions and the asymptotic convergence for a nonlinear delay differential equation arising from the modeling of platelet production. We consider four different cell compartments corresponding to different cell maturity levels: stem cells, megakaryocytic progenitors, megakaryocytes, and platelets compartments, and the quantity of circulating thrombopoietin (TPO), a platelet regulation cytokine. Our initial model consists in a nonlinear age-structured partial differential equation system, where each equation describes the dynamics of a single compartment. This system is reduced to a single nonlinear delay differential equation describing the dynamics of the platelet population, in which the delay accounts for a differentiation time. After introducing the model, we prove the existence of a unique steady state for the delay differential equation. Then we determine necessary and sufficient conditions for the existence of oscillating solutions. Next we set up conditions to get local asymptotic stability and asymptotic convergence of this steady state. Finally we present a short analysis of the influence of the conditions at t < 0 on the proof for asymptotic convergence.

Dynamic Boundary Fuzzy Control Design of Semilinear Parabolic PDE Systems With Spatially Noncollocated Discrete Observation.

The problem of dynamic boundary fuzzy control design is investigated in this paper for nonlinear parabolic partial differential equation (PDE) systems with spatially noncollocated discrete observation. Two cases of noncollocated discrete observation in space (i.e., pointwise observation in space and local piecewise uniform observation in space) are considered, respectively. The spatially noncollocated discrete observation makes the control design very difficult. Such design difficulty can be surmounted by an observer-based feedback control technique. It is assumed that the semilinear PDEs are accurately represented by a Takagi-Sugeno fuzzy PDE model. On the basis of the obtained fuzzy PDE model, a fuzzy Luenberger-type PDE state observer with above two cases of noncollocated discrete observation in space is first proposed for exponential estimation of the PDE system state. An observer-based dynamic fuzzy controller is then constructed such that the resulting closed-loop system is exponentially stable. Sufficient conditions on existence of such fuzzy controller are developed by Lyapunov technique with variations of vector-valued Poincaré-Wirtinger inequality, and presented in terms of linear matrix inequalities. Finally, extensive numerical simulation results of two examples are provided to support the proposed design method.

Observer-Based &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$H_{\infty }$&lt;/tex-math&gt; &lt;/inline-formula&gt; Sampled-Data Fuzzy Control for a Class of Nonlinear Parabolic PDE Systems

In this paper, an observer-based $H_{\infty }$ sampled-data fuzzy control problem is addressed for a class of nonlinear parabolic partial differential equation (PDE) systems. With the aid of the modal decomposition technique, a nonlinear ordinary differential equation (ODE) model is initially derived to describe the dominant (slow) dynamics of the PDE system. Subsequently, the resulting nonlinear ODE model is accurately represented by the Takagi–Sugeno (T–S) fuzzy model. Then, based on the T–S fuzzy model, a finite-dimensional observer-based sampled-data fuzzy control design with $H_\infty$ performance is developed for the PDE system via employing a novel time-dependent functional. The outcome of the observer-based $H_{\infty }$ sampled-data fuzzy control problem can be formulated as a bilinear matrix inequality optimization problem. Moreover, an iterative optimization algorithm based on the linear matrix inequalities is given to obtain a suboptimal $H_\infty$ sampled-data fuzzy controller. Finally, simulation results on the Fisher equation and a temperature cooling fin of high-speed aerospace vehicle illustrate that the proposed design method is effective.

A difference scheme for a degenerating convection-diffusion-reaction system modelling continuous sedimentation

Continuously operated settling tanks are used for the gravity separation of solid-liquid suspensions in several industries. Mathematical models of these units form a topic for well-posedness and numerical analysis even in one space dimension due to the spatially discontinuous coefficients of the underlying strongly degenerate parabolic, nonlinear model partial differential equation (PDE). Such a model is extended to describe the sedimentation of multi-component particles that react with several soluble components of the liquid phase. The fundamental balance equations contain the mass percentages of the components of the solid and liquid phases. The equations are reformulated as a system of nonlinear PDEs that can be solved consecutively in each time step by an explicit numerical scheme. This scheme combines a difference scheme for conservation laws with discontinuous flux with an approach of numerical percentage propagation for multi-component flows. The main result is an invariant-region property, which implies that physically relevant numerical solutions are produced. Simulations of denitrification in secondary settling tanks in wastewater treatment illustrate the model and its discretization.

