Solving Conventional PDEs: Separation of Variables and Eigen Theory

In summary, for conventional PDEs like diffusion and waves, the standard approach is to first use separation of variables to transform them into ODEs, and then use eigenvalues and eigenfunctions theory to construct the final solution. Some people argue that either separation of variables or eigen theory can be used, but they are actually intimately related and both are necessary for solving PDEs. However, in the case of nonlinear PDEs, separation of variables may not always work and other methods, such as Green functions or symmetry analysis, may be needed.
  • #1
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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?
 
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  • #2
pivoxa15 said:
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.
 
  • #3
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
 
  • #4
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
 
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What are conventional PDEs?

Conventional PDEs, or partial differential equations, are mathematical equations that involve multiple variables and their partial derivatives. They are used to describe physical phenomena in fields such as physics, engineering, and economics.

What is separation of variables?

Separation of variables is a method used to solve conventional PDEs by breaking down the equation into simpler, one-dimensional equations. This allows for the use of standard techniques to solve each individual equation, which can then be combined to find a solution to the original PDE.

What is eigen theory?

Eigen theory, or eigenvalue theory, is a mathematical concept that is used to solve certain types of conventional PDEs. It involves finding the eigenvalues and eigenfunctions of a differential operator, which can then be used to construct a solution to the PDE.

What are some common applications of solving conventional PDEs?

Conventional PDEs have a wide range of applications in various fields, such as heat transfer, fluid dynamics, and quantum mechanics. They are used to model and predict the behavior of physical systems and processes.

What are the advantages of using separation of variables and eigen theory to solve PDEs?

These methods provide a systematic and efficient approach to solving PDEs, especially those with simple boundary conditions and initial conditions. They also allow for the use of standard mathematical techniques, making the solution process more manageable and less prone to errors.

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