Equations Of Reaction-diffusion Type Research Articles

On the Existence of Stationary Solutions for Some Systems of Non-Fredholm Integro-Differential Equations

We prove the existence of stationary solutions for certain systems of reaction-diffusion type equations in the corresponding H2 spaces. Our method relies on the fixed point theorem when the elliptic problem involves second order differential operators with and without Fredholm property.

Conservation laws for a system of reaction-diffusion type with one spatial variable

For a system of equations of reaction-diffusion type with one spatial variable, some necessary and sufficient conditions are obtained for the existence of nontrivial first-order conservation laws. We establish a theorem about the basis of conservation laws.

Boundary characteristic point regularity for semilinear reaction-diffusion equations: towards an ode criterion

The classical problem of regularity of boundary characteristic points for semilinear heat equations with homogeneous Dirichlet conditions is considered. The Petrovskii $ \left( {2\sqrt {{\log \log }} } \right) $ criterion (1934) of the boundary regularity for the heat equation can be adapted to classes of semilinear parabolic equations of reaction–diffusion type and takes the form of an ordinary differential equation (ODE) regularity criterion. Namely, after a special matching with a boundary layer, the regularity problem reduces to a onedimensional perturbed nonlinear dynamical system for the first Fourier-like coefficient of the solution in an inner region. A similar ODE criterion, with an analogous matching procedures, is shown formally to exist for semilinear fourth order biharmonic equations of reaction-diffusion type. Extensions to regularity problems of backward paraboloid vertices in $ {\mathbb{R}^N} $ are discussed. Bibliography: 54 titles. Illustrations: 1 figure.

Catalytic conversion reactions mediated by single-file diffusion in linear nanopores: Hydrodynamic versus stochastic behavior

We analyze the spatiotemporal behavior of species concentrations in a diffusion-mediated conversion reaction which occurs at catalytic sites within linear pores of nanometer diameter. Diffusion within the pores is subject to a strict single-file (no passing) constraint. Both transient and steady-state behavior is precisely characterized by kinetic Monte Carlo simulations of a spatially discrete lattice-gas model for this reaction-diffusion process considering various distributions of catalytic sites. Exact hierarchical master equations can also be developed for this model. Their analysis, after application of mean-field type truncation approximations, produces discrete reaction-diffusion type equations (mf-RDE). For slowly varying concentrations, we further develop coarse-grained continuum hydrodynamic reaction-diffusion equations (h-RDE) incorporating a precise treatment of single-file diffusion in this multispecies system. The h-RDE successfully describe nontrivial aspects of transient behavior, in contrast to the mf-RDE, and also correctly capture unreactive steady-state behavior in the pore interior. However, steady-state reactivity, which is localized near the pore ends when those regions are catalytic, is controlled by fluctuations not incorporated into the hydrodynamic treatment. The mf-RDE partly capture these fluctuation effects, but cannot describe scaling behavior of the reactivity.

Parallel Multilevel Schwarz and Block Preconditioners for the Bidomain Parabolic-Parabolic and Parabolic-Elliptic Formulations

The aim of this work is to develop parallel multilevel and block preconditioners for the Bidomain model of electrocardiology. The Bidomain model describes the electrical activity of the heart tissue and consists of a system of two parabolic nonlinear partial differential equations (PDEs) of reaction-diffusion type (PP formulation) or alternatively of a system of a parabolic nonlinear PDE and an elliptic linear PDE (PE formulation). In both formulations, the PDEs are coupled with a system of ordinary differential equations, modeling the cellular membrane ionic currents. The first goal of the present study is to construct, analyze, and numerically test a multilevel additive Schwarz preconditioner for the PE formulation of the Bidomain model, extending previous results obtained for the PP formulation. Optimal convergence rate estimates are established and confirmed by 3D numerical test on Linux clusters. The second goal of the present study is to analyze the scalability of multilevel Schwarz block-diagonal and block-factorized preconditioners for both PP and PE formulations of the Bidomain model and to compare them with multilevel Schwarz coupled preconditioners. The 3D parallel numerical tests show that block preconditioners for the PP formulation are not scalable, while they are scalable for the PE formulation, but less efficient than the coupled preconditioners.

Gaussian estimates on networks with applications to optimal control

We study a class of reaction-diffusion type equations on a finite network with continuity assumptions and a kind of non-local , stationary Kirchhoff's conditions at the nodes. A multiplicative random Gaussian perturbation acting along the edges is also included. For such a problem we prove Gaussian estimates for the semigroup generated by the evolution operator, hence generalizing similar results previously obtained in [21]. In particular our main goal is to extend known results on Gaussian upper bounds for heat equations on networks with local boundary conditions to those with non-local ones. We conclude showing how our results can be used to apply techniques developed in [13] to solve a class of Stochastic Optimal Control Problems inspired by neurological dynamics.

On the existence of stationary solutions for some non-Fredholm integro-differential equations

We show the existence of stationary solutions for some reaction-diffusion type equations in the appropriate H^2 spaces using the fixed point technique when the elliptic problem contains second order differential operators with and without Fredholm property.

Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology

The bidomain equations, a continuum approximation of cardiac tissue based on the idea of a functional syncytium, are widely accepted as one of the most complete descriptions of cardiac bioelectric activity at the tissue and organ level. Numerous studies employed bidomain simulations to investigate the formation of cardiac arrhythmias and their therapeutical treatment. They consist of a linear elliptic partial differential equation and a non-linear parabolic partial differential equation of reaction-diffusion type, where the reaction term is described by a set of ordinary differential equations. The monodomain equations, although not explicitly accounting for current flow in the extracellular domain and its feedback onto the electrical activity inside the tissue, are popular since they approximate, under many circumstances of practical interest, the bidomain equations quite well at a much lower computational expense, owing to the fact that the elliptic equation can be eliminated when assuming that conductivity tensors of intracellular and extracellular space are related to each other. Optimal control problems suggest themselves quite naturally for this important class of modelling problems and the present paper is a first attempt in this direction. Specifically, we present an optimal control formulation for the monodomain equations with an extra-cellular current, I e , as the control variable. I e must be determined such that electrical activity in the tissue is damped in an optimal manner. The derivation of the optimality system is given and a method for its numerical realization is proposed. The solution of the optimization problem is based on a non-linear conjugate gradient (NCG) method. The main goals of this work are to demonstrate that optimal control techniques can be employed successfully to this class of highly nonlinear models and that the influence of I e judiciously applied can in fact serve as a successful control for the dampening of propagating wavefronts.

Generic two-phase coexistence, relaxation kinetics, and interface propagation in the quadratic contact process: Analytic studies

The quadratic contact process is implemented as an adsorption–desorption model on a two-dimensional square lattice. The model involves random adsorption at empty sites, and correlated desorption requiring diagonal pairs of empty neighbors. A simulation study of this model [D.-J. Liu, X. Guo, J.W. Evans, Phys. Rev. Lett. 98 (2007) 050601] revealed the existence of generic two-phase coexistence between a low-coverage active steady-state and a completely covered absorbing state. Here, an analytic treatment of model behavior is developed based on truncation approximations to the exact master equations. Applying this approach for spatially homogeneous states, we characterize steady-state behavior as well as the kinetics of relaxation to the steady-states. Extending consideration to spatially inhomogeneous states, we obtain discrete reaction–diffusion type equations characterizing evolution. These are employed to analyze an orientation-dependence of the propagation of planar interfaces between active and absorbing states which underlies the generic two-phase coexistence. We also describe the dynamics and critical forms of planar perturbations of the active state and of droplets of one phase embedded in the other.

Dynamics and rheology of entangled linear polymers

In order to describe the dynamics of the linear polymer in monodisperse melts, we present the so-called reaction–diffusion type equation that contains contributions from reptation and contour-length fluctuation based on the tube model. The relaxation function calculated from this equation is in good agreement with the experimental results of the dielectric relaxation function and also gives the stress relaxation function in binary blends by use of semi-empirical scheme called double reptation. We add advection term to this equation in order to predict the nonlinear rheological properties of steady shear. The calculated shear stress and the first normal stress difference at steady state agree well with the measurements.

Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems

We study in this paper a new class of waveform relaxation algorithms for large systems of ordinary differential equations arising from discretizations of partial differential equations of advection reaction diffusion type. We show that the transmission conditions between the subsystems have a tremendous influence on the convergence speed of the waveform relaxation algorithms, and we identify transmission conditions with optimal performance. Since these optimal transmission conditions are expensive to use, we introduce a class of local transmission conditions of Robin type, which approximate the optimal ones and can be used at the same cost as the classical transmission conditions. We determine the transmission conditions in this class with the best performance of the associated waveform relaxation algorithm. We show that the new algorithm is well posed and converges much faster than the classical one. We illustrate our analysis with numerical experiments.

Asymptotic numerical methods for singularly perturbed fourth-order ordinary differential equations of reaction-diffusion type

Singularly perturbed boundary value problems (BVPs) for fourth-order ordinary differential equations (ODES) with a small positive parameter multiplying the highest derivative of the form −εy iv(x)+b(x)y″(x)−c(x)y(x)=−f(x), x∈D≔(0,1) , y(0)=p, y(1)=q, y″(0)=−r, y″(1)=−, 0≤ε≪1 , are considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter e multiplying the highest derivative, and suitable boundary conditions. In this paper, computational methods for solving this system are presented. In these methods, we first find the zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the fitted operator method, fitted mesh method, and boundary value technique. Error estimates are derived and examples are provided to illustrate the methods.

An asymptotic initial value method for boundary value problems for a system of singularly perturbed second order ordinary differential equations

In this paper a numerical method is suggested to solve a class of boundary value problems for a weakly coupled system of singularly perturbed second order ordinary differential equations of reaction-diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of the well known perturbation method. Then initial value problems and terminal value problems (TVPs) are formulated such that their solutions are the terms of this asymptotic expansion. These problems are happened to be singularly perturbed problems and therefore exponentially fitted finite difference schemes are used to solve these problems. Since the boundary value problem is converted into a set of initial and TVPs and an asymptotic expansion approximation is used, the present method is termed as an asymptotic initial value method. Necessary error estimates are derived and examples provided to illustrate the method. The present method is easy to implement and well suited for parallel computing.

