Question 3.2: Calculate the temperature at 1 cm inside the steel plate of ......

The temperature at the interior point of interest x_{P}=1\operatorname{cm}\operatorname{is}T(x_{P},t) and increases with time. If the piecewise-linear approximation of the applied surface temperature is based on a time step Δt = 1 s, by using Eq. (3.31) with θ_{0,0} = 0 the temperatures at t = l, 2 and 3 seconds are given, respectively, by

T_{1}(x_{P})=T_{i n}+\theta_{0,1}\mu_{1}(\tilde{x}_{P})=30.112681\ ^{\circ}C (3.37a)

T_{2}(x_{P})=T_{i n}+\theta_{0,1}\left[\Delta(\Delta\mu(\tilde{x}_{P}))_{1}\right]+\theta_{0,2}\mu_{1}(\tilde{x}_{P})=31.480690^{\circ}C (3.37b)

T_{3}(x_{P})=T_{i n}+\theta_{0,1}{\big[}\Delta(\Delta\mu({\tilde{x}}_{P}))_{2}{\big]}+\theta_{0,2}{\big[}\Delta(\Delta\mu({\tilde{x}}_{P}))_{1}{\big]}+\,\theta_{0,3}\Delta\mu_{1}(\tilde{x}_{P})=34.613044 °C (3.37c)

where \mu_{M}(\tilde{x}_{P})=\tilde{T}_{\mathrm{X12B20T0}}(\tilde{x}_{P},M\Delta\tilde{t},\tilde{t}_{r e f}=\Delta\tilde{t}), with M = 1, 2 or 3. Also, \tilde{x}_{P}=x_{P}/L=0.2,\mathrm{~}\Delta\tilde{t}=\alpha\Delta t/L^{2}=0.004,, and the first six decimal-places have been considered for the building block solution (A = 6).

The exact temperatures can be obtained from the fdX12B20T0.m MATLAB function of Section 2.3.7 with an accuracy of one part in 10^{15} (A = 15) in such a way as the computational analytical solution can be assumed exact (see Section 2.3.5.1).

They are 30.112681, 31.480690, and 34.613044 °C, respectively, as already given in Example 3.1.

Therefore, the approximate and exact temperatures are the same (as expected) as the applied surface temperature changes linearly with time and, hence, the piecewise-linear approximation cannot perform any approximation in the current case. This is an application of “numerical” intrinsic verification of Eq. (3.31) and, hence, of the fdX12B_0T0_pla.m MATLAB function (D’Alessandro and de Monte 2018).