Numerical solution of Fisher's equation

1974 ◽  
Vol 11 (3) ◽  
pp. 445-457 ◽  
Author(s):  
Jenö Gazdag ◽  
José Canosa
Keyword(s):  
Water Waves ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Step Function ◽  
Local Perturbation ◽  
Fisher’S Equation ◽  
Minimal Speed ◽  
Fisher's Equation

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.

Numerical solution of Fisher's equation

1974 ◽  
Vol 11 (03) ◽  
pp. 445-457 ◽  
Author(s):  
Jenö Gazdag ◽  
José Canosa
Keyword(s):  
Water Waves ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Step Function ◽  
Local Perturbation ◽  
Fisher’S Equation ◽  
Minimal Speed ◽  
Fisher's Equation

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

Approximate travelling waves for generalized KPP equations and classical mechanics

1994 ◽  
Vol 446 (1928) ◽  
pp. 529-554 ◽  
Keyword(s):  
Travelling Waves ◽  
Classical Mechanics ◽  
Initial Distribution ◽  
Travelling Wave ◽  
Step Function ◽  
Initial Position ◽  
Classical Path ◽  
Huygens Principle ◽  
Kpp Equations ◽  
Complete Riemannian Manifolds

We consider the existence of approximate travelling waves of generalized KPP equations in which the initial distribution can depend on a small parameter μ which in the limit μ → 0 is the sum of some δ -functions or a step function. Using the method of Elworthy & Truman (1982) we construct a classical path which is the backward flow of a classical newtonian mechanics with given initial position and velocity before the time at which the caustic appears. By the Feynman–Kac formula and the Maruyama–Girsanov–Cameron–Martin transformation we obtain an identity from which, with a late caustic assumption, we see the propagation of the global wave front and the shape of the trough. Our theory shows clearly how the initial distribution contributes to the propagation of the travelling wave. Finally, we prove a Huygens principle for KPP equations on complete riemannian manifolds without cut locus, with some bounds on their volume element, in particular Cartan–Hadamard manifolds.

Oscillatory instability and rupture in a thin melt film on its crystal subject to freezing and melting

2007 ◽  
Vol 586 ◽  
pp. 423-448 ◽  
Author(s):  
M. BEERMAN ◽  
L. N. BRUSH
Keyword(s):  
Temperature Gradient ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Nonlinear Partial Differential Equations ◽  
Heat Of Fusion ◽  
Fixed Temperature ◽  
Strongly Nonlinear ◽  
Thermocapillary Forces ◽  
Oscillatory Instabilities

Lubrication theory is used to derive a coupled pair of strongly nonlinear partial differential equations governing the evolution of interfaces separating a thin film of a pure melt from its crystalline phase and from a gas. The free melt–gas (MG) interface deforms in response to the local state of stress and the crystal–melt (CM) interface can deform by freezing and melting only. A linear stability analysis of a static, uniform film subject to the effects of MG interface capillary forces, thermocapillary forces, the latent heat of fusion, van der Waals attraction, heat transfer and solidification volume change effects, reveals stationary and oscillatory instabilities. The effect of a temperature gradient (by increasing the gas phase temperature) is to stabilize a film. As the temperature gradient is reduced, the onset of instability is oscillatory and is at a unique, finite wavenumber. Instability is oscillatory for all marginally stable, non-isothermal cases. Crystals with higher density than the melt are more stable, whereas crystals with lower density are less stable in the presence of an applied temperature gradient. Fully nonlinear numerical solutions show that oscillatory instabilities lead to rupture by growth of standing or travelling waves. Rupture times and the number of oscillations to rupture increase as the temperature gradient is increased. For stationary linearly unstable initial conditions, the CM interface retreats by melting away from the tip region of the encroaching MG interface due to a rise in the heat flux there as the film thins and nears rupture. Larger amplitude disturbances increase the maximum allowable temperature for instability, at a given wavenumber, and decrease the time to rupture at fixed temperature and wavenumber.

