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Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 8

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 5 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | We have the implicit boundary conditions u 0 z u 2n z Uỹ 0 z Uỹ 0 z and the boundedness condition u 0 to bounded. We expand the solution in a Fourier series. This ensures that the boundary conditions at 0 0 2n are satisfied. u 0 z un z eme We substitute the series into the partial differential equation to obtain ordinary differential equations for the un. -n2Un z u n z 0 The general solutions of this equation are un z 1 Cl c2z for n 0 ịci enz c2 e-nz for n 0. The bounded solutions are ce nz for n 0 un z c for n 0 ce- n z . cenz for n 0 We substitute the series into the initial condition at z 0 to determine the multiplicative constants. u 0 0 g un 0 e f 0 un 0 -1 ỉ f 0 e- n d0 fn 2n J0 1854 Thus the solution is u 0 z 52 fn e e- n z . Note that u o z f0 1 í f ớ dớ 2n 0 as z to. Solution 40.8 The decomposition of the problem is shown in Figure 40.1. w g2 x u g2 x v 0 w f y A w 0 w f2 y u 0 A u 0 u 0 v f y A v 0 v f2 y Wy g x Uy g x Vy 0 Figure 40.1 Decomposition of the problem. First we solve the problem for u. uxx uyy 0 0 x a 0 y b u 0 y u a y 0 uy x 0 g1 x u x b g2 x We substitute the separation of variables u x y X x Y y into Laplace s equation. X Y - ỹ -A2 X .