Solving PDE with Green's Function: Finding Solution in Terms of G

[Benny]

Homework Statement

Consider $$\nabla ^2 u = Q\left( {x,y,z} \right)$$ in the half space region z > 0 where u(x,y,o) = 0. The relevant Green's function is G(x,y,z|x',y',z').

Find the solution to the PDE in terms of G. If $$Q\left( {x,y,z} \right) = x^2 e^{ - z} \delta \left( {x - 2} \right)\delta \left( {y + 1} \right)\delta \left( {z - 4} \right)$$, find the solution in terms of G.

Homework Equations

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The Attempt at a Solution

I'm using the result that the answer to a general problem of this sort will be the integral of the product of the Green's function and the 'source term'. So I find

$$u\left( {x,y,z} \right) = \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {\int\limits_0^\infty {G\left( {x,y,z|x',y',z'} \right)Q\left( {x',y',z'} \right)dz'dy'dx'} } }$$

Using the given expression for Q,

$$u\left( {x,y,z} \right) = \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {\int\limits_0^\infty {G\left( {x,y,z|x',y',z'} \right)\left( {x'} \right)^2 e^{ - z'} \delta \left( {x' - 2} \right)\delta \left( {y' + 1} \right)\delta \left( {z' - 4} \right)dz'dy'dx'} } }$$

I don't know if I've made a mistake somewhere so it'd be great if someone could check my answer. Also, can this be simplified? The integration region includes x' = 2, y' = -1 and z' = 4 so does the integral become G evaluated at x' = 2, y' = -2 and z' = 4 (multiplied by (x')^2exp(-z') evaluated at the same points)? Ie. G(x,y,z|2,-1,4).

Any help would be good thanks.

 

[Astronuc]

The approach looks correct. Basically using Green's function to solve Poisson's equation, with a point source at x' = 2, y' = -1 and z' = 4.

Is one supposed to use the Green's function to solve the problem, or just set up the intgeral?

Tutorial on Green's function.
http://www.boulder.nist.gov/div853/greenfn/tutorial.html


http://www.engr.unl.edu/~glibrary/home/body_index.html


http://farside.ph.utexas.edu/teaching/em/lectures/node31.html

 

[Benny]

Thanks for the help. The first part of the question involved deriving the Green's function for the 3D laplacian but the question statement itself isn't clear cut on whether an answer in terms of G is acceptable or if the expression for the G must be used. So presumably, an integral expression, followed by a simplification of the integral using the properties of the delta function should be sufficient.

Do you by any chance have links to websites explaining the method of images? The book I have doesn't explain it too well for my purposes. I can do basic questions using that method but my solution method is more or less based on familiarity with examples rather than understanding so any links would be good, thanks.