In this study, the singular boundary method (SBM) is extended to solve 3D inhomogeneous elliptic boundary value problems. Before applying the SBM, the multiple reciprocity method (MRM) removes the inhomogeneous terms and establishes an equivalent high-order homogeneous problem, which is then solved by the SBM, a boundary-only fundamental-solution based collocation method. It is worth-noting that the conventional MRM is not applicable in most cases because polyharmonic operators cannot deal with most of the source terms. Thus, in this study a recursive multiple reciprocity method is developed via different operators. The high-order fundamental solutions are involved as kernel functions in solving a high-order homogeneous problem by the SBM. It is of great importance to remove the source singularity of the kernels when the collocation points coincide with the source points. In this study, simple formulas are analytically derived to desingularize the source singularity of kernel functions. The accuracy and efficiency of the proposed method are illustrated via several numerical examples governed by different partial differential equations.

在这项研究中，扩展了奇异边界方法（SBM）来解决3D不均匀椭圆边界值问题。在应用SBM之前，多重互易方法（MRM）消除了不均匀项，并建立了一个等效的高阶齐次问题，然后通过SBM（基于边界的基本解决方案并置方法）解决了该问题。值得注意的是，传统的MRM在大多数情况下不适用，因为多谐波运算符无法处理大多数源项。因此，在这项研究中，通过不同的算子开发了一种递归多重互易方法。SBM将高阶基本解作为内核函数来解决高阶齐次问题。当搭配点与源点重合时，删除内核的源奇点非常重要。在这项研究中，通过分析得出简单的公式以将核函数的源奇异点去单数化。通过几个由不同偏微分方程控制的数值示例，说明了该方法的准确性和效率。