Eigenpair derivative with respect to boundary shape

This work focuses on how changes of boundary shapes affect eigenvalues and eigenfunctions in continuous systems. The governing equation, plus its boundary conditions for the eigenpair derivatives, is derived using the delta-function method, which can facilitate the differentiation of boundary conditions with respect to boundary shapes. Even though the eigenproblem equation with its corresponding boundary conditions is homogeneous, the governing equation with its boundary conditions for the eigenpair derivatives may be nonhomogeneous. A transformation method is proposed to transform the differential equation with nonhomogeneous boundary conditions into a new problem with homogeneous boundary conditions so that the eigenfunctions form a complete set for this new problem,The explicit results for the eigenpair derivatives are given, and an example is presented to illustrate the method and its validity.