Change of variables heat equation

[Dustinsfl]

\[
\alpha^2T_{xx} = T_t + \beta(T - T_0)
\]
where \(\beta\) is a constant and \(T_0\) is the temperature of the surrounding medium. The initial temperature distribution is \(T(x, 0) = f(x)\) and the ends \(x = 0\) and \(x = \ell\) are maintained at \(T_1\) and \(T_2\) when \(t > 0\).

Show that the substitution \(T(x, t) = T_0 + U(x, t)e^{-\beta t}\) reduces the problem to the following one:
\[
\alpha^2U_{xx} = U_t
\]
with new initial conditions and boundary conditions for \(U\).

With that substitution, I obtain:
\begin{align}
\alpha^2U_{xx} &= -\beta(U_t - T_0U)\\
\alpha_1^2U_{xx} &= U_t - T_0U
\end{align}
What is going wrong?

 

[topsquark]

Re: Change of varibles heat equation

\[
\alpha^2T_{xx} = T_t + \beta(T - T_0)
\]
where \(\beta\) is a constant and \(T_0\) is the temperature of the surrounding medium. The initial temperature distribution is \(T(x, 0) = f(x)\) and the ends \(x = 0\) and \(x = \ell\) are maintained at \(T_1\) and \(T_2\) when \(t > 0\).

Show that the substitution \(T(x, t) = T_0 + U(x, t)e^{-\beta t}\) reduces the problem to the following one:
\[
\alpha^2U_{xx} = U_t
\]
with new initial conditions and boundary conditions for \(U\).

With that substitution, I obtain:
\begin{align}
\alpha^2U_{xx} &= -\beta(U_t - T_0U)\\
\alpha_1^2U_{xx} &= U_t - T_0U
\end{align}
What is going wrong?

I'm not quite sure of the problem here. For example,
T = T_0 + U(x, t)e^{- \beta t} \implies T_t = U_t e^{- \beta t} - \beta U e^{- \beta t}

Do the same for U_x and U_xx, then sub into the original equation. There are a ton of cancellations which gives you the final answer.

Are you having problems with the derivatives or is it something else?

-Dan

 

[Dustinsfl]

Re: Change of varibles heat equation

I'm not quite sure of the problem here. For example,
T = T_0 + U(x, t)e^{- \beta t} \implies T_t = U_t e^{- \beta t} - \beta U e^{- \beta t}

Do the same for U_x and U_xx, then sub into the original equation. There are a ton of cancellations which gives you the final answer.

Are you having problems with the derivatives or is it something else?

-Dan

I forgot to use the product rule