Dynamic linear equations in the presence of small perturbations of an interface

We consider solutions to the initial boundary value problems for both heat and wave equations in two dimensions. In this paper, we aim to advance the development of high-order terms in the asymptotic expansions of current boundary perturbations caused by small perturbations of the shape of a $C^2$-boundary inhomogeneity. Our derivations are rigorously derived by the field expansion (FE) method (formal derivation) and proved by layer potential techniques. It extends those already derived for small-volume inhomogeneities to develop efficient algorithms for determining certain properties of the shape of an inhomogeneity based on boundary measurements.