# PDE model of metal rod at temperature=1 plunged into a bath of temperature=0

Consider a metal rod ($0 < x < l$), insulated along its sides but not at its ends, which is initially at temperature=1. Suddenly both ends are plunged into a bath of temperature=0. Write the PDE, boundary conditions, and initial condition that model the temperature of the rod. Write the formula for the temperature $u(x,t)$ at position $x$ and time $t$ for later times. In this problem assume identity

$$1=(4/\pi)[sin(\pi x/l)+(1/3)sin(3\pi x/l)+(1/5)sin(5\pi x/l)+...]$$

holds for all $0 < x < l$

## 1 Answer

This appears to be an example of a heat equation.

Let $u(x,t)$ denote the temperature at $x$ at a given time $t$ of a rod of length $l$.

We may thus describe the heat distribution as $$\frac{\partial u}{\partial t} = a^2\frac{\partial^2 u}{\partial x^2}$$ where $a^2$ is known as the thermal diffusivity, which depends on the material which the bar is made of. You can read more about this here.

The above PDE must be considered subject to boundary conditions \begin{align}u(0,t) &= 0 ~~~~~~~ t>0 \\ u(l,t) &= 0 ~~~~~~~ t>0\end{align} with initial condition $$u(x,0) = 1 ~~~~ 0 \le x\le l$$

You may now solve this PDE in order to obtain the solution of $u(x,t)$ such that your given boundary conditions are met. I would suggest Seperation of Variables.

• I would like to add to this answer that the equation would be solved with the help of the fourier series identity. You have the answer 1 = (identity of sin and cosin), you will have to find the values of x based on that. I will search for an example and post it as an answer if more clarification is needed. – Engin007 Oct 24 '16 at 7:58