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Generalised Fourier transforms and moving boundary value problems

广义傅里叶变换和移动边值问题

基本信息

批准号：
EP/E022960/1
负责人：
Beatrice Pelloni
金额：
$ 28.54万
依托单位：
University of Reading
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FE022960%2F1
关键词：
Generalised Fourier transforms moving boundary

项目摘要

The aim of the proposed research is the analysis and the numerical solution of equations which describe the evolution of some quantity, for example a fluid or a temperature, from a known initial state. In the case we study, this evolution is restricted in a certain region, and as time progresses we have some information on the state of the system at the boundary of the region. For example, we might know that on the boundary the temperature, or the velocity of the fluid, is always zero. These type of problems are called boundary value problems, and are ubiquitous in the mathematical description of our physical reality. In the cases we want to study, in addition to these data, we know that the boundary moves with time in a way that is either prescribed, or is part of the problem to be solved. In the latter case, such problems are called free-boundary value problems, and arise for example in studying the formation of ice interfaces in flowing water.We consider such problems for an important class of equations, that describe many physical evolution phenomena. These equations are called integrable. In order to understand the behaviour of their solutions in a time-dependent domain, we start with the simpler linear case.We expect to be able to give a general method to study such problems based on recent advances in the study of integrable equations. These are based also on a more strictly mathematical study of certain transforms that generalise the Fourier transform, which one of the most important tools applied mathematicians have at their disposal for studying differential equations.In both the linear and the nonlinear case, an important part of the research is the numerical evaluation of the formulas obtained. These formulas appear to be very convenient for the purpose of numerical evaluation, because of certain decay properties that translate into fast numerical convergence to a good approximation of the theoretical solution. Hence we plan to devise robust and accurate numerical algorithms for the evaluation of these representation formulas.

拟议研究的目的是分析和方程的数值解，这些方程描述了从已知初始状态开始的某些量的演变，例如流体或温度。在我们研究的情况下，这种演化被限制在一定的区域内，随着时间的推移，我们有一些关于该区域边界处系统状态的信息。例如，我们可能知道，在边界上，流体的温度，或速度，总是为零。这些类型的问题被称为边值问题，并且在我们物理现实的数学描述中无处不在。在我们要研究的情况下，除了这些数据，我们还知道边界随时间移动的方式要么是规定的，要么是待解决问题的一部分。在后一种情况下，这样的问题被称为自由边界值问题，并出现在例如在研究冰的界面在流动的water.We考虑一类重要的方程，描述了许多物理演化现象的形成这样的问题。这些方程称为可积的。为了理解它们的解在含时区域中的行为，我们从较简单的线性情形开始，期望能够根据可积方程研究的最新进展，给出研究这类问题的一般方法。这些也是基于一个更严格的数学研究某些变换，概括傅立叶变换，其中一个最重要的工具，应用数学家在他们的处置研究微分方程。在线性和非线性的情况下，一个重要的部分，研究是数值评估的公式获得。这些公式似乎是非常方便的数值计算的目的，因为某些衰减特性，转化为快速的数值收敛到一个很好的近似的理论解。因此，我们计划设计强大的和准确的数值算法，这些代表性的公式进行评估。