How to compute an energy function of PDE ?

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The discussion centers on computing an energy function for a partial differential equation (PDE) defined as f(x,y,z)'' = Δf(x,y,z) + f(x,y,z) * (1 - f(x,y,z)^2), where f(x,y,z) represents a three-dimensional vector field. The goal is to derive an energy function that remains constant throughout the system's evolution, despite potential variations in localized areas. The conversation highlights the relationship between the second derivative, the Laplacian, and the physical interpretation of forces acting on the system.

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  • Understanding of partial differential equations (PDEs)
  • Familiarity with vector fields in three dimensions
  • Knowledge of the Laplacian operator and its applications
  • Concept of energy conservation in dynamic systems
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  • Research methods for deriving energy functions from PDEs
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Mathematicians, physicists, and engineers working with partial differential equations, particularly those interested in energy conservation and dynamic system analysis.

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Hi,

I have a PDE of the form

f(x,y,z)'' = Δf(x,y,z) + f(x,y,z) * (1 - f(x,y,z)^2)

where f(x,y,z) is a 3 dimensional vector-field.
Now I want to compute an energy function for it such that for any state (f(x,y,z) and its first derivative f(x,y,z)') I can compute its corresponding energy (which should stay constant over the whole domain during evolution of the system, but may vary for parts of it).

How would I do this ? Can someone help me out here ?

Thanks and cheers
 
Last edited:
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What does f(x,y,z)'' mean?
 
Xiuh said:
What does f(x,y,z)'' mean?

second derivative of f
The PDE can be interpreted as describing the "force" f(x,y,z)'' (acting on the "velocity" f(x,y,z)') being dependent on the "position" f(x,y,z) of itself (in the second term) and its direct neighborhood (the laplacian in the first term). This should be closely related to PDEs commonly encountered in physics, but I coudn't find a energy-function for this version anywhere.
 
Last edited:

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