Existence and uniqueness of solutions of infinite systems of semilinear parabolic differential-functional equations in arbitrary domains in ordered banach spaces

We consider the Fourier first initial-boundary value problem for a weakly coupled infinite system of semilinear parabolic differential-functional equations of reaction-diffusion type in arbitrary (bounded or unbounded) domain. The right-hand sides of the system are functionals of unknown functions of the Volterra type. Differential-integral equations give examples of such equations. To prove the existence and uniqueness of the solutions, we apply the monotone iterative method. The underlying monotone iterative scheme can be used for the computation of numerical solution.

Global influence domains of stable solutions with internal layers

The initial-boundary-value problem is considered for a non-stationary equation of reaction-diffusion type with non-linearity having two stable zeros. The conditions imposed ensure the existence of a stable stationary solution with internal transition layer (a stable step-like contrast structure). The question of which initial functions belong to the influence domain of such a solution is studied.

Approximation of Singularly Perturbed Parabolic Reaction-Diffusion Equations with Nonsmooth Data

AbstractIn this paper we consider the Dirichlet problem on a rectangle for singularly perturbed parabolic equations of reaction-diffusion type. The reduced (for ε = 0) equation is an ordinary differential equation with respect to the time variable; the singular perturbation parameter ε may take arbitrary values from the half-interval (0,1]. Assume that sufficiently weak conditions are imposed upon the coefficients and the right-hand side of the equation, and also the boundary function. More precisely, the data satisfy the Hölder continuity condition with a small exponent α and α/2 with respect to the space and time variables. To solve the problem, we use the known ε-uniform numerical method which was developed previously for problems with sufficiently smooth and compatible data. It is shown that the numerical solution converges ε-uniformly. We discuss also the behavior of local accuracy of the scheme in the case where the data of the boundary-value problem are smoother on a part of the domain of definition.

Localized State in a Nonlinear Resistor Network

A simple model equation is proposed to understand a mechanism of localized states in dissipative systems. This is a very simplified model for filament-like discharge patterns and can be expressed as a nonlinear resistor network model through spatial discretization. The model equation is piecewise linear and can be solved exactly. We propose another simple and solvable model equation to understand a mech- anism of localized states in dissipative systems. The model equation is related to filament structures in discharge patterns. There are varieties of spatio-temporal pat- terns such as standing and moving striations in gas-discharge systems. A reaction diffusion-type equation for the current density and the electric potential was pro- posed to study spatially periodic patterns and localized patterns transverse to the current flow. 18), 19) We consider a simpler model equation. §2. Model equation and the solution Our model is related to arc discharge. The current density is high and the temperature becomes ∼ 10 4 K in the localized arc region, and strong light is emitted from the filament-like region. Many complicated ionization and reaction processes occur both on the electrodes and in the gas region between the electrodes. However, we consider only the energy balance or heat equation. The temperature T (r ,t )i s assumed to obey

Schwartzian derivative for multidimensional maps and flows

A generalization of Schwartzian derivative to maps and flows in the space and in infinite-dimensional spaces is introduced. It is used to study the type of stability loss (soft or hard) for fixed points and periodic trajectories of diffeo-morphisms and flows. In particular, an example of a partial differential equation of reaction-diffusion type is presented for which the conditions of soft loss of stability of a spatially homogeneous solution are verified.

Boundedness of Temperature and Stability of the Steady-State Temperature Profile in Thermistors

Thermistors are electrical devices with temperature dependent electrical conductivity. The temperature in the ceramic material of a thermistor increases when the current through it increases. This increase in the temperature results in a sharp decrease in the conductivity of the thermistor, and hence a decrease in the current through it. Having this property, thermistors can be used as current surge protectors. The evolution of temperature along a thermistor can be modeled by a reaction-diffusion equation. In this paper, it is proved that the mathematical model of temperature evolution in a thermistor has a unique, bounded, and nonnegative solution. It is further proved that the steady-state temperature profile of a thermistor is unique and asymptotically stable. Finally, a sufficient condition for the bounded-input bounded-output (BIBO) stability of a class of systems represented by nonlinear partial differential equations of reaction-diffusion type, to which the thermistor model belongs, is derived.

Microwave heating and joining of ceramic cylinders: A mathematical model

A thin cylindrical ceramic sample is placed in a single mode microwave applicator in such a way that the electric field strength is allowed to vary along its axis. The sample can either be a single rod or two rods butted together. We present a simple mathematical model which describes the microwave heating process. It is built on the assumption that the Biot number of the material is small, and that the electric field is known and uniform throughout the cylinder''s cross-section. The model takes the form of a non-linear parabolic equation of reaction-diffusion type, with a spatially varying reaction term that corresponds to the spatial variation of the electromagnetic field strength in the waveguide. The equation is analyzed and a solution is found which develops a hot spot near the center of the cylindrical sample and which then propagates outwards until it stabilizes. The propagation and stabilization phenomenon concentrates the microwave energy in a localized region about the center where elevated temperatures may be desirable.