Model order reduction of nonlinear parabolic PDE systems with moving boundaries using sparse proper orthogonal decomposition: Application to hydraulic fracturing

Developing reduced-order models for nonlinear parabolic partial differential equation (PDE) systems with time-varying spatial domains remains a key challenge as the dominant spatial patterns of the system change with time. To address this issue, there have been several studies where the time-varying spatial domain is transformed to the time-invariant spatial domain by using an analytical expression that describes how the spatial domain changes with time. However, this information is not available in many real-world applications, and therefore, the approach is not generally applicable. To overcome this challenge, we introduce sparse proper orthogonal decomposition (SPOD)-Galerkin methodology that exploits the key features of ridge and lasso regularization techniques for the model order reduction of such systems. This methodology is successfully applied to a hydraulic fracturing process, and a series of simulation results indicates that it is more accurate in approximating the original nonlinear system than the standard POD-Galerkin methodology.

B-SPLINE COLLOCATION SIMULATION OF NON-LINEAR TRANSIENT MAGNETIC NANOBIO-TRIBOLOGICAL SQUEEZE-FILM FLOW

A mathematical model is presented for magnetized nanofluid bio-tribological squeeze-film flow between two approaching disks. The nanofluid comprises a suspension of metal oxide nanoparticles with an electrically-conducting base fluid, making the nanosuspension responsive to applied magnetic field. The governing viscous momentum, heat and species (nanoparticle) conservation equations are normalized with appropriate transformations which renders the original coupled, non-linear partial differential equation system into a more amenable ordinary differential boundary value problem. The emerging model is shown to be controlled by a number of parameters, viz nanoparticle volume fraction, squeeze number, Hartmann magnetic body force number, disk surface transpiration parameter, Brownian motion parameter, thermophoretic parameter, Prandtl number and Lewis number. Computations are conducted with a B-spline collocation numerical method. Validation with previous homotopy solutions is included. The numerical spline algorithm is shown to achieve excellent convergence and stability in non-linear bio-tribological boundary value problems. The interaction of heat and mass transfer with nanofluid velocity characteristics is explored. In particular, smaller nanoparticle (high Brownian motion parameter) suspensions are studied. The study is relevant to enhanced lubrication performance in novel bio-sensors and intelligent knee joint (orthopaedic) systems.

Travelling Waves Solution of the Unsteady Flow Problem of a Collisional Plasma Bounded by a Moving Plate

The extension of the previous paper [Can. J. Phys. Vol. 88, (2010), 501–511] has been made. Therefore, the effect of the neutral atoms collisions with electrons and with positive ions is taken into consideration, which was ignored, for the sake of simplicity, in the earlier work. Thus, we will have multi-collision terms (electron–electron, electron–ion, electron– neutral) instead of one term, as was studied before for the sake of facilitation. These collision terms are needed to obtain the real physical situation. The new procedures will increase the ability of the research applications. This study is based on the solution of the BGK (Bhatnager–Gross–Krook) model of the nonlinear partial differential Boltzmann equations coupled with Maxwell’s partial differential equations. The initial-boundary value problem of the Rayleigh flow problem applied to the system of the plasma (positive ions + electrons+ neutral atoms), bounded by a moving plate, is solved. For this purpose, the traveling wave solution method is used to get the exact solution of the nonlinear partial differential equations system. The ratios between the different contributions of the internal energy changes are predicted via the extended Gibbs equation for both dia-magnetic and para-magnetic plasma. The results are applied to a typical model of laboratory argon plasma. 3D-Graphics illustrating the calculated variables are drawn to predict their behavior and the results are discussed.

