| course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
52 |
|
Paper 2, Section II, B |
2013 |
The steady-state temperature distribution $u(x)$ in a uniform rod of finite length satisfies the boundary value problem
$$\begin{gathered}
-D \frac{d^{2}}{d x^{2}} u(x)=f(x), \quad 0<x<l \\
u(0)=0, \quad u(l)=0
\end{gathered}$$
where $D>0$ is the (constant) diffusion coefficient. Determine the Green's function $G(x, \xi)$ for this problem. Now replace the above homogeneous boundary conditions with the inhomogeneous boundary conditions $u(0)=\alpha, \quad u(l)=\beta$ and give a solution to the new boundary value problem. Hence, obtain the steady-state solution for the following problem with the specified boundary conditions:
$$\begin{aligned}
&-D \frac{\partial^{2}}{\partial x^{2}} u(x, t)+\frac{\partial}{\partial t} u(x, t)=x, \quad 0<x<1 \\
&u(0, t)=1 / D, \quad u(1, t)=2 / D, \quad t>0
\end{aligned}$$
[You may assume that a steady-state solution exists.]