# Coupled Differential Equation

4 views (last 30 days)
Graig on 21 Apr 2011
Dear Friend,
I am solving a coupled differential equation in matlab to simulate the laser rate equation. The number of differential equation depends on the number of modes I will put through input, usually it is a very high number say 500-600.
I have two for loops inside another for loop.
I use the usual trick: dx/dt = Ax (say this is the differential equation)
so x2 = x1 + dt(Ax) (I solve it this way giving an initial condition on x1)
The problem is, this equation will be valid as long as abs(dx/x)<<1. And we need 'dt' for this purpose very small, which will eventually increases the iteration of my for loop. Now when I do that, I got an out of memory error.
Is there a way to get rid of it. I was thinking of extracting few outputs from the iteration (not all the output), but also it didn't work.
-Graig
##### 2 CommentsShowHide 1 older comment
Graig on 21 Apr 2011
No, I was not using ODE solution. I am not sure, whether I can use ODE for such a high number of differential equation. Will glad to know about it.

Andrew Newell on 26 Apr 2011
Here is an example where someone is using ode45 to solve an even bigger problem. To avoid out-of-memory errors, make A into a sparse matrix if you can.
EDIT: If I understand your notation, M is actually a vector with components M_q, and ditto for K and B. So you should create a new vector
X = [M; N];
and then define your function as follows:
f = @(x) [-gamma*x(1:end-1) + x(end)*B.*(x(1:end-1)+1) - c*K.*x(1:end-1); P - A*x(end) - x(end)*sum(B.*x(1:end-1))];
(or you could create an easier-to-read function in a separate file).

Jarrod Rivituso on 21 Apr 2011
I would recommend giving these a try:
I've never used them with the number of states you have. You may end up finding that your system is "stiff", depending on the system dynamics. In this case, you may prefer ode15s or other stiff solvers to the commonly used ode45.

Graig on 2 May 2011
Thanks guys. But I don't know how to put this solver in a loop. Need to think about it.
Graig on 5 May 2011
Above that I meant 't_final'. I could solve the equation when I am limited to TSPAN = [0 1.0E-6] when the other parameters e.g., gamma and so on are set to some value. If I increase the final time, meaning 1.0E-6 to say 1.0E-4 it went to the error message. I am bound to keep the input parameters (gamma, P etc) fix. Physically this should work for all time interval.

Graig on 5 May 2011
Here is a discussion I found recently on this topic.