GreensPDE

GreensPDE - c. Eigenfunction expansions for two dimensions...

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Physics 318 R.G. Palmer Electromagnetism 4/9/10 Green’s Function for the Poisson Equation 1. The nonhomogeneous problem for the Poisson Equation 2 Φ( x ) = - ρ ( x ) 0 in a domain V Closed boundary conditions on Φ( x ) at the surface S 2. The Green’s function G ( x , x 0 ) 2 G ( x , x 0 ) = - 4 πδ ( x - x 0 ) 3. The magic rule Φ( x ) = 1 4 πε 0 Z V G ( x 0 , x ) ρ ( x 0 ) d 3 x 0 - 1 4 π I S ± Φ( x 0 ) ∂G ∂n 0 ( x 0 , x ) - G ( x 0 , x ) Φ ∂n 0 ( x 0 ) ² da 0 4. Homogeneous boundary conditions for G ( x , x 0 ) We can choose homogeneous boundary conditions for G ( x , x 0 ) on S . Then G ( x , x 0 ) = G ( x 0 , x ) * Often we use the Dirichlet BC’s ( G ( x , x 0 ) = 0 for x on S ), giving Φ( x ) = 1 4 πε 0 Z V G D ( x , x 0 ) ρ ( x 0 ) d 3 x 0 - 1 4 π I S Φ( x 0 ) ∂G D ∂n 0 ( x , x 0 ) da 0 (Dirichlet Green’s Fn) or the Neumann BC’s ( ∂G ( x , x 0 ) /∂n = - 4 π/S for x on S ), giving Φ( x ) = h Φ i S + 1 4 πε 0 Z V G N ( x , x 0 ) ρ ( x 0 ) d 3 x 0 + 1 4 π I S G N ( x , x 0 ) Φ ∂n 0 ( x 0 ) da 0 (Neumann Green’s Fn) 5. Techniques for constructing Green’s function a. Method of images. [J2.6] b. Fundamental solution (1 / | x - x 0 | ) + solutions for Laplace’s equation. [J1.10]
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Unformatted text preview: c. Eigenfunction expansions for two dimensions and using the matching method for the third. [J3.9, J3.11] d. Eigenfunction expansions for three dimensions (see below). [J3.12] 6. Eigenfunction expansions 2 lmn ( x ) + lmn lmn ( x ) = 0 (eigenfunctions) Z V * l m n ( x ) lmn ( x ) d 3 x = l l m m n n (orthonormal) 2 ( x ) + ( x ) =- ( x ) (nonhomogeneous problem) Use the same boundary conditions on lmn ( x ) as for G ( x , x ) G ( x , x ) = 4 X l X m X n lmn ( x ) * lmn ( x ) lmn-...
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