Green Functions: Finding Solutions for Equations of All Types & Dimensions

[JohanL]

I need a list of Green functions,
for different types of equations and dimensions.

I have tried to use google but with no success.

 

[rcgldr]

I got a bunch of hits with a google for "green's theorem", but I'm not sure if there are other functions you're looking for.

 

[FredGarvin]

Green functions are not the same thing as Green's Theorem.

I can't honestly say that I have seen anything tabulated. I can't say that I have ever looked for anything like that though either. I was always under the impression that green fuctions are dependent on boundary conditions and the Diff. Eq's being used. I'll keep my eyes open to see if I find anything. You might try searching with "Green's functions" as well. I have heard them referred to in both ways.

EDIT: The first hit I got using the "green's function" search...
http://mathworld.wolfram.com/GreensFunction.html

 

[JohanL]

I can't honestly say that I have seen anything tabulated. I can't say that I have ever looked for anything like that though either. I was always under the impression that green fuctions are dependent on boundary conditions and the Diff. Eq's being used. I'll keep my eyes open to see if I find anything. You might try searching with "Green's functions" as well. I have heard them referred to in both ways.

Thx. Yes they are dependent on boundary conditions. But for the more simple cases i was sure to find tables of green functions.

Something like

$$-\frac {1} {2\pi}ln(\rho_1- \rho_2) ;\frac {i} {4}H_0[k(\rho_1- \rho_2)];\frac {1} {2\pi}K_0[k(\rho_1- \rho_2)]$$

for Laplace, Helmholtz and modified Helmholtz in the plane when G goes to 0 as r goes to infinity.
And also in 3 dimensions, for a sphere with homegenous diricihlet on the boundary, the diffusion equation, the wave equation etc.

Green functions are very useful when solving P.D.E and therefor i thought i could find some good tables. But none of my mathematical handbooks have this and i haven't found any on internet yet either.

 

[pmb_phy]

I need a list of Green functions,
for different types of equations and dimensions.

I have tried to use google but with no success.

There exists no such list since a "Green Function" is any function whose Laplacian equals $\delta$(|x-x'|). There is an infinity of such functions.

Pete

 

[jarvis]

Green functions are not confined to differential equations containing the Laplacian, they work under an arbitrary differential operator. The Green function is by definition the solution to a differential equation under application of a unit impulse source term. The beautiful thing about Green functions is that once I know the solution for a unit impulse, I can obtain the solution for an arbitrary source by a convolution of the Green function with that source. This is of course predicated on linearity of the solutions.

 

[WMGoBuffs]

Check out the first chapter or two of Jackson Electrodynamics for everything you would ever care to know about green functions.

 

[JohanL]

There exists no such list since a "Green Function" is any function whose Laplacian equals $\delta$(|x-x'|). There is an infinity of such functions.

Thats the Green function for the Laplace operator.
How is a general green function different from general solutions to ODEs, integrals etc? There are tables of those.
I found a short table of green functions in Arfken.

 

[JohanL]

Check out the first chapter or two of Jackson Electrodynamics for everything you would ever care to know about green functions.

Thx, but that's probably Laplace and Poisson differential operator only?

 

[Rade]

I need a list of Green functions, for different types of equations and dimensions.

You may find these books useful:http://www.worldscibooks.com/physics/p067.html
https://www.amazon.com/gp/product/1584881100/?tag=pfamazon01-20
Also, try these links:
http://www.amath.washington.edu/courses/351-summer-2002/notes/GreensFunctions.pdf
http://home.comcast.net/~jamesverebeck/sgreen.pdf
http://webphysics.davidson.edu/Faculty/wc/WaveHTML/node6.html#tablePDEs

 

[JohanL]

Thanks for your answer!