Systems Of Diffusion Type Research Articles

A computational method for singularly perturbed reaction–diffusion type system of integro-differential equations with discontinuous source term

A computational method for singularly perturbed reaction–diffusion type system of integro-differential equations with discontinuous source term

Blow-up prevention by sub-logistic sources in Keller-Segel cross diffusion type system

Blow-up prevention by sub-logistic sources in Keller-Segel cross diffusion type system

Efficiency of parallel anisotropic mesh adaptation for the solution of the bidomain model in cardiac tissue

Electrocardiology models are nonlinear reaction–diffusion type systems, where the numerical simulation requires extremely fine meshes to accurately compute the heart’s electrical activity. Anisotropic mesh adaptation methods have been proven to be efficient for simulating cardiac dynamic by many authors and showed a considerable improvement in the numerical accuracy while reducing the computational expenses. However, the efficiency of these techniques in parallel computing environments has not been shown yet, especially when compared to the performance of parallel uniform meshes. In this paper, we demonstrate the efficiency of a parallel anisotropic mesh adaptation method for the solution of the bidomain model in cardiac tissue. The technique is based on an efficient error estimator appropriate for second or higher order numerical solutions. To demonstrate the effectiveness of the developed methodology, comparisons between the numerical simulations on parallel adapted meshes with those on parallel uniform meshes are presented. The computational efficiency is assessed by computing spiral and scroll waves in cardiac tissue.

Convex restrictions in physical design

In a physical design problem, the designer chooses values of some physical parameters, within limits, to optimize the resulting field. We focus on the specific case in which each physical design parameter is the ratio of two field variables. This form occurs for photonic design with real scalar fields, diffusion-type systems, and others. We show that such problems can be reduced to a convex optimization problem, and therefore efficiently solved globally, given the sign of an optimal field at every point. This observation suggests a heuristic, in which the signs of the field are iteratively updated. This heuristic appears to have good practical performance on diffusion-type problems (including thermal design and resistive circuit design) and some control problems, while exhibiting moderate performance on photonic design problems. We also show in many practical cases there exist globally optimal designs whose design parameters are maximized or minimized at each point in the domain, i.e., that there is a discrete globally optimal structure.

High-Power Formate/Dioxygen Biofuel Cell Based on Mediated Electron Transfer Type Bioelectrocatalysis

A high-power mediated electron transfer type formate (HCOO–)/dioxygen (O2) biofuel cell is reported herein. The cell utilizes a Ketjen Black modified waterproof carbon cloth as the electrode material. The bioanode comprises tungsten-containing formate dehydrogenase and a viologen-functionalized polymer, whereas the biocathode comprises bilirubin oxidase and 2,2′-azinobis(3-ethylbenzothiazoline-6-sulfonate). In addition, a gas diffusion type system was employed for the biocathode to realize a high-speed O2 supply. These electrodes exhibited a large current density of 20 mA cm–2 in the quiescent steady state for both HCOO– oxidation and O2 reduction. Finally, these electrodes were coupled to construct an HCOO–/O2 biofuel cell without a separator. The cell exhibited a maximum power density of 12 mW cm–2 at a cell voltage of 0.78 V under quiescent conditions and an open-circuit voltage of 1.2 V. We show the great potential of HCOO– for the fuel of biofuel cells.

Application of kinetic flux vector splitting scheme for solving multi-dimensional hydrodynamical models of semiconductor devices

In this article, one and two-dimensional hydrodynamical models of semiconductor devices are numerically investigated. The models treat the propagation of electrons in a semiconductor device as the flow of a charged compressible fluid. It plays an important role in predicting the behavior of electron flow in semiconductor devices. Mathematically, the governing equations form a convection–diffusion type system with a right hand side describing the relaxation effects and interaction with a self consistent electric field. The proposed numerical scheme is a splitting scheme based on the kinetic flux-vector splitting (KFVS) method for the hyperbolic step, and a semi-implicit Runge–Kutta method for the relaxation step. The KFVS method is based on the direct splitting of macroscopic flux functions of the system on the cell interfaces. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge–Kutta time stepping method. Several case studies are considered. For validation, the results of current scheme are compared with those obtained from the splitting scheme based on the NT central scheme. The effects of various parameters such as low field mobility, device length, lattice temperature and voltage are analyzed. The accuracy, efficiency and simplicity of the proposed KFVS scheme validates its generic applicability to the given model equations. A two dimensional simulation is also performed by KFVS method for a MESFET device, producing results in good agreement with those obtained by NT-central scheme.

