Question

Under ideal assumptions, what is the two-dimensional heat equation?

(a) ut  = c∇^2 u = c(uxx  + uyy)

(b) ut  = c^2 uxx

(c) ut  = c^2 ∇^2 u = c^2 (uxx  + uyy)

(d) ut  = ∇^2 u = (uxx  + uyy)

I have been asked this question in an interview for internship.

The question is from Derivation and Solution of Two-dimensional Heat Equation in section Applications of Partial Differential Equations of Engineering Mathematics

Answer 1

The correct answer is (c) ut  = c^2 ∇^2 u = c^2 (uxx  + uyy)

To explain: Consider a rectangular plate (thermally conductive material), with dimensions a × b. The plate is heated and then insulated.

We let u (x, y, t) = temperature of plate at position (x, y) and time t.

For a fixed t, the height of the surface z = u (x, y, t) gives the temperature of the plate at time t and position (x, y).

Under ideal assumptions (e.g. uniform density, uniform specific heat, perfect insulation, no internal heat sources etc.) one can show that u satisfies the two-dimensional heat equation,

ut  = c^2 ∇^2 u = c^2 (uxx  + uyy)        for 0 < x < a, 0 < y < b.