In this paper, numerical simulations of nonlinear Fisher’s equation in oneand twodimensions have been considered. The derivatives and integrals are replaced by the necessary matrices, and the resulting algebraic system of equations was advanced by the popular fourth-order exponential time differencing Runge-Kutta (ETDRK4) schemes proposed by Cox and Matthew [Exponential time differencing for st...

In this paper we present an intuitive explanation for the non-uniqueness of the traveling wave speed in the Fisher equation, showing a similar non-uniqueness property in the case of inviscid traveling waves. More precisely, we prove that traveling waves of the Fisher equation with wave speed c > 0 converges to the inviscid traveling wave with speed c > 0 as the diffusion vanishes. A complete di...

The Wright–Fisher equation, which was introduced as a model to study demography in the presence of diffusion, has had a renaissance as a model for the migration of alleles in the genome. Our goal in this paper is to give a careful analysis of the fundamental solution to the Wright–Fisher equation, with particular emphasis on its behavior for a short time near the boundary.

Nonlinear reaction-diffusion systems are known to exhibit very many novel spatiotemporal patterns. Fisher equation is a prototype of diffusive equations. In this contribution we investigate the integrability properties of the generalized Fisher type equation to obtain physically interesting solutions using Lie symmetry analysis. In particular, we report several travelling wave patterns, static ...

An information-theoretic foundation for compound Poisson approximation limit theorems is presented, in analogy to the corresponding developments for the central limit theorem and for simple Poisson approximation. It is shown that the compound Poisson distributions satisfy a natural maximum entropy property within a natural class of distributions. Simple compound Poisson approximation bounds are...

Reaction-diffusion problems are often described at a macroscopic scale by partial derivative equations of the type of the Fisher or KolmogorovPetrovsky-Piscounov equation. These equations have a continuous family of front solutions, each of them corresponding to a different velocity of the front. By simulating systems of size up to N = 10 particles at the microscopic scale, where particles reac...

More recently, Wazwaz [An analytic study of Fisher’s equation by using Adomian decomposition method, Appl. Math. Comput. 154 (2004) 609–620] employed the Adomian decomposition method (ADM) to obtain exact solutions to Fisher’s equation and to a nonlinear diffusion equation of the Fisher type. In this paper, He’s homotopy perturbation method is employed for the generalized Fisher’s equation to o...