Geometric Subspace Updates with Applications to Online Adaptive Nonlinear Model Reduction

In many scientific applications, including model reduction and image processing, subspaces are used as ansatz spaces for the low-dimensional approximation and reconstruction of the state vectors of interest. We introduce a procedure for adapting an existing subspace based on information from the least-squares problem that underlies the approximation problem of interest such that the associated least-squares residual vanishes exactly. The method builds on a Riemmannian optimization procedure on the Grassmann manifold of low-dimensional subspaces, namely the Grassmannian Rank-One Update Subspace Estimation (GROUSE). We establish for GROUSE a closed-form expression for the residual function along the geodesic descent direction. Specific applications of subspace adaptation are discussed in the context of image processing and model reduction of nonlinear partial differential equation systems.

A New Simulation Framework for Soil–Root Interaction, Evaporation, Root Growth, and Solute Transport

Core Ideas We present a locally mass‐conservative flow, transport, and root–soil interaction model. We include a sustainable, flexible research software framework for plant‐scale model development. This is an improved model concept for fluid dynamic processes, evaporation, root water uptake. We have developed a general model concept and a flexible software framework for the description of plant‐scale soil–root interaction processes including the essential fluid mechanical processes in the vadose zone. The model was developed in the framework of non‐isothermal, multiphase, multicomponent flow and transport in porous media. The software is an extension of the open‐source porous media flow and transport simulator DuMux to embedded mixed‐dimensional coupled schemes. Our coupling concept allows us to describe all processes in a strongly coupled form and adapt the complexity of the governing equations in favor of either accuracy or computational efficiency. We have developed the necessary numerical tools to solve the strongly coupled nonlinear partial differential equation systems that arise with a locally mass conservative numerical scheme even in the context of evolving root architectures. We demonstrate the model concept and its features, discussing a virtual hydraulic lift experiment including evaporation, root tracer uptake on a locally refined grid, the simultaneous simulation of root growth and root water uptake, and an irrigation scenario comparing different models for flow in unsaturated soil. We have analyzed the impact of evaporation from soil on the soil water distribution around a single plant's root system. Moreover, we have shown that locally refined grids around the root system increase computational efficiency while maintaining accuracy. Finally, we demonstrate that the assumptions behind the Richards equation may be violated under certain conditions.

The space–time CESE scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

The effects of bottom topography and horizontal temperature gradients on the shallow water flows are theoretically investigated. The considered systems of partial differential equations (PDEs) are non-strictly hyperbolic and non-conservative due to the presence of non-conservative differential terms on the right hand side. The solutions of these model equations are very challenging for a numerical scheme. Thus, our primary goal is to introduce an improved numerical scheme which can handle the non-conservative differential terms efficiently and accurately. In this paper, the space–time conservation element and solution element (CESE) method is extended to approximate these model equations. The proposed scheme has capability to overcome all difficulties posed by this nonlinear system of PDEs. The performance of the scheme is analyzed by considering several case studies of practical interest and the results of suggested scheme are compared with those of central NT scheme. The accuracy of the scheme is verified qualitatively and quantitatively.

Model predictive control of a dynamic nonlinear PDE system with application to continuous casting

Setting the value of the water flow rate in the secondary cooling zone of continuous casting systems plays a crucial role in the slab quality. The standard method to achieve it is unsuitable when the casting speed changes. Therefore, this paper focuses on model predictive control (MPC) for the continuous casting process, based on a heat transfer model described by a dynamic nonlinear partial differential equation (PDE). Realising this MPC is difficult, because some parameters in the nonlinear PDE are unknown. Hence the MPC scheme is designed in the following two steps. First, we introduce a new iterative algorithm to identify the unknown parameters with the help of measured data. Both synthetic and real data are used to illustrate the validity of this method. The experimental results show that this new algorithm reduces the number of iterations and running time as compared to the methods of Landweber and Cao. Second, the dynamic optimisation problem of the MPC strategy is described according to metallurgical principles. To obtain a more stable temperature, an adaptive step size quasi-Newton method is presented to solve this dynamic optimisation problem. The reliability and accuracy of our MPC approach is confirmed by simulation and real steel data.

