(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider [tex]\nabla ^2 u = Q\left( {x,y,z} \right)[/tex] 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 [tex]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)[/tex], find the solution in terms of G.

2. Relevant equations

...

3. 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

[tex]

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'} } }

[/tex]

Using the given expression for Q,

[tex]

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'} } }

[/tex]

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.

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# Homework Help: PDE Green's function

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