Modeling with PDE, MA 461

The Problem
The uniform metal plate with thermal diffusivity 1m2/s is represented by a rectangle with
opposite corners at (−0.5, −0.8) and (0.5, 0.8). It is assumed that the plate has a hollow crack
or cavity represented by the rectangle with opposite corners at (−0.05, −0.4) and (0.05, 0.4).
The left side of the plate is heated to 100◦C, while on the right side the heat is flowing out
at a constant rate of 10W/m2
; all other boundaries (the top, bottom, and interior sides of
the cavity) are assumed insulated. The entire plate is initially at 0◦C.
The PDE for the temperature u(x, y, t) at time t seconds at a point in the plate with
coordinates (x, y) takes the form
∂t = ∆u.
The region is bounded on the outside by the boundary of the large rectangle, and on the
inside by the boundary of the inner rectangle. The boundary conditions are given by the
Dirichlet condition



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