Solving Non-Linear Diffusion Eqn w/ Eigenfunction Expansion

[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

 

[gtoguy97]

Yes, in this case it is reasonable to assume that T_i(0) = 1/3. This is because you have assumed that R(x,t) = \sum^{J}_{1} X_i(x)T_i(t). Since you know that R(x,0) = 1, then it follows that \sum^{J}_{1} X_i(x)T_i(0) = 1. Therefore, if you assume that all of the T_i(0) are equal, then you can conclude that T_i(0) = 1/3.