What comes on top of a generator of a PDE?

In summary, a PDE generator is a mathematical operator that generates a partial differential equation (PDE) from a given set of variables. Its purpose is to provide a systematic way of generating PDEs for scientific research, allowing for a more structured and organized approach to studying complex systems. PDE generators are commonly used in various fields, such as physics, biology, and social sciences, to model and analyze different systems. Examples include the heat equation, wave equation, and Schrodinger equation. However, PDE generators have limitations, such as potential inaccuracies in describing physical phenomena, high computational costs, and potential oversimplification of complex systems.
  • #1
Omega0
205
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From some principles in nature we are using in physics the calculus of variations. Let me call it a generator for PDE's. My question: Are there levels above? What I mean is: Is there mathematics where you have principles where the solutions are generators for the generators for PDEs ?
What about generators for the generators for the generators?
 
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  • #2
Omega0 said:
From some principles in nature we are using in physics the calculus of variations. Let me call it a generator for PDE's.
The heat equation for example does not come from a variational principle
 

Related to What comes on top of a generator of a PDE?

1. What is a PDE generator?

A PDE generator is a mathematical operator that generates the solutions to a partial differential equation (PDE). It is often represented as a differential operator, such as ∂/∂x or ∂²/∂x², and acts on a function to produce the PDE solution.

2. How does a PDE generator work?

A PDE generator works by applying a differential operator to a function, which transforms the function into a PDE. The PDE generator essentially "generates" the PDE by representing the relationships between the function's variables and their derivatives.

3. What is the purpose of a PDE generator?

The purpose of a PDE generator is to simplify the process of solving a PDE. Instead of solving the PDE directly, the generator can be used to produce the PDE solution by applying the appropriate differential operator to a function.

4. What are some common PDE generators?

Some common PDE generators include the Laplace operator (∇²), the heat operator (∂/∂t - κ∇²), and the wave operator (∂²/∂t² - c²∇²). These operators are used to generate solutions for different types of PDEs, such as the Laplace equation, heat equation, and wave equation.

5. Can a PDE generator be used for any type of PDE?

No, a PDE generator is specific to the type of PDE it is designed for. Different types of PDEs require different generators, and not all PDEs can be solved using a generator. Additionally, some PDEs may not have a closed-form solution and therefore cannot be solved using a PDE generator.

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