Is there a theory for infinite dimensional PDEs?

In summary, the conversation discusses the topic of infinite dimensional partial differential equations (PDEs), specifically in relation to a quantum mechanical real Klein-Gordon field. The example provided involves an infinite number of variables and can be solved using a separation attempt method. It is mentioned that this may be considered a functional differential equation and there may be more rigorous examples with a countable number of variables.
  • #1
jostpuur
2,116
19
Is there any established theory concerning infinite dimensional PDE?
 
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  • #2
Do you mean that the function has infinitely many variables, or that it is an infinite dimensional function of a finite number of variables?
 
  • #3
Infinitely many variables.

For example a quantum mechanical real Klein-Gordon field, if I have understood correctly, can be pretty much described by the infinite dimensional non-homogenous heat equation (the Shrodinger's equation, with certain constants and with the harmonic potential). Something like this

[tex]
i\partial_t \Psi(t,\phi) = \sum_{k\in\mathbb{R}^3} \Big(-\alpha \partial^2_{k} + \beta |k|^2\Big)\Psi(t, \phi)
[/tex]

where

[tex]
\Psi:\mathbb{R}\times\mathbb{R}^{\mathbb{R}^3}\to\mathbb{C}.
[/tex]

It can be solved by a separation attempt

[tex]
\Psi(t,\phi) = \prod_{k\in\mathbb{R}^3} \Phi_k(t) \Psi_k (\phi(k)),
[/tex]

where

[tex]
\Phi_k,\;\Psi_k:\mathbb{R}\to\mathbb{C}
[/tex]

This is total honest pseudo mathematics, motivated by physics, don't complain about it! :biggrin:

In fact his is a very vague example with uncountable set of variables. There could be more rigor examples with only countably many variables.
 
Last edited:
  • #4
It could be these are supposed to be called functional differential equations, but I'm not sure. Some quick google hits were slightly confusing.
 

What is an infinite dimensional PDE?

An infinite dimensional PDE is a partial differential equation that involves infinitely many variables. These equations are typically used to describe phenomena that occur in continuous spaces, such as fluid dynamics or quantum mechanics.

How do infinite dimensional PDEs differ from finite dimensional PDEs?

The main difference is that finite dimensional PDEs involve a finite number of variables, while infinite dimensional PDEs involve infinitely many variables. This means that the solutions to infinite dimensional PDEs are typically more complex and may require specialized techniques for their analysis.

What are some examples of infinite dimensional PDEs?

Some common examples include the heat equation, wave equation, and Schrödinger equation. These equations are used to model a wide range of phenomena in fields such as physics, engineering, and economics.

How are infinite dimensional PDEs solved?

There are various methods for solving infinite dimensional PDEs, including numerical methods, perturbation methods, and variational methods. The specific approach used depends on the specific equation and problem being studied.

What are the applications of infinite dimensional PDEs?

Infinite dimensional PDEs have a wide range of applications in many different fields, such as physics, engineering, biology, and finance. They can be used to model and understand complex systems and phenomena, and to make predictions and inform decision-making.

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