Chpter 9: Green s functions for time-independent problems Introductory emples One-dimensionl het eqution Consider the one-dimensionl het eqution with boundry conditions nd initil condition We lredy know tht the solution of this problem is given by u t = k u (1) u(, t) = () u(, t) = (3) u(, ) = f() (4) u(, t) = n sin nπ e k(nπ/) t (5) where n re the Fourier sine series coefficients of f(). f() = n = n sin nπ After replcing (6) in (5) we my epress the solution s [ u(, t) = f( ) sin nπ which my be written ( fter interchnging nd ) ( u(, t) = f( ) We define the quntity f() sin nπ (6) sin nπ sin nπ nπ sin e k(nπ/) t ] sin nπ e k(nπ/) t ) nπ sin e k(nπ/) t (7) s the influence function for the initil condition. For every point this quntity shows the influence of the initil temperture t on the temperture t position nd time t. Further insight my be obtined by considering the het eqution with sources with boundry conditions (-3) nd the initil condition (4). u t = k u + Q(, t) (8) 1
Q: Show tht the solution of (8), (-3), (4) my be epressed s u(, t) = where G(, t;, t ) is given by f( )G(, t;, ) + t Q(, t )G(, t;, t )dt (9) G(, t;, t ) = sin nπ nπ sin e k(nπ/) (t t ) (1) The function G(, t;, t ) defined by (1) is clled the Green s function for the het eqution problem (8), (-3), (4). At t =, G(, t;, t ) epresses the influence of the initil temperture t on the temperture t position nd time t. In ddition, G(, t;, t ) shows the influence of the source/sink term Q(, t ) t position nd time t on the temperture t position nd time t. Notice tht the Green s function depends only on the elpsed time t t since G(, t;, t ) = G(, t t ;, ) Green s functions for boundry vlue problems for ODE s In this section we investigte the Green s function for Sturm-iouville nonhomogeneous ODE (u) = f() subject to two homogeneous boundry conditions. The simplest emple is the stedy-stte het eqution with homogeneous boundry conditions d = f() The method of vrition of prmeters Consider the liner nonhomogeneous problem u() =, u() = (u) = f() (11) where u = u(), < < b stisfies homogeneous boundry conditions nd is the Sturm-iouville opertor d ( p d ) + q If u 1 nd u re two linerly independent solutions of the homogeneous problem (u) =, the generl solution of the homogeneous problem is u = c 1 u 1 + c u where c 1 nd c re rbitrry constnts. To solve the nonhomogeneous problem, we use the method of vrition of prmeters nd serch for prticulr solution of (11) of the form v 1 ()u 1 () + v ()u () (1)
where v 1 () nd v () re functions to be determined such tht (1) stisfies (11). Since we hve only one eqution to be stisfied nd two unknown functions, we impose n dditionl constrint Using (13), we obtin from (1) nd the eqution (11) is stisfied if dv 1 u 1 + dv u = (13) du = v du 1 1 + v du From (13) nd (14) we obtin where Remrk: The quntity dv 1 pdu 1 + dv pdu = f() (14) dv 1 = fu c dv = fu 1 c ( ) du c = p u 1 u du 1 du W = u 1 u du 1 = u 1 u is clled the Wronskin of u 1 nd u nd stisfies the differentil eqution dw = 1 dp p W Q: Using the Wronskin, show tht epression (17) is constnt. Then, integrting (15) nd (16) we obtin v 1 () = 1 c v () = 1 c such tht the generl solution of the nonhomogeneous problem (11) is du 1 du (15) (16) (17) f( )u ( ) + c 1 (18) f( )u 1 ( ) + c (19) v 1 ()u 1 () + v ()u () () = c 1 u 1 () + c u () u 1() f( )u ( ) + u () f( )u 1 ( ) (1) c c The constnts c 1 nd c re determined by the boundry conditions. A simple emple. Consider the following problem with homogeneous boundry conditions d u = f() u() =, u() = Two linerly independent solutions of the homogeneous differentil eqution re u 1 () = () 3
Then, W = u 1 du u du1 Using the boundry conditions, u () = (3) = such tht from (18) nd (19) we obtin v 1 () = 1 v () = 1 f( )( ) + c 1 (4) u() = c = f( ) + c (5) nd replcing in (4), (5) it follows tht u() = c 1 = 1 f( )( ) v 1 () = 1 f( )( ) 1 v () = 1 nd the solution of the nonhomogeneous problem is This solution my be written s f( )( ) = 1 f( )( ) (6) f( ) (7) v 1 ()u 1 () + v ()u () (8) = f( )( ) where the Green s function G(, ) is given by G(, ) = The method of eigenfunction epnsion for Green s functions f( ) (9) f( )G(, ) (3) { ( ), < (31) ( ), > Consider generl Sturm-iouville nonhomogeneous ODE (u) = f(), < < b subject to two homogeneous boundry conditions. We know tht the eigenfunctions Φ n () of the relted eigenvlue problem (Φ) = λσφ (for n rbitrry weight function σ) subject to the sme homogeneous boundry conditions form complete set, such tht u() my be epressed s generlized Fourier series of eigenfunctions Term-by-term differentition of (3) implies (together with the linerity of ) n Φ n () (3) 4
