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

In summary, a student is attempting to find the solution to a problem involving the half space region z > 0 where u(x,y,o) = 0. They find the solution in terms of G and use the result to simplify a previous problem.
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Benny
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Homework Statement



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.

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

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

What is a Green's Function and how is it used in solving PDEs?

A Green's Function is a mathematical tool used to solve differential equations. It is a function that satisfies a certain differential equation and boundary conditions, and can be used to find the solution of a more complicated differential equation by breaking it down into simpler parts.

What types of problems can be solved using Green's Function?

Green's Function can be used to solve a variety of problems, including problems in electromagnetics, fluid dynamics, quantum mechanics, and heat transfer. It is particularly useful for solving boundary value problems involving partial differential equations.

How is Green's Function related to the fundamental solution of a PDE?

The fundamental solution of a PDE is a Green's Function that is used to solve a specific type of PDE, such as the Laplace equation or the heat equation. The fundamental solution is a Green's Function that satisfies certain properties, such as being symmetric and having a singularity at the origin.

Can Green's Function be used for nonlinear PDEs?

Yes, Green's Function can be used to solve nonlinear PDEs, although the process may be more complicated. In these cases, the Green's Function may not be a closed-form solution and numerical methods may be needed to approximate the solution.

What are the advantages of using Green's Function to solve PDEs?

Using Green's Function to solve PDEs has several advantages. It allows for the decomposition of a complex problem into simpler parts, it is a powerful tool for finding solutions to boundary value problems, and it can be used to solve both homogeneous and non-homogeneous PDEs. Additionally, Green's Function can be used to solve problems in different physical domains without having to derive a new solution for each domain.

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