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Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 1

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 6 part 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 45.7 Solutions Solution 45.1 The Green function problem is Gt kGxx 8 x e J t T G x t e T 0 for t T G 0 as x x We take the Fourier transform of the differential equation. Gt KW2G F J x e J t T G u t e T 0 for t T Now we have an ordinary differential equation Green function problem for G. The homogeneous solution of the ordinary differential equation is e-K 2t The jump condition is G u 0 e T F í x e . We write the solution for G and invert using the convolution theorem. G F J x e H t T G F J x e F T e-x2 4K t-F H t T G r ố x y e J-GGỹ e-y2 4K -T dyH t T 2n J-rc V K t T G 1 e- x- 2 4K t-T H t T ỵ 4nK t T We write the solution of the diffusion equation using the Green function. u i i G x t e T s e T de dT i G x t e 0 f e de J0 J- J- u 1 x e- x-í 2 4K t-T s e T dedT 1 i e- x- 2 4Kt f e de Jo y 4nn t T 7-rc ự4nKt J- 1974 Solution 45.2 1. We apply Fourier transforms in x and y to the Green function problem. Gtt - c2 Gxx Gyy ỗ t T J x ổ y n Gtt c2 a2 p2 G ố t T 71 e-ia -L v z Tr TT -1-1 I I rr . I . - r . II r - Ì A f I . I I I . This gives us an ordinary differential equation Green function problem for G a p t . We find the causal solution. -I-1 . .1 I .1 . . f n i r r I That is the solution that satisfies G a p t 0 for t T. G sin i a a2 p2c t T 1 y v v_Le-i i n cựa2 p2 4n2 H t T Now we take inverse Fourier transforms in a and p. Z. rro ei a x-ị y-n z _ X y - 2 2 - sin 2 p2c t T da dpH t T We make the change of variables a p cos p p p sin p and do the integration in polar coordinates. 1 p2n pro gip x- cosộ y-n sinỘ G cj i Jo ----------------------p-----------sin pc t T p dp dpH t T .