Safari File Edit wew History Bookrmarks Window Help ODE 8 (2. 14 o 25 points I P

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Safari File Edit wew History Bookrmarks Window Help ODE 8 (2. 14 o 25 points I Previous Answers zuonE08 12.5015. use the superposition principle to solve Laplace's equation (1) in section 12.5 o, o c c a, o y b (11) for a square plate subject to the given boundary conditions. u(0, y) 1, u(r, y) 1 u(x, 0) 0, u(x, t) 1 sinh(nx) sin(ny) where (nx) An cosh(nx) B u(x, y) n sinh (ny)sin(nx) x sinh Need Help? A O points l Previo Answers zuomat 12.1 02s classify the given partial differential equation as hyperbolic, parabolic, or elliptic. Q hyperbolic

Explanation / Answer

The initial conditions for this problem are
u(0,y) = 1, u(pi,y)=1, u(x,0)=0, u(x,pi)=1.
We have already the general solution for Laplace's equation, given by
u(x,y) = Sum_{n=1}^{infty} [(An cosh(ny) + Bn sinh(ny)) sin(nx) + (Abn cosh(nx) + Bbn sinh(nx))sin(ny)].
We impose the first condition u(0,y) = 1, obtaining
u(0,y) = Sum_{n=1}^{infty} (Abn cosh(0) + Bbn sinh(0))sin(ny) = 1,
Sum_{n=1}^{infty} Abn sin(ny) = 1,
The second condition says that
u(pi, y) = Sum_{n=1}^{infty} [(An cosh(ny) + Bn sinh(ny)) sin(n*pi) + (Abn cosh(n*pi) + Bbn sinh(n*pi))sin(ny)] =
= Sum_{n=1}^{infty} (Abn cosh(n*pi) + Bbn sinh(n*pi))sin(ny) = 1,
the third one says that
u(x,0)= Sum_{n=1}^{infty} [An sin(nx) ] =0,
==> An = 0,
and the fourth, that
u(x,pi) = Sum_{n=1}^{infty} [(An cosh(n*pi) + Bn sinh(n*pi)) sin(nx) ] = 1.
Then we have An=0. We put this in all the remaining three equations, obtaining the following system of equations:
Sum_{n=1}^{infty} Abn sin(ny) = 1,
Sum_{n=1}^{infty} (Abn cosh(n*pi) + Bbn sinh(n*pi))sin(ny) = 1,
Sum_{n=1}^{infty} Bn sinh(n*pi) sin(nx) = 1.
We can put the first in the second, obtaining that Bbn = 0. Therefore,
Abn = 1/sin(ny),
Bn = 1/sinh(n*pi).

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