A splitting scheme based on the space–time CE/SE method for solving multi-dimensional hydrodynamical models of semiconductor devices

Numerical solutions of the hydrodynamical model of semiconductor devices are presented in one and two-space dimension. The model describes the charge transport in semiconductor devices. Mathematically, the models can be written as a convection–diffusion type system with a right hand side describing the relaxation effects and interaction with a self consistent electric field. The proposed numerical scheme is a splitting scheme based on the conservation element and solution element (CE/SE) method for hyperbolic step, and a semi-implicit scheme for the relaxation step. The numerical results of the suggested scheme are compared with the splitting scheme based on Nessyahu–Tadmor (NT) central scheme for convection step and the same semi-implicit scheme for the relaxation step. The effects of various parameters such as low field mobility, device length, lattice temperature and voltages for one-space dimensional hydrodynamic model are explored to further validate the generic applicability of the CE/SE method for the current model equations. A two dimensional simulation is also performed by CE/SE method for a MESFET device, producing results in good agreement with those obtained by NT-central scheme.

Regularity and structure of pullback attractors for reaction–diffusion type systems without uniqueness

In this paper, we study the pullback attractor for a general reaction–diffusion system for which the uniqueness of solutions is not assumed. We first establish some general results for a multi-valued dynamical system to have a bi-spatial pullback attractor, and then we find that the attractor can be backwards compact and composed of all the backwards bounded complete trajectories. As an application, a general reaction–diffusion system is proved to have an invariant (H,V)-pullback attractor A={A(τ)}τ∈R. This attractor is composed of all the backwards compact complete trajectories of the system, pullback attracts bounded subsets of H in the topology of V, and moreover⋃s⩽τA(s)is precompact in V,∀τ∈R.A non-autonomous Fitz-Hugh–Nagumo equation is studied as a specific example of the reaction–diffusion system.

Semi-discrete a priori error analysis for the optimal control of the unsteady Navier–Stokes equations with variational multiscale stabilization

In this work, the optimal control problems of the unsteady Navier–Stokes equations with variational multiscale stabilization (VMS) are considered. At first, the first order continuous optimality conditions are obtained. Since the adjoint equation of the Navier–Stokes problem is a convection diffusion type system, then the same stabilization is applied to it. Semi discrete a priori error estimates are obtained for the state, adjoint state and control variables. Crank–Nicholson time discretization is used to get the fully discrete scheme. Numerical examples verify the theoretical findings and show the efficiency of the stabilization for higher Reynolds number.

Synchronization and control of coupled reaction–diffusion systems of the FitzHugh–Nagumo type

We consider a FitzHugh–Nagumo reaction–diffusion type system (FHN). The dynamics of the reaction part induces a unique repulsive stationary point (0,0) and a unique attractive limit cycle. After a description of the asymptotic behaviour of the FHN system, we deal with the synchronization and control analysis of N coupled FHN systems.

Polymorphic and regular localized activity structures in a two-dimensional two-component reaction–diffusion lattice with complex threshold excitation

Space–time dynamics of the system modeling collective behaviour of electrically coupled nonlinear units is investigated. The dynamics of a local cell is described by the FitzHugh–Nagumo system with complex threshold excitation. It is shown that such a system supports formation of two distinct kinds of stable two-dimensional spatially localized moving structures without any external stabilizing actions. These are regular and polymorphic structures. The regular structures preserve their shape and velocity under propagation while the shape and velocity as well as other integral characteristics of polymorphic structures show rather complex temporal behaviour. Both kinds of structures represent novel sorts of spatially temporal patterns which have not been observed before in typical two-component reaction–diffusion type systems. It is demonstrated that there exist two types of regular structures: single and bound states and three types of polymorphic structures: periodic, quasiperiodic and even chaotic ones. The partition of the parameter plane into regions corresponding to the existence of these different types of structures is carried out. High multistability of regular structures is indicated. The interaction of regular structures is investigated. The correspondence between the structures and trajectories in multidimensional phase space associated with the system is given. Bifurcation mechanisms leading to the loss of stability of regular structures as well as to a transition from one type of polymorphic structure to another are indicated. The mechanisms of formation of regular and polymorphic structures are discussed.

The entropy dissipation method for spatially inhomogeneous reaction–diffusion-type systems

We study the long-time asymptotics of reaction–diffusion-type systems that feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expressions for (or estimates of) the rates of convergence towards an (entropy minimizing) equilibrium state in terms of the constants of diffusion and reaction and with respect to conserved quantities. Our method, the so-called entropy approach, seeks to quantify convergence to equilibrium by using functional inequalities, which relate quantitatively the entropy and its dissipation in time. The entropy approach is well suited to nonlinear problems and known to be quite robust with respect to model variations. It has already been widely applied to scalar diffusion–convection equations, and the main goal of this paper is to study its generalization to systems of partial differential equations that contain diffusion and reaction terms and admit fewer conservation laws than the size of the system. In particular, we successfully apply the entropy approach to general linear systems and to a nonlinear example of a reaction–diffusion–convection system arising in solid-state physics as a paradigm for general nonlinear systems.

