System Of Nonlinear Reaction-diffusion Equations Research Articles

Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix

We complete the group classification of systems of two coupled nonlinear reaction-diffusion equations with general diffusion matrix begun in author’s previous works. Namely, all nonequivalent equations with triangular diffusion matrix are classified. In addition, we describe symmetries of diffusion systems with nilpotent diffusion matrix and additional terms with first-order derivatives.

A reaction-diffusion system of $\lambda$ – $\omega$ type Part I: Mathematical analysis

We study two coupled reaction-diffusion equations of the boundary. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions [15] and compactness arguments. We also present a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations.

An attractor of a nonlinear system of reaction-diffusion equations in $$\mathbb{R}^n $$ and estimates of its ε-entropyand estimates of its ε-entropy

An attractor of a nonlinear system of reaction-diffusion equations in $$\mathbb{R}^n $$ and estimates of its ε-entropyand estimates of its ε-entropy

Petrov-Galerkin methods for systems of nonlinear reaction-diffusion equations

Petrov-Galerkin method is proposed for solving systems of nonlinear reaction-diffusion equations. A new concept of ordered pair of supersolution and subsolution is introduced, and then the existence of numerical solutions is studied. A new monotone iteration is provided for solving the resulting problem. The convergence of the Petrov-Galerkin scheme is discussed. The numerical results show the advantages of such method.

On the Generation of Discontinuous Shearing Motions of a Non-Newtonian Fluid

The system under study models unsteady, one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian fluid with fading memory under isothermal conditions. The flow, in a channel, is driven by a constant pressure gradient, is symmetric about the center line, and satisfies a no-slip boundary condition at the wall. The non-Newtonian contribution to the stress is assumed to obey a differential constitutive law (due to Oldroyd, Johnson & Segalman), the key feature of which is a non-monotone relation between the total steady shear stress and strain rate. In a regime in which the Reynolds number is much smaller than the Deborah (or Weissenberg) number, one obtains a degenerate, singularly perturbed system of nonlinear reaction-diffusion equations. It is shown that if the driving pressure gradient exceeds a critical value (the local shear stress maximum of the steady stress vs. strain rate relation), then the solution to the governing system, starting from rest at \(\), tends as \(\) to a particular discontinuous steady state solution (the “top-jumping” steady state), except in a small neighborhood of the discontinuity. This discontinuous steady state is shown to be nonlinearly stable in a precise sense with respect to perturbations yielding smooth initial data. Such discontinuous steady states have been proposed to explain “spurting” flows, which exhibit a large increase in mean flow rate when the driving pressure is raised above a critical value.

Conversion estimates for gas-solid reactions: reversible kinetics independent of solid concentrations

The mathematical formulation of gas-solid reactions are usually described by a system of nonlinear reaction-diffusion equations and closed-form solutions are not readily available for most cases. Back reactions are usually neglected in the analysis of solid conversion estimates for these systems. When reversible reactions are considered, the kinetics are assumed to be of first order in both the reactant and product gas concentrations, independent of the solid concentrations. In such a situation, a number of authors have investigated the two-stage model. This model uses the pseudo-steady-state assumption in conjunction with a special restriction that will allow the nonlinear problem to be mathematically tractable. In this work, we consider in detail the case where the form of kinetics is independent of both the reactant and product solid concentrations. First, we apply the pseudo-steady-state assumption and investigate the conditions under which the two-stage model need not be assumed. An extension, by means of the cumulative gas concentration method, is then made to allow for arbitrary values for all parameters and reaction orders in both the reactant and product gas concentrations. Expressions for the estimates of full conversion time for a range of kinetic models have been established.

Group Analysis of System of Reaction-Diffusion Equations, Symmetry of Brusselator

General formulas of group classification of the system of nonlinear reaction-diffusion equations(SNRDE) with two unknown functions and four independent variables are given. Group classifying equations of the traveling wave system of equations corresponding to SNRDE are also obtained. When these results are applied to analyzing Brusselator, we discovered that the system of differential equations describing it has only invariant group of spacetime translation.

