Solving Heat Flow Problem: Initial Boundary Value Problem

[hydralisks]

Homework Statement

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

Homework Equations

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

2) Fourier series equation

The Attempt at a Solution

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

I calculate a₀=1/6

a_n = 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!

 

[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.