Hi. Having problems with this tricky Heat Equation Question. Managed to do part (a) and would appreciate verification that it's right.(adsbygoogle = window.adsbygoogle || []).push({});

But I can't manage to finish off the second part. I've started it off so please do advice me. Thanks a lot!

QUESTION:

-----------------------------------------

(a)

Show that the steady solution (which is independent of t) of the heat equation,

[itex]

\frac{{\partial ^2 \theta }}

{{\partial x^2 }} = \frac{1}

{{\alpha ^2 }}\frac{{\partial \theta }}

{{\partial t}}

[/itex] where [itex]\alpha[/itex] is a constant, on the interval:

[itex]

- L \leqslant x \leqslant L

[/itex] with conditions: [itex]\begin{gathered} \theta ( - L,t) = 0 \hfill \\

\theta (L,t) = T \hfill \\

\end{gathered} [/itex] is: [itex]\theta = \theta _0 (x) = \frac{{T(1 + \frac{x}

{L})}}

{2}[/itex]

(b)

Use methods of separation of variables to show that the unsteady solution for

[itex]\theta = (x,t)[/itex] withconditions: [itex]\theta ( - L) = T,{\text{ }}\theta (L) = 0,{\text{ }}\theta (x,0) = \theta _0 (x){\text{ is}}[/itex]:

[itex]

\theta (x,t) = \frac{T}

{2}\left[ {1 - \frac{x}

{L}} \right] - 2T\sum\limits_{n = 1}^\infty {\left[ {e^{ - \alpha ^2 n^2 \pi ^2 t/L^2 } \frac{{( - 1)^n \sin (n\pi x/L)}}

{{n\pi }}} \right]}[/itex]

My attempt:

Part (A)::

-----------------------

Using separation of variables:

[itex]\begin{gathered}

Let:\theta = (x,t) = X(x)T(t) \hfill \\

X(x) = Ax + B \hfill \\

T(t) = C \hfill \\

so:X(x)T(t) = (Ax + B)(C) \hfill \\

\end{gathered} [/itex]

now to use the conditions:

----------------------------------

[itex]\begin{gathered}

when:\theta ( - L,t) = 0, \hfill \\

A(x + L) + B = 0 \hfill \\

A(0) + B = 0,so:B = 0 \hfill \\

\end{gathered} [/itex]

[itex]\begin{gathered}

when:\theta (L,t) = T \hfill \\

A(x + L) + 0 = T \hfill \\

A(2L) = T \hfill \\

A = \frac{T}

{{2L}} \hfill \\

\end{gathered}[/itex]

[itex]\begin{gathered}

so: \hfill \\

\theta _0 (x) = \frac{T}

{{2L}}(x + L) = \frac{{T(1 + \frac{x}

{L})}}

{2} \hfill \\

\end{gathered}[/itex]

That's part (A) done. is my method to approach the final answer correct?

Part (B)::

-----------------------

I'm having problems here. Can't quite finish the question off. Please could you guide me out here. Thanks.

Using separation of variables:

[itex]Let:\theta = (x,t) = X(x)T(t)[/itex]

[itex]\begin{gathered}

unsteady - solution: - \rho ^2 < 0 \hfill \\

X(x) = Ae^{i\rho x} + Be^{ - ipx} = A\cos px + B\sin px \hfill \\

T(t) = Ce^{ - \alpha ^2 \rho ^2 t} \hfill \\

so:X(x)T(t) = (A\cos \rho x + B\sin \rho x)(Ce^{ - \alpha ^2 \rho ^2 t} ) \hfill \\

\end{gathered}[/itex]

now to use the conditions:

------------------------------

[itex]\begin{gathered}

\theta ( - L) = T:so: \hfill \\

(A\cos \rho (x + L) + B\sin \rho (x + L))(e^{ - \alpha ^2 \rho ^2 t} ) = T \hfill \\

(A\cos \rho ( - L + L) + B\sin \rho ( - L + L))(e^{ - \alpha ^2 \rho ^2 t} ) = T \hfill \\

(A\cos (0) + B\sin (0))(e^{ - \alpha ^2 \rho ^2 t} ) = T \hfill \\

\left[ {A(e^{ - \alpha ^2 \rho ^2 t} )} \right] = T \hfill \\

\end{gathered}[/itex]

[itex]\begin{gathered}

\theta (L) = 0:so: \hfill \\

(A\cos (2\rho L) + B\sin (2\rho L))(e^{ - \alpha ^2 \rho ^2 t} ) = 0 \hfill \\

\end{gathered}[/itex]

[itex]

\begin{gathered}

\theta (x,0) = \theta _0 (x) = \frac{{T(1 + \frac{x}

{L})}}

{2}:so: \hfill \\

(A\cos \rho (x + L) + B\sin \rho (x + L))(e^{ - \alpha ^2 \rho ^2 (0)} ) = \frac{{T(1 + \frac{x}

{L})}}

{2} \hfill \\

(A\cos \rho (x + L) + B\sin \rho (x + L))(1) = \frac{{T(1 + \frac{x}

{L})}}

{2} \hfill \\

\end{gathered}[/itex]

I'm stuck here. Any ideas on how to approach the final answer (as shown on the question)????

Thanks so much.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Heat Equation Problem

**Physics Forums | Science Articles, Homework Help, Discussion**