( ) (u) = n Φ n () = n (Φ n ) = n λ n σφ n In the lst reltion bove we used (Φ n ) = λ n σφ n. Therefore, (u) = f() n λ n σφ n = f() nd using the orthogonlity of the eigenfunctions (with weight σ) we obtin the coefficients n = f()φ n λ n φ nσ (33) Replcing (33) in (3) we obtin ( fter interchnging nd ) For this problem, where the Green s function G(, ) is f()φ n() λ n φ nσ Φ n() = f( ) f( )G(, ) Φ n ()Φ n ( ) λ n φ nσ (34) Φ n ()Φ n ( ) G(, ) = λ n φ nσ (35) Remrk: Formul (35) implies tht the Green s function is symmetric: G(, ) = G(, ) The Dirc Delt Function nd its reltionship to Green s function In the previous section we proved tht the solution of the nonhomogeneous problem subject to homogeneous boundry conditions is (u) = f() f( )G(, ) In this section we wnt to give n interprettion of the Green s function. et s, < s < b represent n rbitrry fied point. We consider perturbtion of the source term f() + δ ɛ () where δ ɛ () is continuous function with the following properties: 1. δ ɛ() = 1. δ ɛ () = if s + ɛ or s ɛ 5
A possible choice for δ ɛ () is the ht function, < s ɛ 1 δ ɛ () = ɛ ( s + ɛ), s ɛ < < s 1 ɛ ( s ɛ), s < < s + ɛ, s + ɛ < b (36) Denote u ɛ () the solution of the perturbed problem (u ɛ ) = f() + δ ɛ () subject to the sme homogeneous boundry conditions s u. The vrition u = u ɛ u in the solution u due to the perturbtion of the source term stisfy the following problem: ( u) = δ ɛ () with homogeneous boundry conditions, such tht we hve δ ɛ ( )G(, ) = s+ɛ s ɛ δ ɛ ( )G(, ) = G(, ɛ ) s+ɛ s ɛ δ ɛ ( ) where ɛ is point in the intervl ( s ɛ, s + ɛ). The eistence of such point is consequence of the men vlue theorem, since G(, ) is continuous. Notice tht s+ɛ s ɛ δ ɛ ( ) = δ ɛ ( ) = 1 If we let ɛ, we obtin The limit G(, s ) lim δ ɛ = δ( s ) (37) ɛ represents n infinitely concentrted pulse t s, tht is zero everywhere, ecept t = s {, δ( s ) = s (38), = s We define the Dirc delt function δ( s ) s n opertor with the property (38) such tht for every continuous function f() we hve The Dirc delt function hs unit re f() = f( s )δ( s ) s δ( s ) s = 1 nd is even in the rgument s δ( s ) = δ( s ) Then G(, s ) stisfies [G(, s )] = δ( s ) nd the homogeneous boundry conditions t = nd = b, such tht we obtin the following interprettion: the Green s function G(, s ) is the response t due to concentrted source t s. The symmetry of the Green s function G(, s ) = G( s, ) 6
implies tht the response t due to concentrted source t s is the sme s the response t s due to concentrted source t. This property is known s Mwell s reciprocity. Jump conditions Net we show how the Green s function my be obtined by directly solving subject to homogeneous boundry conditions. Emple consider the stedy-stte problem [G(, )] = δ( ) (39) u () = f() u() =, u() = The Green s function to this problem is then solution to d G(, ) = δ( ) (4) G(, ) =, G(, ) = (41) It follows then tht d G(, ) =,, therefore G(, ) must be liner function on ech intervl < nd > { + b, < G(, ) = (4) c +, > The constnts, b, c, d need to be determined. From the boundry conditions we hve such tht we obtin G(, ) = = G(, ) = c = d G(, ) = { b, < To find b nd d, first we require tht G(, ) must be continuous t Net, for ny ɛ >, integrting (4) in ( ɛ, + ɛ) d( ), > (43) G(, ) = G( +, ) b = d( ) (44) d G( + ɛ, ) d G( ɛ, ) = 1 then pssing to the limit s ɛ, we get the jump condition Solving (44-45) we get such tht the Green s function (43) is d G(+, ) d G(, ) = 1 d b = 1 (45) G(, ) = d =, b = { ( ), < ( ), > which is the sme epression we obtined using the vrition of prmeters. (46) 7
Nonhomogeneous boundry conditions The Green s function my be used to solve problems with nonhomogeneous boundry conditions e.g., u = f() (47) u() = α, u() = β (48) The solution u() my be written s the sum u = u 1 + u where u 1 () solves the nonhomogenous ODE with homogeneous boundry conditions u 1 = f(), u 1 () = u 1 () = nd u () solves the homogenous ODE with nonhomogeneous boundry conditions Then, such tht u =, u () = α, u () = β u 1 () = u 1 () + u () = f( )G(, ) u () = α + β α is the solution to the nonhomogeneous problem (47-48). f( )G(, ) + α + β α 8