# Applying initial conditions to functions that are part of an Eigenfunction Expansion

1. Jan 6, 2012

### ryan.j

I'm trying to solve a non-linear time-dependent diffusion equation to find R(x,t). To do so, I'm positing that :

R(x,t)=$\sum^{J}_{1}$X$_{i}$(x)T$_{i}$(t)

which allows me to arrive at something that looks like :

dT$_{i}$/dt=A$_{i}$T$_{i}$(t)-B*T$_{i}$(t)$^{2}$

The problem I'm having, through sheer lack of knowledge, is ascribing initial conditions to T$_{i}$(t).

I know that R(x,0) = 1. Taking the case where J = 3, for example, can I simply say that T$_{i}$(0) = 1/3? If not, is there a way to determine the initial conditions for each T$_{i}$(t), given that I know that they need to sum to 1?

Thank you kindly for any help.
-ryan