Numerical methods of synthesis of an optimal control for stochastic dynamical systems of diffusion type

Several outlines of numerical methods of synthesis of optimal trajectories in the problem of control of a stochastic dynamic system of diffusion type are proposed. Both the systems with complete information on the state and the systems with information constraints, which mean that each component of the control strategy may depend only on the set of the components of the state vector measured precisely stipulated in advance, are considered. It is shown that in the problem with complete information on the state, one of the methods converges for just one iteration. For linear and quasilnear systems with quadratic cost function, the proposed methods are brought out to the regular computation procedures.

Euler–Poincaré flows and leibniz structure of nonlinear reaction–diffusion type systems

In this paper we present Euler–Poincaré formulation of the Fisher, Fitzhugh–Nagumo, Burgers–Huxley and extended Fitzhugh–Nagumo and extended Burgers–Huxley type nonlinear reaction–diffusion systems. All these flows are related to infinite dimensional almost Poisson manifolds and the corresponding Lie–Poisson structures yield Leibniz brackets, a bracket endowed with both symmetric and skewsymmetric parts. The symmetric part contributes the diffusion part of the ssystem. The properties exhibited by the reaction–diffusion systems defined in this way are in general very different from the standard Hamiltonian mechanics since the dynamics are controlled by the standard Poisson brackets. Moreover, all the nonlinear reaction–diffusion systems under consideration are Euler–Poincaré flows on the dual of Kirillov’s superalgebra associated to the Bott–Virasoro group.

Numerical methods for controlled and uncontrolled multiplexing and queueing systems

We deal with a very useful numerical method for both controlled and uncontrolled queuing and multiplexing type systems. The basic idea starts with a heavy traffic approximation, but it is shown that the results are very good even when working far from the heavy traffic regime. The underlying numerical method is a version of what is known as the Markov chain approximation method. It is a powerful methodology for controlled and uncontrolled stochastic systems, which can be approximated by diffusion or reflected diffusion type systems, and has been used with success on many other problems in stochastic control. We give a complete development of the relevant details, with an emphasis on multiplexing and particular queueing systems. The approximating process is a controlled or uncontrolled Markov chain which retains certain essential features of the original problem. This problem is generally substantially simpler than the original physical problem, and there are associated convergence theorems. The non-classical associated ergodic cost problem is derived, and put into a form such that reliable and good numerical algorithms, based on multigrid type ideas, can be used. Data for both controlled and uncontrolled problems shows the value of the method.

A Monte Carlo Method for Sensitivity Analysis and Parametric Optimization of Nonlinear Stochastic Systems: The Ergodic Case

For high-dimensional or nonlinear problems there are serious limitations on the power of available computational methods for the optimization or parametric optimization of stochastic systems of diffusion type. The paper develops an effective Monte Carlo method for obtaining good estimators of systems sensitivities with respect to system parameters, when the system is of interest over a long period of time. The value of the method is borne out by numerical experiments, and the computational requirements are favorable with respect to competing methods when the dimension is high or the nonlinearities “severe.” The method is a type of “derivative of likelihood ratio” method. For a wide class of problems, the cost function or dynamics need not be smooth in the state variables; for example, where the cost is the probability of an event or “sign” functions appear in the dynamics. Under appropriate conditions, it is shown that the invariant measures are ifferentiable with respect to the parameters. Since the basi...

Singularly perturbed systems of diffusion type and feedback control

Asymptotic approximations describing the behaviour of linear systems of diffusion type (convective or non-convective) with a small diffusivity, to which a feedback control of distributed or boundary type based on point sensors is applied, are constructed and proven to be correct. As a consequence one can find a near-optimal feedback control for a cost minimization problem with a quadratic performance index measuring the deviation of the stationary state from an ideal state, under the restriction of a prescribed exponential degree of stability of the stationary state.

Well-Posedness of Some Evolution Problems in the Theory of Automatic Feed-Back Control for Systems with Distributed Parameters

The subject of this paper is the evolution, in time, of systems of diffusion type controlled by an automatic feed-back mechanism using a finite number of observators and control-inputs. For the initial-boundary value problem for a functional partial differential equation of parabolic type, which the state variable has to satisfy, the following types of questions are considered: existence, uniqueness, regularity and continuous dependence on data of a solution. Answers to these questions are given in function spaces of Holder and Sobolev type.

Suboptimal controls of a second-order system subjected to discontinuous random perturbations

This paper doala with tho suboptimal control of a stochastic second-order system. It is assumed that the system is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and tho second kind by a homogeneous process with independent increments, finite second-order moments, mean zero and no continuous sample paths. The stochastic system is approximated by a diffusion type system, and the optimal feedback control laws for the diffusion type system are chosen as the suboptimal controls for the given system. It is shown that in order to construct tho suboptimal controls, a non-linear partial differential equation and a linear partial integro-difforontial equation, have to be solved. A finite difference algorithm for the solution of these equations is proposed, and its efficiency and applicability are demonstrated with examples.