Find the temperature w(x, t) in a semi-infinite laterally insulated bar extending from x = 0 along the x-axis to infinity, assuming that the initial temperature is 0, w(x, t) → 0 as x → ∞ for every fixed t ≥ 0, and w(0, t) = f(t). Proceed as follows.

Set up the model and show that the Laplace transform leads to

and