A Software Tool for Analytical Solutions of Laplace's Equation

一种提供拉普拉斯方程解析解的软件工具比数值技术有几个优点。工程和物理许多领域的学生将受益于研究解的参数依赖性的机会。位势理论的一些方面，如高曲面和导数变量，对数值技术来说是困难的，但对解析方法来说却很简单，因此对新结构和新装置的分析和设计将变得更加精确。使用解析解的主要障碍是将形状与解相匹配的困难，以及需要使用代数和微积分进行大量乏味的符号计算。通过实现一个接口，该接口引导用户通过一系列图形菜单，以各种格式（包括图形、计算机子程序和文字处理元素）找到所需的解决方案，从而克服了这些困难。项目的第一阶段致力于研究出能够使用基于计算机的符号数学包（Mathematica）生成拉普拉斯方程解的技术，并测试程序的最小版本，以确定该接口是否适合初级工程学生。

A software tool which presents analytical solutions of Laplace's equation offers several advantages over numerical techniques. Students in many areas of engineering and physics will benefit from the opportunity to study parametric dependence of the solutions. Aspects of potential theory that are difficult for numerical techniques, such as highly curved surfaces and derivative variables, are simple for analytical methods, so both analysis and design of new structures and devices will become more accurate. The principal obstacles to the use of analytical solutions have been the difficulty of matching the shape to a solution, and the need to perform a large number of tedious symbolic calculations using algebra and calculus. These difficulties are overcome in the proposed work by implementing an interface which guides the user through a sequence of graphical menus leading to the desired solution in a variety of formats, including, graphs, computer subprograms, and word processing elements. Phase I of the project is devoted to working out the techniques which will enable the solutions of Laplace's equation to be generated using a computer based symbolic mathematics package (Mathematica), and to testing a minimal version of the program for the suitability of the interface for Junior level engineering students.