Mathematical Sciences: Numerical Methods and Computer Software for Solving Integral Equations

本课题研究的是几类线性和非线性积分方程的数值解。具体研究的积分方程类型包括线性和非线性边界积分方程、非线性Urysohn方程和非线性Fredholm方程。这样的方程出现在许多纯粹和应用领域，研究的目标之一是开发多用途代码，这些代码可以安装到软件程序中，使来自许多不同学科的研究人员能够有效和准确地求解这些积分方程。通常，数学物理或数学生物学问题的解可以用积分方程的形式表示。然而，这并不意味着问题已经解决；相反，得到的积分方程必须用解析法或数值法求解。在这个建议中，研究人员将改进现有的数值格式，并开发新的格式，以生产多用途的软件，用于几种类型的积分方程在几个空间维度上的近似解。

The research project is concerned with the numerical solution of several classes of linear and nonlinear integral equations. Specific types of integral equations slated for study include linear and nonlinear boundary integral equations, nonlinear Urysohn equations and nonlinear Fredholm equations. Such equations arise in a host of pure and applied areas, and one of the goals of the research is the development of multi-purpose codes that can be fitted into software routines to enable researchers from many different disciplines to solve these integral equations efficiently and accurately. Very often the solution of a problem in mathematical physics or mathematical biology can be expressed in the form of an integral equation. However this does not mean that the problem is solved; rather, the resulting integral equation must be solved either analytically or numerically. In this proposal the researchers will improve existing numerical schemes and develop new ones in order to produce multi-purpose software for the approximate solution of several types of integral equations in several space dimensions.