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 †’ ˆž for every fixed t ‰¥ 0, and w(0, t) = f(t). Proceed as follows.

Show that in Prob. 11,

and the convolution theorem gives Duhamel€™s formula

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Vax/c Wo (x, s) = e-Vazje W(x, t) = — т) f(t awo dт. дт 0.