Mathematical Sciences: Mathematical Problems in Various Applied Fields

[401405] Keller将对应用数学中的一些不同问题进行研究。本文将研究下列问题：(1)将导出一个精确的非反射边界条件的结果推广到麦克斯韦方程下的电磁波和线性弹性理论下的弹性波；(2)建立了包含非反射边界条件的标量波动方程散射问题的有限元公式；(3)利用渐近方法对迭代数值法第n步误差进行分析；(4)弹性薄片的折叠和绳索落在水平面上时的盘绕；(5)共振相互作用波的研究。微扰、渐近和数值方法将被发展并用于研究应用数学中的许多问题。这些问题将包括绳子在水平面上的盘绕，薄片在水平面上的折叠，采用非局部精确非反射边界条件的数值方法的发展，使用渐近方法分析各种迭代计算方案的误差，以及非线性波（如流体表面的重力波）的共振相互作用。

9401405 Keller Research will be conducted on a number of different problems in applied mathematics. The following problems will be studied: (1) the extension of a result deriving an exact nonreflecting boundary condition to electromagnetic waves governed by Maxwell's equations and to elastic waves governed by the linear theory of elasticity; (2) the development of a finite element formulation of scattering problems for the scalar wave equation incorporating the nonreflecting boundary condition; (3) the use of asymptotic methods to analyze the error at the nth step of an iterative numerical method; (4) the folding of elastic sheets and the coiling of ropes when they fall onto a horizontal surface, and (5) the study of resonantly interacting waves. Perturbation, asymptotic, and numerical methods will be developed and utilized to study a number of problems in applied mathematics. These problems will include the coiling of a rope falling on a horizontal plane, the folding of a sheet falling on a horizontal plane, the development of numerical methods employing exact nonreflecting boundary conditions which are nonlocal, the use of asymptotic methods to analyze the error in various iterative computational schemes, and the resonant interaction of nonlinear waves, such as gravity waves on the surface of a fluid.