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Q3.14 The steady temperature distribution in a square plate. The temperature,
in a plate is described by the heat conduction equation (written in non-
dimensional variables)

where the plate is with the temperature on the boundary
given by
and

Seek the first term of a composite expansion by writing

where is the relevant boundary-layer variable. Determine
completely and and then show that

where
Q3.15 Mathieu’s equation for n = 2. See E3.6; find the asymptotic expansion of as
far as the term in the case n = 2 (and there will be two versions of this,
depending on the choice of either sin or cos).
Q3.16 Mathieu’s equation based on Floquet theory. Write the Mathieu equation

as an equation in by setting where is a
constant; is periodic with period or (Note that the transitional
curves are now Seek a solution

in the case n = 1; because we now have the solution will be that which
is valid near the transitional curves. Show that

where is a free parameter.