Math Help

Dirichlet Applications - Heat Conduction in a Plate

In a Nut Shell: The governing equation for heat conduction in a plate is:

∂u/∂t = k [∂²u/∂x²+ ∂²u/∂y²] ----------------------- (1)

where u = u(x,y,t) = the temperature in the plate at any time t

x,y = the location in the plate

t = the time at which the temperature at x is u(x,y,t)

and k is the thermal diffusivity of the material

The desired outcome is to predict the temperature distribution, u(x,y,t), in the plate

as a function of time, t.

For steady-state heat conduction, ∂u/∂t = 0. So the steady-state temperature

distribution in the plate is governed by Laplace’s equation:

∂²u/∂x²+ ∂²u/∂y² = 0 -------------------------------------- (2)

Strategy: Use the method of separation of variables to solve (2) subject to the

boundary conditions.

Consider a thin plate with dimensions (a by b) shown below. The objective is to find

the steady state temperature distribution given boundary conditions on each edge of

the plate.

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