The Perturbation Method due to the Small Change in the Shape of the Boundary

Abstract

A general method is presented for estimating the effect caused by a small change in the boundary shape (or the hypersurface shape in the many-dimensional space, the time-coordinate being included) upon the solutions or eigenvalues of linear partial differential equations (elliptic, parabolic or hyperbolic). When the given boundary is slightly deformed from a shape favorable for obtaining the solutions, this method gives the approximate solutions or eigenvalues based upon the exact or unperturbed ones for the case of the favorable shape, where many solutions can be obtained easier. The following illustrative examples are given, showing the validity of the method presented: (1) The eigenfrequencies of the lateral vibration of a uniform membrane fixed at its boundary close to a circle in shape, e.g. elliptic, oval, cinquefoil-shaped, etc. (2) The eigenvalue problem concerning the heat conduction of a “corned-beef-can” -shaped uniform matter, close to a rectangular parallelepiped in shape and with its tempera...