Solutions Of Reaction-diffusion Equations Research Articles

Traveling wave speed and solution in reaction-diffusion equation in one dimension

By Painleve analysis, traveling wave speed and solution of reaction-diffusion equations for the concentration of one species in one spatial dimension are in detail investigated. When the exponent of the creation term is larger than the one of the annihilation term, two typical cases are studied, one with the exact traveling wave solutions, yielding the values of speeds, the other with the series expansion solution, also yielding the value of speed. Conversely, when the exponent of creation term is smaller than the one of the annihilation term, two typical cases are also studied, but only for one of them, there is a series development solution, yielding the value of speed, and for the other, traveling wave solution cannot exist. Besides, the formula of calculating speeds and solutions of planar wave within the thin boundary layer are given for a class of typical excitable media.

Target Patterns and Spirals in Planar Reaction-Diffusion Systems

Summary. Solutions of reaction-diffusion equations on a circular domain are considered. With Robin boundary conditions, the primary instability may be a Hopf bifurcation with eigenfunctions exhibiting prominent spiral features. These eigenfunctions, defined by Bessel functions of complex argument, peak near the boundary and are called wall modes. In contrast, if the boundary conditions are Neumann or Dirichlet, then the eigenfunctions are defined by Bessel functions of real argument, and take the form of body modes filling the interior of the domain. Body modes typically do not exhibit pronounced spiral structure. We argue that the wall modes are important for understanding the formation process of spirals, even in extended systems. Specifically, we conjecture that wall modes describe the core of the spiral; the constant-amplitude spiral visible outside the core is the result of strong nonlinearities which enter almost immediately above threshold as a consequence of the exponential radial growth of the wall modes.

Existence of bounded trajectories via upper and lower solutions

The paper deals with the boundary value problem on the whole line $u'' - f(u,u') + g(u) = 0 $ $u(\infty,1) = 0$, $ u(+\infty) = 1$ $(P)$ where $g : R \to R$ is a continuous non-negative function with support $[0,1]$, and $f : R^2\to R$is a continuous function. By means of a new approach, based on a combination of lower and upper-solutions methods and phase-plane techniques, we prove an existence result for$(P)$ when $f$ is superlinear in $u'$; by a similar technique, we also get a non-existence one.As an application, we investigate the attractivity of the singular point $(0,0)$ in the phase-plane $(u,u')$. We refer to a forthcoming paper [13] for a further application in the field offront-type solutions for reaction diffusion equations.

Analytic solutions of reaction diffusion equations and implications for the concept of an air parcel

In the atmosphere, hydroxyl radical concentrations can be estimated by considering the relative change in the concentration of two hydrocarbons of differing reactivity. This approach is based on three assumptions: (1) that background concentrations of the two hydrocarbon are zero; (2) that transport processes will influence all hydrocarbon concentrations equally; that is, hydrocarbon changes can be separated into the product of a chemical term and a transport term; and (3) that hydrocarbons have the same spatial and temporal emission pattern. In this paper, analytical solutions to a steady state reaction diffusion equation are derived. The general solutions to this problem are nonseparable, with the degree of nonseparability defined by a single parameter that is a simple function of the system's intrinsic timescales. When this parameter is evaluated for diffusivities typical of the boundary layer (∼102 m2/s) and hydrocarbon reactivities that are sufficiently slow to be practically useful, it can be readily shown that for all practical purposes, separability can be assumed. This separability influences the spatial distribution of loss but not the net global loss. Thus even under nonseparable conditions, although the apparent local loss rate may be considerably less than the actual kinetic loss rate implied by hydrocarbon reactivity, when the apparent local loss rates are integrated to deduce a global loss rate, there will be no underestimate in the global loss rate, since the chemical loss rate is a linear function of hydrocarbon concentrations. It is conjectured that when much higher dispersion rates common in photochemical transport models (∼105–106 m2/s) are invoked, local photochemical balances may be perturbed when chemical loss rates are either spatially inhomogeneous or influenced by reactant concentrations. Thus in highly diffusive models the influence of highly reactive chemical species may be extended further from the source regions than is realistic, even though the globally averaged loss rates would still be consistent with the magnitude of the globally averaged sink.

