Generalised Fourier transforms and moving boundary value problems

所提出的研究的目的是分析和数值求解描述某些量的演化的方程，例如，流体或温度，从已知的初始状态。在我们研究的情况下，这种演化被限制在某个区域内，随着时间的推移，我们获得了关于该区域边界处系统状态的一些信息。例如，我们可能知道，在边界上，温度或流体的速度总是为零。这类问题被称为边值问题，在我们的物理现实的数学描述中无处不在。在我们想要研究的情况下，除了这些数据，我们知道边界随着时间以一种规定的方式移动，或者是需要解决的问题的一部分。在后一种情况下，这类问题被称为自由边值问题，例如在研究流动水中冰界面的形成时就出现了这样的问题。我们将这类问题考虑为一类重要的方程，它描述了许多物理演化现象。这些方程称为可积方程。为了了解它们在含时区域中的解的性态，我们从更简单的线性情形入手，期望能够基于可积方程研究的最新进展，给出一种研究这类问题的一般方法。这些也是基于对推广了傅里叶变换的某些变换的更严格的数学研究，傅立叶变换是应用数学家用于研究微分方程的最重要的工具之一。在线性和非线性情况下，研究的一个重要部分是对所获得的公式进行数值评估。这些公式对于数值计算来说似乎是非常方便的，因为某些衰减性质可以转化为快速的数值收敛到理论解的良好近似值。因此，我们计划设计稳健和准确的数值算法来评估这些表示公式。

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.