[SOLVED]Change of variables heat equation

dwsmith

Well-known member
$\alpha^2T_{xx} = T_t + \beta(T - T_0)$
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

Well-known member
MHB Math Helper
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

dwsmith

Well-known member
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