Existence and stability of travelling wave solutions for an evolutionary ecology model

1990 ◽  
Vol 115 (1-2) ◽  
pp. 1-18 ◽  
Author(s):  
Roger Lui
Keyword(s):  
Population Size ◽  
Evolutionary Ecology ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Fisher’S Equation ◽  
Wave Solutions ◽  
Fisher's Equation ◽  
Existence And Stability ◽  
The Stability

SynopsisMonotone travelling wave solutions are known to exist for Fisher's equation which models the propagation of an advantageous gene in a single locus, two alleles population genetics model. Fisher's equation assumed that the population size is a constant and that the fitnesses of the individuals in the population depend only on their genotypes. In this paper, we relax these assumptions and allow the fitnesses to depend also on the population size. Under certain assumptions, we prove that in the second heterozygote intermediate case, there exists a constant θ*>0 such that monotone travelling wave solutions for the reaction–diffusion system exist whenever θ > θ*. We also discuss the stability properties of these waves.

scholarly journals ASYMPTOTICAL ANALYSIS OF SOME COUPLED NONLINEAR WAVE EQUATIONS

2011 ◽  
Vol 16 (1) ◽  
pp. 97-108 ◽  
Author(s):  
Aleksandras Krylovas ◽  
Rima Kriauzienė
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Linear Equations ◽  
Travelling Wave ◽  
Nonlinear Wave Equations ◽  
Linear Wave ◽  
Resonance Case ◽  
Non Linear ◽  
Asymptotical Analysis

We consider coupled nonlinear equations modelling a family of travelling wave solutions. The goal of our work is to show that the method of internal averaging along characteristics can be used for wide classes of coupled non-linear wave equations such as Korteweg-de Vries, Klein – Gordon, Hirota – Satsuma, etc. The asymptotical analysis reduces a system of coupled non-linear equations to a system of integro – differential averaged equations. The averaged system with the periodical initial conditions disintegrates into independent equations in non-resonance case. These equations describe simple weakly non-linear travelling waves in the non-resonance case. In the resonance case the integro – differential averaged systems describe interaction of waves and give a good asymptotical approximation for exact solutions.

scholarly journals Recurrence of travelling waves in transitional pipe flow

2007 ◽  
Vol 584 ◽  
pp. 69-102 ◽  
Author(s):  
R. R. KERSWELL ◽  
O. R. TUTTY
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Pipe Flow ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Transition Process ◽  
Travelling Wave ◽  
Shear Stresses ◽  
Wall Shear ◽  
Dividing Surface

The recent theoretical discovery of families of unstable travelling-wave solutions in pipe flow at Reynolds numbers lower than the transitional range, naturally raises the question of their relevance to the turbulent transition process. Here, a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface in phase space which separates initial conditions which uneventfully relaminarize and those which lead to a turbulent evolution. This dividing surface (a separatrix if turbulence is a sustained state) is then minimally the union of the stable manifolds of all these travelling waves. Evidence for recurrent travelling-wave visits is found in both 5D and 10D long periodic pipes, but only for those travelling waves with low-to-intermediate wall shear stress and for less than about 10% of the time in turbulent flow at Re = 2400. Given this, it seems unlikely that the mean turbulent properties such as wall shear stress can be predicted as an expansion solely over the travelling waves in which their individual properties are appropriately weighted. Instead the onus is on isolating further dynamical structures such as periodic orbits and including them in any such expansion.

Algebra, analysis and probability for a coupled system of reaction-duffusion equations

1995 ◽  
Vol 350 (1692) ◽  
pp. 69-112 ◽  
Keyword(s):  
Differential Equation ◽  
Large Deviation ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Travelling Wave ◽  
The Other ◽  
Reaction Diffusion Equation ◽  
Local Martingale

This paper is designed to interest analysts and probabilists in the methods of the ‘other’ field applied to a problem important in biology and in other contexts. It does not strive for generality. After § 1 a , it concentrates on the simplest case of a coupled reaction-diffusion equation. It provides a complete treatment of the existence, uniqueness, and asymptotic behaviour of monotone travelling waves to various equilibria, both by differential-equation theory and by probability theory. Each approach raises interesting questions about the other. The differential-equation treatment makes new use of the maximum principle for this type of problem. It suggests a numerical method of solution which yields computer pictures which illustrate the situation very clearly. The probabilistic treatment is careful in its proofs of martingale (as opposed to merely local-martingale) properties. A new change-of-measure technique is used to obtain the best lower bound on the speed of the monotone travelling wave with Heaviside initial conditions. Waves to different equilibria are shown to be related by Doob h -transforms. Large-deviation theory provides heuristic links between alternative descriptions of minimum wave speeds, rigorous algebraic proofs of which are provided. Since the paper was submitted, an alternative method of proving existence of monotone travelling waves has been developed by Karpelevich et al. (1993). We have extended our results in different directions from theirs (one of which is hinted at in § 1 a ), and have found the methods used here well equipped for these generalizations. See the Addendum.

Sign in / Sign up

Export Citation Format

Share Document