Tham khảo tài liệu 'mathematical method in science and engineering episode 17', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | TIME-INDEPENDENT GREEN S FUNCTIONS 593 with the initial conditions 2 0 i 0 0 Cl and C2 in Equation is zero hence the solution will be written as xịt Foe et i sin Jcug-e2 t-t1 c u t Jo x Wp-e2 Lv J Fo sin vGF f-_ _gt p x u q Ê2 ỵ iUq a2 2ae cJq a2 2ae where we have defined One can easily check that x t satisfies the differential equation For weak damping the solution reduces to Fp sin ppt - ĩị Fp _at L l ỉ 5 ---5- . .2 i 2e 0 V 4- a2 0 f 0 2 As expected in the t oo limit this becomes Fp sin Pot - r J Uto Green s Function for the Helmholtz Equation in Three Dimensions The Helmholtz equation in three dimensions is given as V2 kij ip T F r . We now look for a Green s function satisfying V 2 G T 7 6 r T . We multiply the first equation by G r 7 and the second by tp r and then subtract and integrate the result over the volume V to get V i F JJJ 7 7 V2V 7i V2G T T j d3 -f LF r d3 r . 594 GREEN S FUNCTIONS Using Green s theorem III f 2G - d3f II fv G - G F Ads where s is a closed surface enclosing the volume V with the outward unit normal n we obtain T III F T G r I JV C T7 r 1 G F -r VV ri j Ads. Interchanging the primed and the unprimed variables and assuming that the Green s function is symmetric in anticipation of the corresponding boundary conditions to be imposed later we obtain the following remarkable formula y G -F FCr d3 19. F V F i Ads Il G r -Ads . S L -I J J S Boundary conditions The most frequently used boundary conditions are i Dirichlet boundary conditions where G is zero on the boundary. ii Neumann boundary conditions where the normal gradient of G on the surface is zero Vc-nl 0. 1 boundary iii General boundary conditions Vg 0 boundary where Vf r is a function of the boundary point T . For any one of these cases the Green s function is symmetric and the surface term in the above equation vanishes thus giving V T III G FCr d3- 1. Green s Functions in Three Dimensions with
