PDE, heat equation with mixed boundary conditions

[Hakkinen]

Homework Statement

solve the heat equation over the interval [0,1] with the following initial data and mixed boundary conditions.

Homework Equations

\partial _{t}u=2\partial _{x}^{2}u
u(0,t)=0, \frac{\partial u}{\partial x}(1,t)=0

with B.C

u(x,0)=f(x)
where f is piecewise with values:
0, 0<x\leq \frac{1}{2}
3, \frac{1}{2}<x<1

The Attempt at a Solution

after separation of variables where u(x,t)=h(x)\phi (t):

h''(x)=-\frac{\lambda }{2}h(x)
\phi'(t)=-\lambda \phi(t)

gen. solution to h is
h(x)=a\sin \sqrt{\frac{\lambda }{2}} the constant with the cos term is 0 from initial value

I'm stuck trying to find the eigenvalue
h'(1)=\frac{\lambda }{2}a\cos\sqrt{\frac{\lambda }{2}}=0
\sqrt\frac{\lambda }{2}=\arccos 0

I'm not sure what expression with n to use for arccos of 0. npi/2 won't work, or (n+1)pi/2, is this the right procedure though?

Any help is greatly appreciated!

EDIT:

I'm trying \frac{\pi }{2}+n\pi now to solve for the eigenvalue

 

[Hakkinen]

So for the general solution of u I have u(x,t)=\sum_{n=1}^{\infty}A_{n}\sin [\frac{\pi}{2}(1+2n)x]\exp -2t[\frac{\pi}{2}(1+2n)]^2

and the coefficient A_n given by

A_{n}=\frac{12}{\pi(1+2n)}\cos \pi(1+2n)

There was another cosine term with pi/2 in the argument that was always zero for any n, but is this an acceptable way to leave the coefficient expression?