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jostpuur
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Is there any established theory concerning 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.
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.
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.
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.
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.