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Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 5 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Let Rk be the relative error at that point incurred by taking k terms. Rk 8 - .o 1 sinh nn n3 Vn k 2 n3 sinh 2nn n n sum 4 v i _odd n __ 8 v .o 1 sinh nn n3 od n n3 sinh 2nn Ex 1 sinh nn n k 2 n3 sinh 2nn n sumi Lnn R __ odd n_ k o 1 sinh nn 2- Odj n n3 sinh 2nn Since R1 0.0000693169 we see that one term is sufficient for 1 or 0.1 accuracy. Now consider ộx 1 2 1 . ệx x y _ -82 -4 cos nnx n2 n2 n 1 odd n sinh nny sinh 2nn x 1 2 1 _0 Since all the terms in the series are zero accuracy is not an issue. Solution 37.35 The solution has the form i ar-n-1 P c s ớ sin mự r a ị rnP cos Ỡ sin m r a. The boundary condition on at r _ a gives us the constraint aa n-1 - fian _ 0 p _ aa 2n-1. 1814 Then we apply the boundary condition on ộr at r a. - n 1 aa-n-2 - n a-2n-1an-1 1 an 2 a 7 - 2n 1 __an 2 r-n- mínnữtìẦiìnímíH ỉ- Í 2n 1r Pn cos P Sin m r a a n 1 2n 1 r P n cos u sìn mộ r a Solution 37.36 We expand the solution in a Fourier series. Ộ 2a0 r an r cos n0 bn r sln n0 n 1 n 1 We substitute the series into the Laplace s equation to determine ordinary differential equations for the coefficients. d 8 j 1 ỡ2 dr di r2 du2 0 a0 a0 0 a n an n2an 0 bn bn n2bn 0 The solutions that are bounded at r 0 are to within multiplicative constants ao r 1 an r rn bn r rn. Thus ộ r 0 has the form ộ r Ỡ 2c0 cnrn cos nớ dnrn sln nớ n 1 n 1 We apply the boundary condition at r R. ộr R 0 ncnRn-1 cos nớ yndnRn-1 sln nớ n 1 n 1 .