Approximate general solutions for certain systems of nonlinear reaction-diffusion equations

Approximate general solutions for certain systems of nonlinear reaction-diffusion equations

Existence of coupled quasi-solutions of systems of nonlinear reaction-diffusion equations

Systems of nonlinear parabolic initial boundary value problems arise in many applications, such as epidemies, ecology, biochemistry, biology, chemical and nuclear engineering. Constructive methods of proving existence results for such problems, which can also provide numerical procedures for the computation of solutions, are of greater value than theoretical existence results. The method of upper and lower solutions coupled with monotone iterative technique has been employed successfully to prove existence of multiple solutions of nonlinear reaction-diffusion equations, in special cases, by various authors [335, 10, 11, 15, 181. Recently, in [6, 171 weakly coupled systems of reaction-diffusion equations, when the nonlinear terms are independent of gradient terms, are discussed and some special type of results are obtained. We, in this paper, investigate general systems of nonlinear reaction-diffusion problems when the nonlinear terms possess a mixed quasi-monotone property. We discuss a very general situation and obtain coupled extremal quasi-solutions, which in special cases reduce to minimal and maximal solutions. We shall also indicate how one-step cyclic monotone iterative schemes can be generated which yield accelerated rate of convergence of iterates. This work is in the spirit of our recent paper [12] for elliptic systems.

Invariant Regions and asymptotic behaviour for the numerical solution of reaction-diffusion systems by a class of alternating direction methods

In this paper we study the asymptotic behaviour of the numerical solution of systems of nonlinear reaction-diffusion equations, with homogeneous Dirichlet boundary conditions. We construct a class of alternating direction methods. In order to obtain a good simulation of the analytical solution, we require the difference schemes to be of positive type; this fact enables us to prove that, if an invariant setS exists for the analytical solutions,S is also invariant for the numerical solution and, moreover, to find a time-independent error estimate, if the nonlinear termF satisfies a monotonicity condition.

Stability of invariant regions for an alternating direction method for systems of nonlinear reaction-diffusion equations

In this paper, we consider an alternating direction method for the numerical solution of systems of nonlinear reaction-diffusion equations, with homogeneous Dirichlet boundary conditions, and we consider some properties of the difference scheme. If the system admits a bounded invariant regionS of the phase space, we prove that, under certain conditions on the mesh,S is also invariant for the difference equations. Moreover we find an error bound which tends to diamS ast→+∞ and which is 0 (h) fort fixed. Finally, we derive a time-independent error bound for a special case of mild nonlinearity.

Decay to spatially homogeneous states for the numerical solution of reaction-diffusion systems

The aim of this paper is to discuss the behaviour of the numerical solution of systems of nonlinear reaction-diffusion equations with homogeneous Neumann boundary conditions. We show that, under certain conditions, the solution obtained using known finite difference methods reproduces the behaviour of the exact solution. In particular we prove that the numerical solution decays as time increases to a spatially homogeneous vector, which is a suitably «weighted» mean value of the numerical solution itself.

Evidence of Soliton-Like Behavior of Solitary Waves in a Nonlinear Reaction-Diffusion System

Many systems of nonlinear reaction-diffusion equations have been found in which stable traveling solitary waves annihilate one another on collision. A two-component reaction-diffusion system with this property is constructed from the equations in a model of spreading cortical depression, in which the components are the concentrations of potassium and calcium ions in the extracellular compartment of nervous tissue. The solitary wave orbit is studied in relation to the source functions in the reaction diffusion system. The trajectories of the solutions are studied at various spatial points during the collision of the annihilating waves. During the collision interaction, parts of the plane of values of the components are visited that have not arisen during the passage of the solitary waves. The source functions are modified off the solitary wave trajectory, and it is found that two solitary waves emerge from the collision whose wave forms and speeds are identical to those of the colliding waves. The numerica...

Solitons in a Reaction-Diffusion System

Solitary waves in reaction-diffusion systems usually annihilate on collision. A nonlinear system of reaction-diffusion equations has been constructed which has solitons: solitary waves whose interaction in a collision results in the emergence of two solitary waves identical to the colliding waves.

Stability and Convergence of Finite Difference Methods for Systems of Nonlinear Reaction-Diffusion Equations

We consider the initial-boundary value problem for the general reaction-diffusion system (*) $U_t = D\nabla ^2 U + \Sigma _j M_j U_{xj} + F$, where U is a vector, $D,M_j $, and F depend on $(x,t,U)$, and D and $M_j$ are diagonal matrices with $D \geqq 0$. It is known that if there is a box S in U-space such that F does not point out of S, then S is invariant for (*). This invariance gives rise to an a priori estimate which in turn yields the existence of a smooth solution.In this paper we construct a family of locally second-order accurate finite difference schemes for (*). We prove that, under certain conditions on the mesh, S is also invariant for the difference equations. This enables us to derive an error bound which is $O(h)$ for t fixed, and which tends to diam S as $t \to \infty $. Next, we obtain a time-independent error bound by imposing a monotonicity condition on F, and we examine the implications of this condition for the solution of (*). Finally we derive a time-independent error bound for a ...

