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

the journal of integral equations and applications/journal of integral equations and applications

An integration by parts formula for the bilinear form of the  hypersingular boundary integral operator for the transient heat equation in  three spatial dimensions