Re: lsode for matrix DGLs?

[Matthias Brennwald]

On 22.06.2007, at 21:19, John W. Eaton wrote:

> On 22-Jun-2007, Matthias Brennwald wrote:
>
> | I guess
> | Johns approach of reshaping my matrices to vectors should do the  
> trick.
>
> Only if you have T as a scalar and the dimension of the vector
> returned by F is the same as the dimension of the state variable X.
>
> I don't recall ever encountering a differential equation in which the
> independent variable (T) is a vector.  Can you give us some more
> detail about the problem you are trying to solve?  How does T enter
> the problem?
>
> jwe

Ok, here's what I'm thinking about: I have a model consisting of N  
boxes ('compartments' or 'reservoirs') filled with water. The water  
in each box contains M solvents. The boxes are connected to each  
other by a network of pipes, and water can flow from one box to  
another through this network, therefore exchanging solvents between  
the boxes. Furhtermore, the solvents are radioactive. For instance,  
if a solvent A decays to solvent B, I'll have to consider the  
decrease in the concentration of A and the corresponding increase in  
the concentration of B.

So, I thought I'd set up my variables as follows:

- t is time
- C(i,j) is the concentration of solvent i in box j (i=1...M,  
j=1...N). This is a function of time t.
- F(k,l) is the water flux from box k to box l (k, l = 1...N, with  
the diagonal elements zero).
- R(u,v) the production rate of solvent v from the radioactive decay  
of u (u,v = 1...M, with R(u,v) <= 0 for u=v and R(u,v) >= 0 for u != v).

So, t is a scalar, and C, F and R are matrices.

With these variables, it should be possible to write down a  
differential equation for C that completely describes the dynamics of  
the solutes in the different boxes. dC/dt is then a matrix, too.

In Octave, the time steps will be stored in a vector. If the length  
of this vector is P, the solution of the differential equation will  
consist of P matrices of size M x N, i.e. a M x N x P array.

Matthias


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Matthias Brennwald
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