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations.

For any essentially nonlinear system of reaction-diffusion equations of the generic form δc i /δt = D i ∇ 2 c i + Q i ( c, x , t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distribution n -tuple function c = ( c 1 ( x , t ), …, c n ( x , t )) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion initial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentration n -tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c = c(x) . Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for an n -species symbiosis model.

Large Time Behavior of Solutions of Systems of Nonlinear Reaction-Diffusion Equations

We discuss the asymptotic behavior of solutions of weakly coupled parabolic equations describing systems undergoing diffusion, convection and nonlinear interaction in a bounded spatial domain, $\Omega $. If the system admits a bounded invariant region $\Sigma $ of phase space then we isolate a parameter $\sigma $ which depends upon the size of $\Omega $, the lower bound for the diffusion matrix, the magnitude of convection and a measure of the strength or sensitivity of the reaction. For $\sigma > 0$ we show that every solution with initial values in $\Sigma $ and subject to homogeneous Neumann boundary conditions decays exponentially to a spatially homogeneous function of time. This limiting function is a solution of an ordinary differential equation whose $\omega $-limit sets are determined by the reaction mechanism alone. This result may be interpreted as giving a sufficient condition for the validity of the “lumped parameter” approximation of distributed systems by solutions of ordinary differential e...

Effects of diffusion on the stability of the equilibrium in multi-species ecological systems.

Continuous population distributions that undergo self-diffusion, migrational cross-diffusion and interaction in a region of (1-, 2- or 3-dimensional) space are described dynamically by a governing system of nonlinear reaction-diffusion equations. It is shown that the constants associated with migrational cross-diffusion are ordinarily nonnegative or nonpositive, contingent on the type of species interaction. A simple sign relationship obtains between the latter diffusivity constants and the rate constants for species interaction in the neighborhood of a spatially uniform equilibrium state, and this relationship of signs serves to simplify the general stability theory for the growth or decay of arbitrary perturbations on a spatially uniform equilibrium state. The stability of the equilibrium state is analyzed and discussed in detail for the case of a generic two-species model, where the self-diffusion and migrational cross-diffusion of species act to either stabilize or destabilize the equilibrium, depending essentially on the character of the species interaction and also on the magnitude of the Helmholtz eigenvalues associated with the region and boundary conditions. In particular, for a prey-predator or host-parasite model, self-diffusion usually helps to stabilize the equilibrium state and migrational cross-diffusion can only act as an additional stabilizing influence, as evidenced generally by the experiments on such two-species systems. Sufficient conditions are derived for stability of the equilibrium state in the general case for an arbitrarily large number of interacting species. It is shown that the equilibrium state is always stable if all species undergo significant self-diffusion and the Helmholtz eigenvalues are suitably large.

Solutions to systems of nonlinear reaction-diffusion equations.

General criteria which either preclude time-periodic dissipative structure solutions or imply asymptotically steady solutions are derived for generic systems of reaction-diffusion equations ∂ci/∂t = Di▿2ci + Qi(c) subject to boundary conditions of practical interest, where the enumerator index i runs 1 to n, ci = ci(x, t) denotes the concentration or density of the ith participating molecular or biological species, Di is the diffusivity constant for the ith species, and Qi(c), an algebraic function of the n-tuple c = (c1,…, cn), expresses the local rate of production of the ith species due to chemical reactions or biological interactions. It is demonstrated that certain functionals of c which decrease monotonically with time can often be found, as examplified here for Volterra and Verhulst-Volterra n-species model systems, and thus time-periodic dissipative structure solutions are precluded for such systems of reaction-diffusion equations. It is shown that all solutions to a generic system of reaction-diffusion equations evolve dynamically to a unique steady state, lim ⁡ t → ∞ c i ( x , t ) = ĉ i ( x ) , if the diffusivity constants are all sufficiently large in magnitude. A necessary condition for the existence of a periodic solution (either spatially uniform or non-uniform) is formulated in terms of the curl of Q(c) in c-space. Finally, necessary and sufficient conditions are derived for the existence of time-periodic dissipative structure solutions in cases of “weak diffusion” with the reaction rate terms dominant in the governing equations.

Solutions to systems of nonlinear reaction-diffusion equations

Solutions to systems of nonlinear reaction-diffusion equations