Non-Homogeneous Heat Equation (Insulated Bar Question)

[speedycatz]

Homework Statement

Find U(x,t)

dU/dt = d2U/dx2 + sin x

Boundary Conditions:
dU/dx (0,t) = 0

and

U(1,t) = 0

Initial Condition: U(x,0) = cos 7*π*x

2. The attempt at a solution

I start off with:
d2(Un)/dx2 = λnUn (as an initial value problem)

[d(Un)/dx](0) = 0; [d(Un)/dx](1) = 0

My teacher told me to use the orthogonality later on, but at this point I'm stuck.
Can anyone enlighten me? Thanks!

 

[lanedance]

ok, so so far you have assumed separatino of variables, that is u(x,t) = U(x)T(t) and

so now you'll need to find the eigenfunctions of the homogenous boundary value problem
$$U_n''(x) = \lambda_n U_n(x)$$
$$U_n(0) = 0$$
$$U_n'(1) = 0$$

now solve the ODE above for 3 separate cases λn <0, =0, >0 and see which values of λn are allowable for the given boundary conditions. This yields the eigenvalues & eigenfunctions for the boundary value problem.

 

[speedycatz]

ok, so so far you have assumed separatino of variables, that is u(x,t) = U(x)T(t) and

so now you'll need to find the eigenfunctions of the homogenous boundary value problem
$$U_n''(x) = \lambda_n U_n(x)$$
$$U_n(0) = 0$$
$$U_n'(1) = 0$$

now solve the ODE above for 3 separate cases λn <0, =0, >0 and see which values of λn are allowable for the given boundary conditions. This yields the eigenvalues & eigenfunctions for the boundary value problem.

I get like $$U_n = cos(n \pi x)$$ and $$\lambda _n = -(n \pi)^2$$
Is that correct?

 

[lanedance]

do they satisfy you boundary conditions?

look ok for the 2nd, not so sure about the first