# Heat Flow Problem

1. May 9, 2009

### hydralisks

1. The problem statement, all variables and given/known data

Find a formal solution to the given initial boundary value problem.

du/dt=5(d^2u/dx^2) 0<x<1 t>0
u(0,t)=u(1,t)=0 t>0
u(x,0)=(1-x)(x^2) 0<x<1

2. Relevant equations

1) u(x,t) = a0/2 + sum[an*e^(-b(n pi/L)^2*t) * cos(n pi x/L)

2) Fourier series equation

3. The attempt at a solution

(1-x)(x^2) = a0/2 + sum(an * cos(n pi x) with cn = an

I calculate a0=1/6

an = 2* integral[(1-x)(x^2)(cos n pi x)dx] from 0 to 1

I'm wondering if this is write so far? And if so, how do I proceed from here? Do I just plug everything back into the general u(x,t) equation?

Thanks!

2. May 9, 2009

### nickjer

You might not want to use the cosine series. Since one of your BC's is u(0,t) = 0. You will never satisfy that condition when cos(0) = 1. Try using the sine series instead.