Green s Identities and Green s Functions
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1 LECTURE 7 Geen s Identities and Geen s Functions Let us ecall The ivegence Theoem in n-dimensions Theoem 7 Let F : R n R n be a vecto field ove R n that is of class C on some closed, connected, simply connected n-dimensional egion R n Then F dv F n ds whee is the bounday of and n is the unit vecto that is outwad nomal to the suface at the point As a special case of Stokes theoem, we may set F φ with φ a C 2 function on We then obtain 2 2 φ dv Anothe special case of Stokes theoem comes fom the choice 3 F φ ψ Fo this case, Stokes theoem says 4 Using the identity φ ψ dv φ ds φ ψ nds 5 φf φ F + φ F we find 4 is equivalent to 6 Equation 6 is known as Geen s fist identity φ ψ dv + φ 2 ψ dv φ ψ n ds Revesing the oles of φ and ψ in 6 we obtain 7 ψ φ dv + ψ 2 φ dv Finally, subtacting 7 fom 6 we get 8 φ 2 ψ ψ 2 φ dv Equation 8 is known as Geen s second identity ψ φ n ds φ ψ ψ φ n ds Now set ψ o + ɛ 78
2 7 GREEN S IENTITIES AN GREEN S FUNCTIONS 79 and inset this expession into 8 We then get φ 2 dv o + ε o + ε 2 φ dv + o + ε φ φ nds o + ε Taking the limit ɛ 0 and using the identities lim ɛ 0 2 o + ε lim ɛ 0 o + ε lim ɛ 0 o + ε we obtain 9 4πδ n o o o 4πφ o o 2 φ dv + φ φ o Equation 9 is known as Geen s thid identity o nds Notice that if φ satisfies Laplace s equation the fist tem on the ight hand side vanishes and so we have φ o 4π 0 φ φ o o nds ds 4π φ n o φ o n Hee n is the diectional deivative coesponding to the suface nomal vecto n Thus, if φ satisfies Laplace s equation in then its value at any point o is completely detemined by the values of φ and on the bounday of φ n
3 GREEN S FUNCTIONS AN SOLUTIONS OF LAPLACE S EQUATION, II 80 Geen s Functions and Solutions of Laplace s Equation, II Recall the fundamental solutions of Laplace s equation in n-dimensions { log, if n 2 Φ n, ψ, θ,, θ n 2, if n > 2 n 2 Each of these solutions eally only makes sense in the egion R n O; fo each possesses a singulaity at the oigin We studied the case when n 3, a little moe closely and found that we could actually wite { 2 2 4πδ 3 0, if 0, if 0 In fact, using simila aguments one can show that 3 2 Φ c n δ n whee c n is the suface aea of the unit sphee in R n Thus, the fundamental solutions can actually be egaded as solutions of an inhomogeneous Laplace equation whee the diving function is concentated at a single point Let us now set n 3 and conside the following PE/BVP 4 2 Φ f, Φ h whee is some closed, connected, simply connected egion in R 3 Let o be some fixed point in and set 5 G, o 4π o + φ o, o whee φ o, o is some solution of the homogeneous Laplace equation 6 2 φ o, o 0 Then 7 2 G, o δ 3 o Now ecall Geen s thid identity 8 Φ 2 Ψ Ψ 2 Φ dv If we eplace ψ in 8 by G, o we get 9 Φ Ψ Ψ Φ n ds Φ o Φδ3 o dv Φ 2 GdV G 2 Φ dv + Φ G G Φ n ds Gf dv + h G n G Φ n ds Gf dv + h G n ds G Φ n ds Up to this point we have only equied that the function φ o satisfies Laplace s equation We will now make ou choice of φ o moe paticula; we shall choose φ o, o to be the unique solution of Laplace s equation in satisfying the bounday condition 20 4π o φ o, o so that G, o 0
