  Many problems in physics, biology, and economics depend upon the duration of
time required for a diffusing particle to cross a boundary. As such,
calculations of the distribution of first passage time, and in particular the
mean first passage time, is an active area of research relevant to many
disciplines. Exact results for the mean first passage time for diffusion on
simple geometries, such as lines, discs and spheres, are well--known. In
contrast, computational methods are often used to study the first passage time
for diffusion on more realistic geometries where closed--form solutions of the
governing elliptic boundary value problem are not available. Here, we develop a
perturbation solution to calculate the mean first passage time on irregular
domains formed by perturbing the boundary of a disc or an ellipse. Classical
perturbation expansion solutions are then constructed using the exact solutions
available on a disc and an ellipse. We apply the perturbation solutions to
compute the mean first exit time on two naturally--occurring irregular domains:
a map of Tasmania, an island state of Australia, and a map of Taiwan. Comparing
the perturbation solutions with numerical solutions of the elliptic boundary
value problem on these irregular domains confirms that we obtain a very
accurate solution with a few terms in the series only. Matlab software to
implement all calculations is available on GitHub.
