How to Derive Green's Function for the Laplacian in 3D?

In summary, the conversation is about understanding the Green's function for the Laplacian operator under infinite boundary condition. The equations 9.173, 9.174, and 9.175 are discussed, with 9.174 being the result of integrating the gradient operator in spherical coordinates and 9.175 being the indefinite integral of 9.174. Haberman's book on PDE provides an explicit derivation of this concept.
  • #1
centry57
8
0
Here are some pages of Arfken's “Mathematical Methods for Physicists ”
I don't How to work out the Green's function!
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attachment.php?attachmentid=21048&stc=1&d=1255190271.png

Can anyone explain (9.174)and(9.175) for me ?
I'm hoping for your help, Thank you !
 

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  • #2
This is about finding the Green's function for the Laplacian operator under infinite boundary condition. Basically, 9.173 gives the relation between the Green's function and the unit excitation. In 3D, the unit excitation is assumed located at the center of a sphere. If you write out the gradient operator in 9.173 in the spherical coordinate on the radial component, and perform surface integration, you will get 9.174. 9.175 is the indefinite integral of 9.174.

Haberman's book on PDE Section9.5.6 has the explicit derivation.
 

FAQ: How to Derive Green's Function for the Laplacian in 3D?

What is Green's function?

Green's function is a mathematical tool used in physics and engineering to solve differential equations. It represents the response of a system to an impulse or point source input.

How is Green's function related to differential equations?

Green's function is the solution to a specific type of differential equation known as an inhomogeneous boundary value problem. It is used to find the general solution to a given differential equation.

What are some applications of Green's function?

Green's function has various applications in different fields such as electromagnetics, fluid dynamics, solid mechanics, and quantum mechanics. It is used to solve problems related to heat transfer, acoustics, and wave propagation, among others.

Is Green's function unique?

No, Green's function is not unique. Different boundary conditions and input functions can lead to different Green's functions for the same problem. However, it is unique for a given set of boundary conditions and input function.

How is Green's function calculated?

Green's function can be calculated using various methods such as the method of images, separation of variables, and integral transforms. The specific method used depends on the type of problem and the boundary conditions given.

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