While an integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions is available in the literature, a proof of this formula seems to be missing. Moreover, the available formula contains an integral term including the time derivative of the fundamental solution of the heat equation, whose interpretation is difficult at second glance. To fill these gaps we provide a rigorous proof of a general version of the integration by parts formula and an alternative representation of the mentioned integral term, which is valid for a certain class of functions including the typical tensor-product discretization spaces.

翻译：虽然文献中可以找到三个空间维度的超双线线性高度边界整体操作器双线式双线式集成公式,但似乎缺少这一公式的证明,此外,现有公式包含一个整体术语,包括热方程基本解决方案的时间衍生物,其解释在第二眼很难理解,为填补这些空白,我们提供了严格证据,证明按部分公式集成的通用版本,以及上述整体术语的替代表述,该术语适用于某类功能,包括典型的高压产品离散空间。