0
votes
2answers
57 views

Laplace's equation in polar coords

Question: Suppose that the function u(r, $\phi$) satisfies Laplace’s equation for plane polar co-ordinates (r, $\phi$) i.e. $$ ∇^2u = \frac{1}{r} \frac{∂}{∂r}(\frac{r∂u}{∂r}) + ...
2
votes
1answer
372 views

Heat equation in polar co-ordinates

I was studying the heat equation, when i saw a new variant of it. Here's the statement: "the edge $r=a$ of a circular plate is kept at temperature $f(\theta)$. The plate is insulted so that there is ...
0
votes
0answers
86 views

How to solve following non-linear differential equation?

Let's have an equation $$ \left(\frac{\partial f}{\partial r}\right)^{2} + \frac{1}{r^{2}}\left(\frac{\partial f}{\partial \varphi}\right)^{2} = g(r). $$ How to solve it?
1
vote
0answers
194 views

Drum membrane wave equation general solution (non-symmetrical)

I think I am stuck in solving this problem. It involves a wave equation in a circular membrane, so polar coordinates must be used: $u_{tt}=c²(u_{rr}+{1\over r}u_r+{1\over r²}u_{\theta\theta})$, ...
0
votes
1answer
287 views

Finding a geodesic on a plane using polar coordinates

This is from my homework on PDE. I need to find a geodesic on a plane using polar coordinates. Now, I know $dl^2 = x^2+y^2$ hence $l=\int \sqrt{dx^2+dy^2}$, but I get stuck while converting ...
2
votes
1answer
2k views

Transforming the Laplace operator from Polar to Cartesian coordinates

I'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + ...