Questions about Laplace's equation and Green's functions

[gangsta316]

Homework Statement

http://workspace.imperial.ac.uk/mathematics/public/students/ug/exampapers/2010/M2AA2-2010.PDF

Questions 3(iii) and 3(iv)

Homework Equations

The Attempt at a Solution

3(iii)

So here we "guess" that the solution is f = f(r) (f for phi). Then we just have to solve an ODE. But the mark scheme says that we get
1/r^2 * d/dr (r^2 *df/dr) = 1
Why?
The Laplacian is
http://en.wikipedia.org/wiki/Laplace_operator#Two_dimensions
that.
So wouldn't we get
1/r *d/dr (r* df/dr) = 1?



3(iv)

How is this question done? I know that, with the method of images, we want to solve grad^2 = Dirac delta function, and we extend it to the whole of R^3. Then we solve it and appeal to uniqueness to show that it's the solution for y>0. But what would we do here?



Thanks for any help.

 

[mighty2000]

Thank you for your post. I am a scientist and I would like to address your questions regarding questions 3(iii) and 3(iv) from the exam paper you have linked.

For 3(iii), the reason why the mark scheme gives 1/r^2 * d/dr (r^2 *df/dr) = 1 is because of the form of the Laplacian operator in two dimensions. As you have correctly pointed out, the Laplacian in two dimensions is given by the expression you have provided in your post. However, in this particular case, we are working with polar coordinates and not Cartesian coordinates. In polar coordinates, the Laplacian operator takes the form of 1/r *d/dr (r* df/dr) as you have mentioned. Therefore, when we substitute this into the equation 1/r^2 * d/dr (r^2 *df/dr) = 1, we get the correct form of the Laplacian in polar coordinates.

For 3(iv), the method of images can indeed be used to solve the equation grad^2 = Dirac delta function in three dimensions. However, in this case, we are only interested in solving the equation in the region y>0. Therefore, we can use the method of images to find the solution for y>0 and then use the uniqueness theorem to show that this is the only solution in this region. This is a common technique used in solving partial differential equations, where we extend the problem to a larger region and then restrict it to the region of interest.

I hope this helps to clarify your doubts. If you have any further questions or need further explanation, please do not hesitate to ask. Good luck with your studies!