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- #1

$$

\overline{T} = \alpha T + \beta

$$

Since $\overline{T} = \alpha T + \beta$, the partial derivatives are

\begin{alignat*}{3}

\overline{T}_t & = & \alpha T_t\\

\overline{T}_{xx} & = & \alpha T_{xx}

\end{alignat*}

So $T_t = \frac{1}{\alpha}\overline{T}_t$ and $T_{xx} = \frac{1}{\alpha}\overline{T}_{xx}$.

The diffusion equation is

$$

\frac{1}{\alpha}T_t = T_{xx}.

$$

By substitution, we obtain

$$

\frac{1}{\alpha}\overline{T}_t = \overline{T}_{xx}.

$$

Correct?