Pade Approximation and it's Applications

[LLT]

Pade Approximation states that a power series can be written as a rational function. Which is a series divided by another series.
(An easy example of this will be the geometric series with mod'r' < 1)

I've read books about the abstract bit of this. But I am completely stuck when it goes onto applications.

How does pade approximation help solving PDE and ODE? And sometimes, people optain a power series solution for PDE and ODE, (of polynomials of x) how did they do that?

I would also very much like to understand the Navier-Stokes equations.. but this can come later.

 

[LeBrad]

Pade Approximation states that a power series can be written as a rational function. Which is a series divided by another series.
(An easy example of this will be the geometric series with mod'r' < 1)

I've read books about the abstract bit of this. But I am completely stuck when it goes onto applications.

How does pade approximation help solving PDE and ODE? And sometimes, people optain a power series solution for PDE and ODE, (of polynomials of x) how did they do that?

Pade approximations are very useful in the area of model order reduction. If you have a large linear dynamical system, e.g.
$$\frac{d x(t)}{dt} = Ax(t) + bu(t) \;\;\;\; , \;\;\;\;\;y(t) = c^Tx(t)$$
where A is a big matrix, b is a vector, c is a vector, x is a vector of unknowns, and y is the output, you can use this Pade approximation of the system's transfer function to create a low-order transfer function of some different system which matches some number of derivatives of the original system's transfer function.

The end result of this is that if you give me a system of 10,000 ODEs, I can return to you a system of 20 ODEs which are a very good approximation to the original system over some range of frequencies.

For more information, search for pade approximation along with 'model order reduction'.