Question

This is a heat conducting problem:

u_(t) = u_(xx)
u(x,t)=w(x,t)+ψ(x)
u(x,0)=cos(∏x)
u_(x)(0,t)=0
u_(x)(1,t)=0

Answer 1

Here is what I have been trying:

The homogenous solution (ψ(x) = 0) is of the form \[u(x,t) = G(0)e^{-λ^{2}t}(A*\sinλx + B*\cosλx)\]
->

(i) \[u(x,0) = \cos\pi x = G(0)(A*\sinλx + B*\cosλx)\]
(ii), \[u_{x}(0,t) = 0 = G(0)e^{-λ^{2}t}(Aλ*\cos0 - Bλ*\sin0) = AλG(0)e^{-λ^{2}t}\]
(iii), \[u_{x}(1,t) = 1 = G(0)e^{-λ^{2}t}(Aλ*\cosλ - Bλ*\sinλ)\]

->
\[\cos\pi x = G(0)B*\cosλx, 1 = -BλG(0)e^{-λ^{2}t}\sinλ\]

Since (ii) -> A = 0

Answer 2

Since (ii) -> A = 0
thats right

Answer 3

and u can drop G(0)

Answer 4

you need to apply the other boundary condition(u_x(12,t)=0) in order to determine the possible values for lambda., after which the general solution will be a sum over lamda values.