The delay effects on the behavior of solutions of reaction diffusion equations with delay

The delay effects on the behavior of solutions of reaction diffusion equations with delay

Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball.

Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball.

Exact steady-state of reaction-diffusion equations in the presence of losses

A particular solution of reaction-diffusion equations which is non-zero only in the presence of losses is found for certain combinations of the exponents of the power-law dependences of the diffusion, source and loss terms. The results, which are generalizations of expressions from previous analytic studies, are of principle interest and useful e.g. for interpreting computer simulations of a burning fusion plasma. The study is limited to one-dimensional situations.

Some aspects of the Morris-Lecar model and the myelinated axon models with Morris-Lecar dynamics

The solutions of reaction-diffusion equations of the Morris-Lecar nerve axon model are analyzed for the robust and weak current-voltage relations. The qualitative behavior of the myelinated axon models with Morris-Lecar dynamics is studied with nonzero length nodal segments and with point nodal segments. No excited steady state solutions will appear if the width of the nodes is too short for the model of nonzero length nodal segments. A nontrivial periodic steady state solution exists and is an attractor if the current-voltage relation is robust enough. A weak current-voltage relation will result in the whole membrane potential being pulled back to the rest state as time elapses.

A note on uniform in time error estimates for approximations to reaction-diffusion equations

The approximation of solutions of reaction-diffusion equations that approach asymptotically stable, hyperbolic equilibria is considered. Near such equilibria trajectories of the equation contract and hence it is possible to seek error estimates that are uniformly valid in time. A technique for the derivation of such estimates is illustrated in the context of an explicit Euler finite-difference scheme.

On nonlinear coupled reaction-diffusion equations

This paper is concerned with the existence and uniqueness of solutions of reaction-diffusion equations arising in many applications and often coupled. Our approach to the problem is based on converting the coupled equations into a system of Volterra-Fredholm type integral equations by means of the Green's function representation and using the Banach fixed point theorem. An application to nonlinear second order evolution equations is also indicated.

A Note on the Solution of Reaction-diffusion Equations with Convection

A Note on the Solution of Reaction-diffusion Equations with Convection

Topological techniques in reaction-diffusion equations

We consider the 'stability' problem for steady-state solutions of reaction-diffusion equations of the form u, = u, + f(u), -L 0 for u 0 for u > B, B > A. Using isolating block techniques, together with some new estimates for elliptic functions, we are able to determine the number of such solutions, and the dimension of their unstable manifolds. In particular we show that non-constant steady-state solutions with homogeneous Neumann boundary conditions cannot be stable, but for homogeneous Dirichlet data, stable non-constant solutions can exist. These questions are connected with the problem of 'secondary' bifurcation for such solutions.

Rotating Spiral Wave Solutions of Reaction-Diffusion Equations

We resolve the question of existence of regular rotating spiral waves as a consequence of only the processes of chemical reaction and molecular diffusion. We prove rigorously the existence of these waves as solutions of reaction-diffusion equations, and we exhibit them by means of numerical computations in several concrete cases. Existence is proved via the Schauder fixed point theorem applied to a class of functions with sufficient structure that, in fact, important constructive properties such as asymptotic representations and frequency of rotation are obtained.

A. Introductory and inorganic oscillations. Thermodynamic aspects and bifurcation analysis of spatio-temporal dissipative structures

The thermodynamic prerequisites for the emergence of spatio-temporal patterns of organization in nonlinear chemical systems are reviewed. Analytic expressions for steady-state, time-periodic and travelling wave like solution of reaction-diffusion equations are reported. A comparison with the results based on computer simulations is outlined.