Pemodelan Matematika Perpindahan Panas Konveksi Campuran (Mixed Convection) pada Pelat Horizontal

This study examines and analyzes mathematical model of mixed convection in horizontal plate. The heat transfer uses the model of a two dimensional nonlinear partial differential equations system. Then, this equation is derived first into the dimensionless equation form, and then it is changed into system of nonlinear ordinary differential equations form using similarity transformation. This system of nonlinear ordinary differential equations is solved by using the finite-difference scheme method, also with the mathematics program with software Matlab. The results obtained form this program is to determine skin friction coefficient , wall temperature velocity profiles and temperature profiles

Analytic properties of American option prices under a modified Black–Scholes equation with spatial fractional derivatives

This paper investigates analytic properties of American option prices under the finite moment log-stable (FMLS) model. Under this model the price of American options is characterized by a nonlinear fractional partial differential equation (FPDE) system. Using the technique of approximation, we prove that the American put price under the FMLS model is convex with respect to the underlying price, and specify the impact of the tail index on option prices. We also show numerical examples to support our analytic results.

Numerical solution of a multi-class model for batch settling in water resource recovery facilities

• The numerical solution for several batch settling unit processes in WRRFs is computed. • Multi-class concentration-driven model based on settling velocity distributions used. • Non-linear convection–diffusion equations appear due to the multi-class extension. • IMEX-RK schemes combined with high order shock-capturing technology are applied. In Torfs et al. (2017) a new unified framework to model settling tanks in water resource recovery facilities was proposed providing a set of partial differential equations (PDEs) modelling different settling unit processes in wastewater treatment such as primary and secondary settling tanks (PSTs and SSTs). The extension to a multi-class framework to deal with the distributed properties of the settling particles leads to a system of non-linear hyperbolic-parabolic PDEs whose solutions may contain very sharp transitions. This necessitates the use of a consistent and robust numerical method to obtain well-resolved and reliable approximations to the PDE solutions. The use of implicit–explicit Runge–Kutta (IMEX-RK) schemes, along with the weighted essentially non-oscillatory (WENO) shock-capturing technology for the discretization of the set of equations, is advocated in this work. The versatility of the proposed unified framework is demonstrated through a set of numerical examples for batch settling occurring in both PSTs and SSTs, along with the efficiency and reliability of the numerical scheme.

Temporal clustering for order reduction of nonlinear parabolic PDE systems with time‐dependent spatial domains: Application to a hydraulic fracturing process

A temporally‐local model order‐reduction technique for nonlinear parabolic partial differential equation (PDE) systems with time‐dependent spatial domains is presented. In lieu of approximating the solution of interest using global (with respect to the time domain) empirical eigenfunctions, low‐dimensional models are derived by constructing appropriate temporally‐local eigenfunctions. Within this context, first of all, the time domain is partitioned into multiple clusters (i.e., subdomains) by using the framework known as global optimum search. This approach, a variant of Generalized Benders Decomposition, formulates clustering as a Mixed‐Integer Nonlinear Programming problem and involves the iterative solution of a Linear Programming problem (primal problem) and a Mixed‐Integer Linear Programming problem (master problem). Following the cluster generation, local (with respect to time) eigenfunctions are constructed by applying the proper orthogonal decomposition method to the snapshots contained within each cluster. Then, the Galerkin's projection method is employed to derive low‐dimensional ordinary differential equation (ODE) systems for each cluster. The local ODE systems are subsequently used to compute approximate solutions to the original PDE system. The proposed local model order‐reduction technique is applied to a hydraulic fracturing process described by a nonlinear parabolic PDE system with the time‐dependent spatial domain. It is shown to be more accurate and computationally efficient in approximating the original nonlinear system with fewer eigenfunctions, compared to the model order‐reduction technique with temporally‐global eigenfunctions. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3818–3831, 2017

Sampled‐data fuzzy control for non‐linear parabolic distributed parameter systems with control inputs missing

