Solving Laplace's Equation

jeremy_jackson

Hello, can anyone help solve laplaces equation in plane polar coordinates to be in the form F(r)cos(theta)

Thanks

Reply 1

TeeEm

19

Original post by jeremy_jackson

Hello, can anyone help solve laplaces equation in plane polar coordinates to be in the form F(r)cos(theta)

Thanks

what exactly would you like to know?

Reply 2

CosmicJay

Hey, are you still needing help with this? It's quite an easy problem - I'd be happy to help!

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Reply 3

TeeEm

19

Original post by CosmicJay

Hey, are you still needing help with this? It's quite an easy problem - I'd be happy to help!

Posted from TSR Mobile

easy?

I guess anything is easy if you know how ...

Reply 4

CosmicJay

Original post by TeeEm

easy?

I guess anything is easy if you know how ...

If you know the right ansatz your laughing 😉

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Reply 5

TeeEm

19

Original post by CosmicJay

If you know the right ansatz your laughing 😉

Posted from TSR Mobile

Reply 6

jeremy_jackson

OP

Original post by CosmicJay

Hey, are you still needing help with this? It's quite an easy problem - I'd be happy to help!

Posted from TSR Mobile

hello,

yes i am thanks!

not sure what to write for it to just be in the form f(r)costheta rather than having a sine term as well

thank you

Reply 7

TeeEm

19

Original post by jeremy_jackson

Hello, can anyone help solve laplaces equation in plane polar coordinates to be in the form F(r)cos(theta)

Thanks

post a photo of your question

Reply 8

CosmicJay

Original post by jeremy_jackson

hello,

yes i am thanks!

not sure what to write for it to just be in the form f(r)costheta rather than having a sine term as well

thank you

Hey, it sounds like you know what your doing. I'm assuming you've arrived at a general solution including both sine and cosine? If this is the case then you need to apply boundary conditions - these conditions should lead to your f(r)cos theta solution.

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Reply 9

jeremy_jackson

OP

Original post by CosmicJay

Hey, it sounds like you know what your doing. I'm assuming you've arrived at a general solution including both sine and cosine? If this is the case then you need to apply boundary conditions - these conditions should lead to your f(r)cos theta solution.

Posted from TSR Mobile

where do i get these boundary conditions from?

Reply 10

CosmicJay

This is where I have to echo what TeeEm has said. In the specific question your tackling either boundary conditions should be explicitly given or something in the wording of the question should indicate what the boundary conditions are. They usually come in the form of: when theta = 0...

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Reply 11

jeremy_jackson

OP

Laplace's equation is ∇^2φ = 0, in plane polar coordinates. Find its generalsolution of the form φ(r, θ) = f(r) cos θ and show thatφ(r, θ) = (Ar +Br^-1) cos θ where A and B are arbitrary constants.

Reply 12

jeremy_jackson

OP

Original post by CosmicJay

This is where I have to echo what TeeEm has said. In the specific question your tackling either boundary conditions should be explicitly given or something in the wording of the question should indicate what the boundary conditions are. They usually come in the form of: when theta = 0...

Posted from TSR Mobile

Laplace's equation is ∇^2φ = 0, in plane polar coordinates. Find its generalsolution of the form φ(r, θ) = f(r) cos θ and show thatφ(r, θ) = (Ar +Br^-1) cos θ where A and B are arbitrary constants

Reply 13

TeeEm

19

Original post by jeremy_jackson

Laplace's equation is ∇^2φ = 0, in plane polar coordinates. Find its generalsolution of the form φ(r, θ) = f(r) cos θ and show thatφ(r, θ) = (Ar +Br^-1) cos θ where A and B are arbitrary constants

will try to help you tonight when I finish work

Reply 14

atsruser

11

Original post by jeremy_jackson

Laplace's equation is ∇^2φ = 0, in plane polar coordinates. Find its generalsolution of the form φ(r, θ) = f(r) cos θ and show thatφ(r, θ) = (Ar +Br^-1) cos θ where A and B are arbitrary constants

1. Write the Laplacian in plane polar coords
2. Assume a separable solution of the form

\phi(r,\theta) = R(r)T(\theta)

3. Substitute this into 1) and write the result in the form

P(r)=Q(\theta)

4. Deduce something important about

P(r), Q(\theta)

. (Hint: how can a function of

r

be identically equal to a function of

\theta

?)
5. Solve a couple of DEs in

r,\theta

separately

Reply 15

TeeEm

19

Original post by jeremy_jackson

Laplace's equation is ∇^2φ = 0, in plane polar coordinates. Find its generalsolution of the form φ(r, θ) = f(r) cos θ and show thatφ(r, θ) = (Ar +Br^-1) cos θ where A and B are arbitrary constants

It looks that atsruser has given some structure to the problem.
If you need more detail quote me.

Solving Laplace's Equation

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