  A novel perturbative method, proposed by Panda {\it et al.} [1] to solve the
Helmholtz equation in two dimensions, is extended to three dimensions for
general boundary surfaces. Although a few numerical works are available in the
literature for specific domains in three dimensions such a general analytical
prescription is presented for the first time. An appropriate transformation is
used to get rid of the asymmetries in the domain boundary by mapping the
boundary into an equivalent sphere with a deformed interior metric. The
deformed metric produces new source terms in the original homogeneous equation.
A deformation parameter measuring the deviation of the boundary from a
spherical one is introduced as a perturbative parameter. With the help of
standard Rayleigh-Schr{\"o}dinger perturbative technique the transformed
equation is solved and the general solution is written down in a closed form at
each order of perturbation. The solutions are boundary condition free and which
make them widely applicable for various situations. Once the boundary
conditions are applied to these general solutions the eigenvalues and the
wavefunctions are obtained order by order. The efficacy of the method has been
tested by comparing the analytic values against the numerical ones for three
dimensional enclosures of various shapes. The method seems to work quite well
for these shapes for both, Dirichlet as well as Neumann boundary conditions.
The usage of spherical harmonics to express the asymmetries in the boundary
surfaces helps us to consider a wide class of domains in three dimensions and
also their fast convergence guarantees the convergence of the perturbative
series for the energy. Direct applications of this method can be found in the
field of quantum dots, nuclear physics, acoustical and electromagnetic
cavities.
