The goals of this paper are twofold: selection of pseudo-boundaries for sources nodes in the method of fundamental solutions (MFS), and comparisons of the MFS, the method of particular solutions (MPS) and the MFS-QR of Antunes. To pursue better pseudo-boundaries, we provide new estimates of the condition number (Cond) by the MFS for arbitrary pseudo-boundaries, and propose a new sensitivity index of stability via accuracy. Numerical experiments and comparisons are carried out to verify the analysis made. For five-pedal-flower-like domains, numerical comparisons are made by the sensitivity index. Circular pseudo-boundaries are optimal for highly smooth solutions, but the pseudo-boundaries near the domain boundary may be better for singular solutions. In this paper the gap has been shortened between theoretical analysis and numerical computation of the MFS, to provide some guidance for users. This is the first goal of this paper. The second goal is to compare the MFS, the MPS and the MFS-QR. Characteristics of the MFS-QR are explored. The new basis functions of the MFS-QR are the very particular solutions (PS), and the MFS-QR may be regarded as a special MPS. The MFS-QR is not a variant of the MFS but a variant of the MPS. The MFS-QR also plays a role in bridging from the MFS to the MPS. Both the MFS and the MPS can also be recognized as twins via the MFS-QR in the Trefftz family. The comparisons in this paper are more comprehensive.

本文的目标是双重的：在基本解法（MFS）中选择源节点的伪边界，并对MFS，特殊解法（MPS）和Antunes的MFS-QR进行比较。为了追求更好的伪边界，我们通过MFS为任意伪边界提供了条件数（Cond）的新估计，并通过准确性提出了新的稳定性敏感性指标。进行了数值实验和比较，以验证所做的分析。对于五瓣花状区域，通过灵敏度指数进行数值比较。对于高度平滑的解，圆形伪边界是最佳的，但是对于奇异解，靠近域边界的伪边界可能更好。本文缩短了MFS的理论分析与数值计算之间的差距，为用户提供了一些指导。这是本文的首要目标。第二个目标是比较MFS，MPS和MFS-QR。探索了MFS-QR的特性。MFS-QR的新基本功能是非常特殊的解决方案（PS），并且MFS-QR可以视为特殊的MPS。MFS-QR不是MFS的变体，而是MPS的变体。MFS-QR在从MFS到MPS的桥接中也起着作用。通过Trefftz系列中的MFS-QR，MFS和MPS都可以被识别为双胞胎。本文中的比较更加全面。探索了MFS-QR的特性。MFS-QR的新基本功能是非常特殊的解决方案（PS），并且MFS-QR可以视为特殊的MPS。MFS-QR不是MFS的变体，而是MPS的变体。MFS-QR在从MFS到MPS的桥接中也起着作用。通过Trefftz系列中的MFS-QR，MFS和MPS都可以被识别为双胞胎。本文中的比较更加全面。探索了MFS-QR的特性。MFS-QR的新基本功能是非常特殊的解决方案（PS），并且MFS-QR可以视为特殊的MPS。MFS-QR不是MFS的变体，而是MPS的变体。MFS-QR在从MFS到MPS的桥接中也起着作用。通过Trefftz系列中的MFS-QR，MFS和MPS都可以被识别为双胞胎。本文中的比较更加全面。