calculating coficients of non-fourier series

[mhamini]

Hi

By solving this problem :

Θtt = a²Θxx ; (0<x<L , t>0)
Θ(x,0) = f(x);
Θt(x,0) = 0;
Θx(0,t) + αΘ(0,t) = 0;
Θx(L,t) + αΘ(L,t) = 0;
α is a positive constant;

I have this non-fourier serie now :

Θ(x,0) = ∑(AnCn)e^(αx)+(AnDn)e^(-αx) = f(x)

How can i obtain AnBn & AnDn cofficients?

 

[steve10]

(1) The first boundary condition is not correctly set
Θx(0,t) + αΘ(0,t) = 0
It should be
Θx(0,t) +bΘ(0,t) = 0
where b is NONPOSITIVE. This is very important, or you may end up with more than one solution.

(2) By separation of variables, when you set
Θ(x,t)=X(x)T(t)
you will have two equations:
T''(t)+λ a²T(t)=0
and
X''(x)+λX(x)=0
X'(0) + bX(0) = 0
X'(L) + cX(L) = 0
where b<0 and c>0
Try to solve this eigenvalue problem. You'll see that you have to deal with some transcendental equations to get the eigenvalues. The eigenfunctions are all combinations of sin and cos. You won't have exponential functions.