Module 9: The Method of Green s Functions The method of Green s functions is n importnt technique for solving oundry vlue nd, initil nd oundry vlue prolems for prtil differentil equtions. In this module, we shll lern Green s function method for finding the solutions of prtil differentil equtions. This is ccomplished y constructing Green s theorem tht re pproprite for the second order differentil equtions. These integrl theorems will then e used to show how BVP nd IBVP cn e solved in terms of ppropritely defined Green s functions for these prolems. More precisely, we shll study the construction nd use of Green s functions for the Lplce, the Het nd the Wve equtions.
MODULE 9: THE METHOD OF GREEN S FUNCTIONS Lecture The Lplce Eqution Let e ounded domin in R. Consider the Lplce eqution stisfying the BC u = in αu + β u = B. Here, αx, βx, nd B re given functions evluted on the oundry region. The term u denotes the exterior norml derivtive on. The oundry condition reltes the vlues of u on nd the flux of u through. We ssume tht αx > nd βx > on. If α, β = then is referred to s Dirichlet BC. If α =, β then is referred to s Neumnn BC. If α, β then the condition is known s Roin s type BC or mixed BC. We ssume tht is sudivided into three disjoint susets, nd 3. On, nd 3, u stisfies oundry condition of the first kind Dirichlet type, second kind Neumnn type nd third kind mixed type, respectively. Introducing function wx whose property we shll specify lter nd pplying Green s theorem, we note tht w u u w dx = u w w u ds, 3 where n is the exterior unit norml to. The eqution 3 is the sic integrl theorem from which the Green s function method proceeds in the elliptic cse. Now the function wx is to e determined so tht 3 expresses u t n ritrry point ξ in the region in terms of w nd known functions in nd. Let wx e solution of w = δx ξ, 4 where δx ξ is two-dimensionl Dirc delt function. Using the property of Dirc delt function, we hve u wdx = u δx ξ dx = uξ. 5 In view of, we hve w u dx =. 6
MODULE 9: THE METHOD OF GREEN S FUNCTIONS 3 It now remins to choose oundry conditions for wx on so tht the oundry integrl in 3 involves only wx nd known functions. This cn e ccomplished y requiring wx to stisfy the homogeneous form of the given oundry condition, i.e. αw + β w =. 7 If x on, we hve u w w u = α = α αu w w u αw αu + β u = α B w, 8 where we hve used. If x 3 on, we hve u w w u = β w αu + β u = Bw. 9 β The function wx is clled the Green s function for the oundry vlue prolem -. To indicte its dependence on the point ξ, we denote the Green s function y w = Gx; ξ. In terms of Gx; ξ, tkes the form uξ = α B G ds + BGds. 3 β The Green s function Gx; ξ thus stisfies the eqution G = δx ξ, x, ξ, nd the BC αg + β G =. 3 The Poisson eqution in rectngle: Let : < x <, < y < e rectngulr domin in R with oundry. Consider the following BVP u = fx, y in, 4 with the BC ux, = ux, =, < x <, 5 u, y = u, y =, < y <.
MODULE 9: THE METHOD OF GREEN S FUNCTIONS 4 Let Gx, y; ξ, η e the solution of the BVP G = δx ξ, y δ in 6 with the BC Gx, ; ξ, η = Gx, ; ξ, η =, < x <, 7 G, y; ξ, η = G, y; ξ, η =, < y <, where δx ξ, y η = δx ξδy η. Since is piecewise smooth, we use the divergence theorem to find tht for pir of smooth functions u nd w. w u u wdx = u w w u ds. 8 Eqution 8 is Green s formul for functions of two spce vriles. If u is the solution of the given BVP nd w is replced y G, then the homogeneous BC stisfied y oth u nd G mke the right-hnd side in 8 vnish nd the formul reduces to [ux, yδx ξδy η fx, ygx, y; ξ, η]dxdy =. 9 This yields uξ, η = Gx, y; ξ, ηfx, ydxdy. Applying 8 with ux, y replced y Gx, y; ξ, η nd wx, y replced y Gx, y; ρ, σ, we otin Gξ, η; ρ, σ = Gρ, σ; ξ, η. Applying simple interchnge of vriles, ecomes ux, y = Gx, y; ξ, ηfξ, ηdξdη. Gx, y; ξ, η is clled the Green s function of the given BVP. Formul shows the effect of ll the sources in on the temperture t the point x, y. To construct the Green s function G, recll the two dimensionl eigenvlue prolem ssocited with the BVP 4-5. U xx + U yy + λu =, < x <, < y <, U, y =, U, y =, < y <, Ux, = Ux, = < x <.
MODULE 9: THE METHOD OF GREEN S FUNCTIONS 5 The eigenvlues nd the corresponding eigenfunctions re given y nπ λ nm = mπ nπx +, Unm = sin sin We now seek n expnsion of the form, n, m =,,... Gx, y; ξ, η = = c nm ξ, ηu nm x, y c nm ξ, η sin nπx sin. 3 Putting 3 in 6, we otin Gx, y; ξ, η = = c nm ξ, η U nm x, y λ nm c nm ξ, ηu nm x, y = δx ξδy η. 4 Multiplying oth sides of 4 y U pq. Then integrte over nd use the property of Dirc delt function to otin In view of 3, we now conclude Gx, y; ξ, η = 4 c pq ξ, η = 4 λ pq U pq ξ, η. 5 sin nπξ sin mπη nπ + mπ EXAMPLE. Using Green s function method to solve sin nπx sin. 6 u xx + u yy = π sinπx sinπy, < x <, < y <, ux, =, ux, =, < x <, u, y = u, y = < y <. Here, =, = nd fx, y = π sinπx sinπy. By 3, we hve Gx, y; ξ, η = sinnπξ sinmπη/ nπ + mπ /4 sin nπx sin.
MODULE 9: THE METHOD OF GREEN S FUNCTIONS 6 It now follows from tht ux, y = 4 sinnπξ sinmπη/ π 4n + m sin nπx sin π sinπξ sinπηdξdη = 8 4n + m sinπξ sinnπξdξ sinπη sin dη sin nπx sin = 8 4 + 4 sinπx sinπy = sinπx sinπy. 5 REMARK. We know tht seprtion of vriles cnnot e performed if the PDE nd/or BCs re not homogeneous. The eigenfunction expnsion technique is used to del with IBVPs, where the PDE is nonhomogeneous nd the BCs re zero. Prctice Prolems. Use the Green s function method to find the solution of the Dirichlet BVP: u xx + u yy = x + y, < x <, < y <, ux, = ux, =, < x <, u, y = u, y =, < y <.. Use the Green s function method to find the solution of the Neumnn BVP: u xx + u yy =, < x <, < y <, u x x, = u x x, =, < x <, u, y = u, y =, < y <.