PDE Separation Problem

**jezzyjez** · April 13th 2010, 03:28 AM

Hey I have been given the below question but am really struggling to get even the first part:

We want to find the value of the electrostatic potential $\phi(r,\theta)$ in the quadrant.

$0 \leq r \leq b$ , $0 \leq \theta \leq \frac{\pi}{2}$

where $r$ and $\theta$ are the standard polar co-ordinates and $b$ is a given positive real number. Assuming electrostatic potential is the solution of Laplaces equation in polar co-ordinates

$\frac{\delta^2 \phi}{\delta r^2} +\frac{1}{r}\frac{\delta\phi}{\delta r} + \frac{1}{r^2} \frac{\delta^2\phi}{\delta\theta^2} = 0$

The boundary conditions on $\phi(r,\theta)$ are

$\phi(b,\theta) = \frac{4}{\pi^2} \theta(\pi - \theta),$

$\phi(r,0) = 0, \phi(r,\frac{\pi}{2}) = 1$

(a) State why the differential equation with the given boundary conditions can't be solved by the method of separating variables.

The problem I am having is that I can separate the equation into to parts so I am assuming it is to do with the boundary conditions why it cannot be solved, but I don't know what aspect of the conditions effects this?

**Danny** · April 13th 2010, 05:24 AM

Originally Posted by jezzyjez

Hey I have been given the below question but am really struggling to get even the first part:

We want to find the value of the electrostatic potential $\phi(r,\theta)$ in the quadrant.

$0 \leq r \leq b$ , $0 \leq \theta \leq \frac{\pi}{2}$

where $r$ and $\theta$ are the standard polar co-ordinates and $b$ is a given positive real number. Assuming electrostatic potential is the solution of Laplaces equation in polar co-ordinates

$\frac{\delta^2 \phi}{\delta r^2} +\frac{1}{r}\frac{\delta\phi}{\delta r} + \frac{1}{r^2} \frac{\delta^2\phi}{\delta\theta^2} = 0$

The boundary conditions on $\phi(r,\theta)$ are

$\phi(b,\theta) = \frac{4}{\pi^2} \theta(\pi - \theta),$

$\phi(r,0) = 0, \phi(r,\frac{\pi}{2}) = 1$

(a) State why the differential equation with the given boundary conditions can't be solved by the method of separating variables.

The problem I am having is that I can separate the equation into to parts so I am assuming it is to do with the boundary conditions why it cannot be solved, but I don't know what aspect of the conditions effects this?

Let's suppose it can. So the solution would be of the form

$\phi = R(r) T(\theta)$ .

The boundary conditions become

$\phi(r,0) = R(r) T(0) = 0\;\;\; \Rightarrow\;\;\; T(0) = 0$

and

$\phi(r,\frac{\pi}{2}) = R(r) T(\frac{\pi}{2}) = 1$

but this can't happen since $R(r)$ varies.

**jezzyjez** · April 13th 2010, 05:42 AM

Originally Posted by Danny

Let's suppose it can. So the solution would be of the form

$\phi = R(r) T(\theta)$ .

The boundary conditions become

$\phi(r,0) = R(r) T(0) = 0\;\;\; \Rightarrow\;\;\; T(0) = 0$

and

$\phi(r,\frac{\pi}{2}) = R(r) T(\frac{\pi}{2}) = 1$

but this can't happen since $R(r)$ varies.

I still dont understand the boundary conditions you have mentioned, dont they just mean if $\theta = 0$ the potential is 0 and if $\theta = \frac{\pi}{2}$ potential is 1. surely the other boundary cond is the one I need to look at???

**************EDIT 15:24*********************

**Danny** · April 13th 2010, 06:06 AM

Originally Posted by jezzyjez

I still dont understand the boundary conditions you have mentioned, dont they just mean if $\theta = 0$ the potential is 0 and if $\theta = \frac{\pi}{2}$ potential is 0. surely the other boundary cond is the one I need to look at???

But you said that $\phi = 1$ at $\theta = \frac{\pi}{2}$ not $\phi = 0.$

**jezzyjez** · April 13th 2010, 06:20 AM

Sorry thats what i meant but i still dont fully understand

**zzzoak** · April 13th 2010, 12:17 PM

$<br /> \phi(r,\frac{\pi}{2}) = R(r) T(\frac{\pi}{2}) = 1<br />$
$<br /> R(r) = \frac{1}{T(\frac{\pi}{2})} = const<br />$
and
$R(r) = const$ this is not possible

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