Fourier decomposition and heat equation

[wlwl]

Homework Statement

In the heat equation, we have $T(t,x)=sum of a_k(t)b_k(x)$.
Now I want to find a formula for computing the initial coefficients $a_k(0)$ given the initial temperature distribution $f(x)$.

Homework Equations

We know that in a heat equation , $f(0)=0$, $f(1)=0$

The Attempt at a Solution

Using Fourier decomposition, we know $f(x)=T(0,x)=a_1(0)b_1(x)+a_2(0)b_2(x)+...+a_k(0)b_k(x)$, and b_k(x) are
orthonormal, but then I don't know how to simplify this equation to get a function of $a_k(0)$

 

[lanedance]

conisider inetgrating something like f(x)sin(kx) taking into account the orthogonal nature of the eigenfunctions