Solving Conventional PDEs: Separation of Variables and Eigen Theory

[pivoxa15]

For conventional PDEs like diffusion, waves, it seems the standard way to solving them is in two steps.

1. Use separation of variables method to make them into ODEs
2. Use eigenvalues and eigenfunctions theory on ODEs to construct the final solution consisting of an infinite number of eigenfunctions which statisfies the BC and intintial conditions. Some people say that to solve them you use either separation of variables technique or eigen theory. But to me they are intimately related and both are needed in solving PDEs. Am I correct?

 

[gvk]

For conventional PDEs like diffusion, waves, it seems the standard way to solving them is in two steps.

1. Use separation of variables method to make them into ODEs
2. Use eigenvalues and eigenfunctions theory on ODEs to construct the final solution consisting of an infinite number of eigenfunctions which statisfies the BC and intintial conditions.


Some people say that to solve them you use either separation of variables technique or eigen theory. But to me they are intimately related and both are needed in solving PDEs. Am I correct?

They are intimately related, if they apply to linear equations.
In this case there is no difference amount them as well as with similar method of Green functions.
But if the PDE are nonlinear sometimes you can separate the variables in contrast to the Green function method which does not work.

 

[Feynman]

It solve by which space you choose?
if continuous functions ok no problem
but if you solve on other spaces like sobolev spaces, in this case your method will not be avalable

 

[Chris Hillman]

Hi, pivoxa,

There are many possible methods which can be considered "standard attacks" on various types of PDEs, but certainly separation of variables (which rests upon Sturm-Liouville theory and eigenthings) is one of the more general. Another general method (actually, it can claim to be in some sense the MOST general method) is symmetry analysis, a powerful generalization of dimensional analysis. There are many excellent textbooks on this method; one which emphasizes applications to fluid dynamics is Cantwell, Introduction to Symmetry Analysis.

Chris Hillman