In this study, an H ∞ sampled-data fuzzy control problem is considered for non-linear parabolic partial differential equation (PPDE) systems with control inputs missing. It is assumed that the actuator to the plant is set to zero if the control inputs missing which occurs in a random way satisfies a Bernoulli distribution. With the aid of the modal decomposition technique, a non-linear ordinary differential equation (ODE) system is initially obtained to represent the dominant dynamics of the PPDE system. Subsequently, a Takagi–Sugeno fuzzy model is utilised to accurately describe the resulting non-linear ODE system. Then, using a novel time-dependent functional and the stochastic analysis technique, an H ∞ sampled-data fuzzy controller with the stochastic missing data via linear matrix inequalities is proposed to stabilise exponentially the non-linear PPDE system in the mean square sense and achieve an H ∞ performance for the derived non-linear ODE system. Finally, an example on a temperature cooling fin is given to verify the proposed design strategy.

Finite volume numerical solution to a blood flow problem in human artery

In this paper, we solve a one dimensional blood flow model in human artery. This model is of a non-linear hyperbolic partial differential equation system which can generate either continuous or discontinuous solution. We use the Lax–Friedrichs finite volume method to solve this model. Particularly, we investigate how a pulse propagates in human artery. For this simulation, we give a single sine wave with a small time period as an impluse input on the left boundary. The finite volume method is successful in simulating how the pulse propagates in the artery. It detects the positions of the pulse for the whole time period.

SHAPE DIFFERENTIABILITY OF DRAG FUNCTIONAL AND BOUNDARY VALUE PROBLEM SOLUTIONS FOR FLUID MIXTURE EQUATIONS

Problems of optimal design of various elements of technical structures stimulate mathematical statements of new problems of continuum mechanics and hydrodynamics in particular. This study refers to problems of shape optimization of profiles in a flow of fluid or gas. The paper deals with properties of solutions and their functional of inhomogeneous boundary value problem for nonlinear composite type partial differential equation system, simulating the a mixture of viscous compressible fluids (gases) flowing around an obstacle. Methods of the theory of partial differential equations, functional analysis and, in particular, the results on the solvability of boundary value problems for transport and Stokes equations established the well-posedness of a linear boundary value problem with singular coefficients (the problem of the original problem solution difference). This result allowed to obtain the uniqueness theorem to determine the nature of solutions dependence on the shape of the flow range and to prove domain differentiability of the solutions considered. Domain differentiability of the solution functional reflecting the force of the obstacle resistance to the incident flow is proved. A formula to equate this derivative as a sum of two summands, one of which clearly depends on mapping setting the domain shape, and the other can be expressed in terms of the so-called adjoint state, depending only on the solution of the original problem in a non-deformed domain. The functional derivative formulas may be used as the basis for building a numerical algorithm for finding the optimal shape of the body in a flow of mixture of viscous compressible fluids.

Disturbance Rejection Fuzzy Control for Nonlinear Parabolic PDE Systems via Multiple Observers

A design method of low-dimensional disturbance rejection fuzzy control (DRFC) via multiple observers is proposed for a class of nonlinear parabolic partial differential equation (PDE) systems, where the disturbance is modeled by an exosystem of ordinary differential equations (ODEs) and enters into the PDE system through the control channel. In the proposed scheme, the modal decomposition technique is initially applied to the PDE system to derive a slow subsystem of low-dimensional nonlinear ODEs, which accurately captures the dominant dynamics of the PDE system. The resulting nonlinear slow subsystem is subsequently represented by a Takagi–Sugeno (T–S) fuzzy model. From the T–S fuzzy model and the exosystem, a fuzzy slow mode observer and a fuzzy disturbance observer are constructed to estimate the slow mode and the disturbance, respectively. Furthermore, a nonlinear observation spillover observer is proposed to compensate the effect of observation spillover. Then, based on these observers, a low-dimensional DRFC design is developed in terms of linear matrix inequalities to guarantee the exponential stability of the closed-loop PDE system in the presence of the disturbance. Finally, the effectiveness of the proposed design method is demonstrated on the control of one-dimensional Burgers–KPP–Fisher diffusion–reaction system and the temperature profile of a catalytic rod.