演化域上非线性扩散的精确尖前解

Stuart T Johnston, Matthew J Simpson
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引用次数: 2

摘要

发生在进化域上的扩散过程模型经常被用来描述生物和物理现象,例如在扩张的组织或基质内的扩散。以前对这些模型的研究要么报告数值解,要么要求线性扩散假设来确定精确解。不幸的是,数值解不能揭示模型参数与解特征之间的关系。此外,实验观测通常报告了锐锋的存在,这不是线性扩散所捕获的。在这里,我们通过在增长域上给出退化非线性扩散模型的精确尖锐前解来解决这两个限制。我们通过识别一系列变换,将一个演化域上的非线性扩散过程模型转化为一个固定域上的非线性扩散方程,并对某些扩散函数的选择给出了已知的精确解。我们确定了临界时间尺度和区域增长率的表达式,使得扩散种群永远不会到达区域边界,因此解仍然有效。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Exact sharp-fronted solutions for nonlinear diffusion on evolving domains
Abstract Models of diffusive processes that occur on evolving domains are frequently employed to describe biological and physical phenomena, such as diffusion within expanding tissues or substrates. Previous investigations into these models either report numerical solutions or require an assumption of linear diffusion to determine exact solutions. Unfortunately, numerical solutions do not reveal the relationship between the model parameters and the solution features. Additionally, experimental observations typically report the presence of sharp fronts, which are not captured by linear diffusion. Here we address both limitations by presenting exact sharp-fronted solutions to a model of degenerate nonlinear diffusion on a growing domain. We obtain the solution by identifying a series of transformations that converts the model of a nonlinear diffusive process on an evolving domain to a nonlinear diffusion equation on a fixed domain, which admits known exact solutions for certain choices of diffusivity functions. We determine expressions for critical time scales and domain growth rates such that the diffusive population never reaches the domain boundaries and hence the solution remains valid.
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