4 GREEN S FUNCTIONS AN SOLUTIONS OF LAPLACE S EQUATION, II 8 Then the last integal on the ight hand side of 9 vanishes and so we have 2 Φ o G, o f dv + h G n, o ds Thus, once we find a solution φ o, o to the homogenenous Laplace equation satisfying the bounday condition 2, we have a closed fomula fo the solution of the PE/BVP 4 in tems of integals of G, o times the diving function f, and of G n, o times the function h descibing the bounday conditions on Φ Note that the Geen s function G, o is fixed once we fix φ o which in tun depends only on the natue of the bounday of the egion though condition 20 Example Let us find the Geen s function coesponding to the inteio of sphee of adius R centeed about the oigin We seek to find a solution of φ o of the homogenous Laplace s equation such that 20 is satisfied This is accomplished by the following tick Suppose Φ, ψ, θ is a solution of the homogeneous Laplace equation inside the sphee of adius R centeed at the oigin Fo > R, we define a function R R 2 22 Φ, ψ, θ Φ, ψ, θ I claim that Φ, ψ, θ so defined also satisfies Laplace s equation in the egion exteio to the sphee To pove this, it suffices to show that Φ o 24 Set 2 Φ 2 Φ sinθ + θ 25 u R2 so that 26 and so 27 2 Φ sinθ θ sinθ Φ θ R2 u Φ, ψ, θ u RΦ u, ψ, θ du d u R2 2 sinθ Φ θ u2 R 4 R 2 u u u2 2 R 2 u2 R u u uφ u2 R u 2 Φ u + 2 Φ 2 u u R u u 2 Φ u u R sinθ θ sinθ Φ θ sinθ θ sinθ Φ θ + 2 Φ sin 2 θ ψ 2 2 Φ sin 2 θ ψ 2 u u2 R 2 u u u R Φ Φ sin 2 θ ψ 2 2 Φ sin 2 θ ψ 2 Notice that 28 lim R Φ, ψ, θ Φ, ψ, θ This tansfom is called Kelvin invesion
5 GREEN S FUNCTIONS AN SOLUTIONS OF LAPLACE S EQUATION, II 82 Now let etun to the poblem of finding a Geen s function fo the inteio of a sphee of adius Let R 2 29, ψ, θ R2 2 In view of the peceding emaks, we know that the functions 30 will satisfy, espectively, Φ o Φ 2 R Φ o 3 2 Φ 4πδ 3 o 2 Φ 2 4πR δ3 R 2 2 o Howeve, notice that the suppot of 2 Φ 2 lies completely outside the sphee Theefoe, in the inteio of the sphee, Φ 2 is a solution of the homogenous Laplace equation We also know that on the bounday of the sphee that we have 32 Φ Φ 2 Thus, the function 33 4π o G, o R 4π o 4π R o 4π R o thus satisfies 34 2 G, o δ 3 o fo all inside the sphee and 35 G, o 0 o all on the bounday of the sphee Thus, the function G, o defined by 33 is the Geen s function fo Laplace s equation within the sphee Now conside the following PE/BVP 36 2 Φ f, B Φ R, ψ, θ 0 whee B is a ball of adius R centeed about the oigin Accoding to the fomula 2 and 33, the solution of 36 is given by Φ o G, o f dv + h ψ, θ G B B n, o ds G, o f dv To aive at a moe explicit expession, we set Then B o cosψ sinθ, sinψ sinθ, cosθ ρ cosα sinβ, ρ sinα sinβ, ρ cosβ dv ρ 2 sin 2 θ dρ dα dβ ds ρ 2 sin 2 θ dα dβ
6 GREEN S FUNCTIONS AN SOLUTIONS OF LAPLACE S EQUATION, II 83 and afte a little tigonomety one finds 4π o 4π 2 + ρ 2 2ρ cosψ α sinθ sinβ + cosθ cosβ 4π R o R o R 4π R ρ 2 2R 2 ρ cosψ α sinθ sinβ + cosθ cosβ Thus, Φ, ψ, θ R 2π π R 2π π Rf, ψ, θ 2 sinθddθdψ 4π R ρ 2 2R 2 ρ cosψ α sinθ sinβ + cosθ cosβ f, ψ, θ 2 sinθddθdψ 4π 2 + ρ 2 2ρ cosψ α sinθ sinβ + cosθ cosβ
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