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求解偏微分方程边值问题的有效方法之一是边界元法。本文针对二维线性问题讨论了边界元法与偏微分方程完备解之间的关系。日前已经发展了许多边界元方法,但是所有这些不同方法的基本边界积分方程都是相应微分方程的完备解。基于这一思想,加以发展,便可方便地构造出更多新的边界元方法。它们可以被应用到许多迄今尚不能有效地采用边界元法求解的问题中去。虽然偏微分方程的完备解可能有许多种表达方式,但为了数值离散求解,仍需选择其中的适当形式。对于通常的边值问题,边界元法中的边界积分方程是一种较好的表达式。但对于某些特殊问题,仍然需要加以修正。另一方面,在许多情况下,常规边界元法的边界积分方程较难求出,而其它形式的完备解却较易获得。例如对于很大一部分常系数偏微分方程,可以用广义付氏变换的方法求得问题的完备解。对于这些完备解,用复变函数的方法住往可将其转换成边界积分方程的形式,更利于数值计算和问题的求解。
One of the effective methods for solving the boundary value problem of partial differential equations is the boundary element method. This paper discusses the relationship between the boundary element method and the complete solution of partial differential equations for two-dimensional linear problems. Many boundary element methods have been developed recently, but the basic boundary integral equations of all these different methods are complete solutions of the corresponding differential equations. Based on this idea and development, more new boundary element methods can be easily constructed. They can be applied to many problems that have hitherto failed to be effectively solved by the boundary element method. Although the complete solution of partial differential equations may have many expressions, it is still necessary to choose the appropriate form for numerical discrete solutions. For the general boundary value problem, the boundary integral equation in the boundary element method is a better expression. However, some special issues still need to be corrected. On the other hand, in many cases, the boundary integral equation of the conventional boundary element method is difficult to find, but other forms of complete solutions are easier to obtain. For example, for a large part of constant-coefficient partial differential equations, the complete solution of the problem can be obtained by the method of generalized Fourier transformation. For these complete solutions, the method of using complex variable functions can be converted into the form of boundary integral equations, which is more conducive to numerical calculations and problem solving.