Exact and numerical solutions of a free boundary problem with a reciprocal growth law

IF 1.4 4区数学 Q2 MATHEMATICS, APPLIED

IMA Journal of Applied Mathematics Pub Date : 2024-06-04 DOI:10.1093/imamat/hxae014

N. R. McDonald, Samuel J Harris

引用次数: 0

Abstract

A two-dimensional free boundary problem is formulated in which the normal velocity of the boundary is proportional to the inverse of the gradient of a harmonic function $T$. The field $T$ is defined in a simply connected region which includes the point at infinity where it has a logarithmic singularity. The growth problem in which the boundary expands outward is formulated both in terms of the Schwarz function of the boundary and a Polubarinova-Galin equation for the conformal map of the region from the exterior of the unit disk. An expanding free boundary is shown to be stable and explicit solutions for growing ellipses and a class of polynomial lemniscates are derived. Numerical solution of the Polubarinova-Galin equation is used to compute the evolution of the boundary having other initial shapes.

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具有倒数增长规律的自由边界问题的精确解和数值解

本文提出了一个二维自由边界问题，其中边界的法向速度与谐函数 $T$ 梯度的倒数成正比。场 $T$ 定义在一个简单连接的区域，该区域包括无穷远处的点，在该点有一个对数奇点。边界向外扩展的增长问题是根据边界的施瓦茨函数和该区域从单位盘外部的共形映射的 Polubarinova-Galin 方程提出的。结果表明膨胀的自由边界是稳定的，并推导出了增长椭圆和一类多项式∞的显式解法。Polubarinova-Galin 方程的数值解用于计算具有其他初始形状的边界的演变。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

IMA Journal of Applied Mathematics 数学-应用数学

CiteScore

2.30

自引率

8.30%

发文量

审稿时长

24 months

期刊介绍： The IMA Journal of Applied Mathematics is a direct successor of the Journal of the Institute of Mathematics and its Applications which was started in 1965. It is an interdisciplinary journal that publishes research on mathematics arising in the physical sciences and engineering as well as suitable articles in the life sciences, social sciences, and finance. Submissions should address interesting and challenging mathematical problems arising in applications. A good balance between the development of the application(s) and the analysis is expected. Papers that either use established methods to address solved problems or that present analysis in the absence of applications will not be considered. The journal welcomes submissions in many research areas. Examples are: continuum mechanics materials science and elasticity, including boundary layer theory, combustion, complex flows and soft matter, electrohydrodynamics and magnetohydrodynamics, geophysical flows, granular flows, interfacial and free surface flows, vortex dynamics; elasticity theory; linear and nonlinear wave propagation, nonlinear optics and photonics; inverse problems; applied dynamical systems and nonlinear systems; mathematical physics; stochastic differential equations and stochastic dynamics; network science; industrial applications.