How to compute an energy function of PDE ?

In summary, the conversation discusses 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. The individual is seeking to compute an energy function for this PDE in order to calculate the energy of any given state. The energy should remain constant over the entire domain but may vary for different parts of it. The PDE can be interpreted as describing the force acting on the velocity, which is dependent on the position of itself and its direct neighborhood. However, the individual is having trouble finding an energy function for this specific version of the PDE.
  • #1
xortdsc
98
0
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
 
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  • #2
What does f(x,y,z)'' mean?
 
  • #3
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.
 
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1. What is an energy function in PDE?

An energy function in PDE (Partial Differential Equations) is a mathematical function that represents the total energy of a system described by a set of PDEs. It is usually defined as the integral of the square of the gradient of the solution to the PDE over the domain of interest.

2. Why is computing an energy function important in PDE?

Computing an energy function in PDE allows us to analyze the stability, convergence, and behavior of the solution to the PDE. It also helps in understanding the physical properties of the system and can aid in the design of numerical methods for solving the PDE.

3. What are the main steps in computing an energy function of PDE?

The main steps in computing an energy function of PDE include:

  1. Defining the energy functional based on the PDE and its boundary conditions.
  2. Deriving the Euler-Lagrange equations from the energy functional.
  3. Solving the Euler-Lagrange equations to obtain the energy function.
  4. Verifying the energy function by checking its properties, such as positive definiteness and convexity.
  5. Applying the energy function to analyze the PDE solution or to design numerical methods.

4. Can an energy function be computed numerically?

Yes, an energy function can be computed numerically by discretizing the PDE domain and using numerical methods, such as finite difference, finite element, or spectral methods, to solve the Euler-Lagrange equations and obtain the energy function. However, the accuracy and convergence of the numerical solution may depend on the choice of numerical method and discretization parameters.

5. Are there any limitations to computing an energy function of PDE?

Yes, there are some limitations to computing an energy function of PDE. It may not be possible to compute an exact energy function for all types of PDEs, especially for complex or nonlinear PDEs. In such cases, approximate methods, such as perturbation or asymptotic methods, may be used to obtain an approximate energy function. Additionally, the computation of an energy function may also require significant computational resources and time, depending on the complexity of the PDE and the chosen numerical method.

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