infinite coupled ODE's

[dirk_mec1]

1. The problem statement, all variables and given/known data


\dot{x}_i = \sum_{ n=1}^{\infty} a(n,i) x_n +b(n,i) y_n




\dot{y}_i = \sum_{ n=1}^{\infty} c(n,i) x_n +d(n,i) y_n



\forall i \in \mathbb{N}

2. Relevant equations
a,b,c and d are constants (though dependent on the constants n and i).


3. The attempt at a solution
I want to know how I can solve such an infinite large coupled system of ODE's. Can someone help me?

[Nick Bruno]

well, you have multiple unknowns and 2 equations. u need as many equations as you have unknowns. Maybe you can choose i such that you have enough equations to solve?

[dirk_mec1]

well, you have multiple unknowns and 2 equations. u need as many equations as you have unknowns. Maybe you can choose i such that you have enough equations to solve?

Maybe I wasn't clear enough, i runs from 1 till infinity, in the first post I've denoted that i is part of the natural numbers so

i = 1,2,3,...,inf.

[D H]

He has an infinite number of equations there, Nick.

To the OP: Convert this problem to a set of equations of the form

\dot u_i = \sum_{n=1}^{\infty}e_{n,i} u_n

with the following:

\aligned
u_{2n-1} &= x_n \\
u_{2n} &= y_n \\
e_{2n-1,2m-1} &= a_{n,m} \\
e_{2n-1,2m} &= b_{n,m} \\
e_{2n,2m-1} &= c_{n,m} \\
e_{2n,2m} &= d_{n,m}
\endaligned

Now you have a problem in one infinite-dimensioned vector (u) rather than two (x and y). See if you can take it from there.

[dirk_mec1]

Very smart: creating one infinite dimensional vector but shouldn't there be another summation sign for m?

[D H]

but shouldn't there be another summation sign for m?
No.

Suppose that instead of an infinite number of x and y, we only have two:

\aligned
\dot x_1 &= a_{1,1} x_1 + b_{1,1} y_1 + a_{2,1} x_2 + b_{2,1} y_2 \\
\dot y_1 &= c_{1,1} x_1 + d_{1,1} y_1 + c_{2,1} x_2 + d_{2,1} y_2 \\
\dot x_2 &= a_{1,2} x_1 + b_{1,2} y_1 + a_{2,2} x_2 + b_{2,2} y_2 \\
\dot y_2 &= c_{1,2} x_1 + d_{1,2} y_1 + c_{2,2} x_2 + d_{2,2} y_2
\endaligned

Define the four-vector \vec u = [u_1, u_2, u_3, u_4]^T = [x_1, y_1, x_2, y_2]^T. The above becomes

\aligned
\dot u_1 &= a_{1,1} u_1 + b_{1,1} u_2 + a_{2,1} u_3 + b_{2,1} u_4 \\
\dot u_2 &= c_{1,1} u_1 + d_{1,1} u_2 + c_{2,1} u_3 + d_{2,1} u_4 \\
\dot u_3 &= a_{1,2} u_1 + b_{1,2} u_2 + a_{2,2} u_3 + b_{2,2} u_4 \\
\dot u_4 &= c_{1,2} u_1 + d_{1,2} u_2 + c_{2,2} u_3 + d_{2,2} u_4
\endaligned

This is just a matrix-vector equation: \dot{\vec u} = \mathbf A \vec u if you treat u as a column vector (or \dot{\vec u} = vec u \mathbf A^T if you use row vectors). Each of those a, b, c, and d elements maps to exactly one of the elements of the state matrix A.

This won't change when you go to infinite dimensional space.

[dirk_mec1]

D_H, thanks for clarifying that. Now with this mapping the equation we've got is:


\dot{ u} = \mathbf A u



Normally I would determine eigenvalues and eigenvectors to get an explicit solution because the general solution is:

u(t) = e^{At}


But now I've got an infinite large matrix. I've thought about this and there's isn't an exact solution that can be found, right?

[D H]

Infinite state matrices are the subjects of many books. Long books. Long books with lots of hairy math.

The best I can do is refer you to some. Here is one: http://books.google.com/books?id=G_x-F-l2V2UC

Google the term "infinite dimensional linear system" and you will find many more.