On a Nonlinear Hyperbolic Equation Describing Transmission Lines, Cell Movement, and Branching Random Walks
On a Nonlinear Hyperbolic Equation Describing Transmission Lines, Cell Movement, and Branching Random Walks
复制标题
关于描述传输线、细胞运动和分支随机游走的非线性双曲方程
DOI：
10.1007/978-3-642-93318-9_18
复制
发表时间：
1986
期刊：
影响因子：
--
通讯作者：
H. Othmer
中科院分区：
文献类型：
--
作者：
S. Dunbar;H. Othmer
这是一篇关于一个由各种不同现象引起的非线性双曲型方程的初步和解释性报告。我们从组织细胞和单细胞生物体中运动和繁殖的简单模型，以及对分支随机游动的数学处理，导出方程(1)，作为具有非线性并联电导和沿传输线长度的串联电感的传输线上的电压的方程。此外，在适当的标度和f和g的选择下，这个非线性双曲型方程可以看作是描述耦合van der Pol振子的连续统的方程。在公式(1)中，(‘：2)的值不必很小，但符号(’：2)的选择意味着与其他众所周知的非线性偏微分方程式的类比，下面我们将提到其中的一些类比。这份报告的目的是简要地解释和激励所有这些派生，并提出关于这个方程的解的一些基本结果。此外，其目的是展示由分支随机游走产生的方程的概率解释如何有助于理解和激励结果。新结果的详细证据将在其他地方展示。这篇论文的组织如下。在第一节中，我们通过应用NSF Grant DMS-8301840部分支持的L来从非线性传输线的模型导出方程(1
This is a preliminary and expository report on a nonlinear hyperbolic equation that arises from a variety of distinct phenomena. We derive the equation (1) as the equation for the voltage along a transmission line with nonlinear shunt conductance and a series inductance along the length of the line, from simple models of movement and reproduction in tissue cells and one celled organisms, and from a mathematical treatment of a branching random walk. In addition, with the proper scaling and choice of f and g this nonlinear hyperbolic equation can be viewed as the equation that describes a continuum of coupled van der Pol oscillators. In equation (1) the value of (': 2 need not be small, but the choice of the notation (': 2 suggests analogies with other well known nonlinear partial differential equations, and we will mention some of these analogies below. The purpose of this report is to briefly explain and motivate all of these derivations and to present some basic results about the solutions of this equation. In addition, the purpose is to show how the probabilistic interpretation of the equation arising out of the branching random walk helps in the understanding and motivation of the results. Detailed proofs of the new results will be presented elsewhere. The organization of the paper is as follows. In the first section we derive equation (1) from a model of a nonlinear transmission line by an application of lSupported in part by NSF Grant DMS-8301840