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Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 3

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 6 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Thus the general solution for u is u r O a0 rn an cos nO bn sin nO . n 1 For the boundary condition u 1 Ớ f O we have the equation f O -20 an cos nO bn sin nO . n 1 If f O has a Fourier series then the coefficients are 1 f2n . ao 1 f O dO n Jo an f O cos nO dO n Jo bn f O sin nO dO. n Jo For the boundary condition ur 1 O g O we have the equation g O n an cos nO bn sin nO . n 1 g O has a series of this form only if i 2n I g O dO 0. o 2054 The coefficients are 1 f2n an g ớ cos nớ dớ nn J 0 1 f2n . bn g ớ sin nớ dớ. nn J 0 47.3 Laplace s Equation in an Annulus Consider the problem v2u 1ẳ r X dffi 0 0 - r a -n ớ n r Or Or r2 oớ2 with the boundary condition u a ớ ớ2. So far this problem only has one boundary condition. By requiring that the solution be finite we get the boundary condition u 0 ớ X. By specifying that the solution be C 1 continuous and continuous first derivative we obtain Ou Ou u r -n u r n and dớ r -n dớ r n . We will use the method of separation of variables. We seek solutions of the form u r ớ R r 0 ớ . .