G. I. Shishkin [Comput. Math. Math. Phys. 31, No. 10, 53–63 (1991); translation from Zh. Vychisl. Mat. Mat. Fiz. 31, No. 10, 1498–1511 (1991; Zbl 0786.65073)] has studied an initial boundary problem under certain Dirichlet data. Estimates for the difference between the solution over the numerical solution have been obtained in the foregoing paper. The authors sharpen this result when the problem considered by Shishkin is replaced by a simpler problem. Problems like this arise when one models the transfer of heat over a rectangle in a medium moving with velocity \(x^\alpha \) along the \(x\)-axis and conducting heat only across the flow.