PDE: If u is a solution to a certain bound problem, question about laplacian u

In summary, the Laplacian, denoted by Lu=0, represents the Laplace equation, which is a type of partial differential equation (PDE). A function u that satisfies this equation is called a harmonic function. If a solution of a different PDE also satisfies the Laplace equation, it can be expressed as a sum or series of harmonic functions due to the linearity and superposition principle of the Laplacian operator. This informal explanation is what the conversation was discussing, and the question was asked for further clarification on any special facts about the Laplacian of a solution to a PDE.
  • #1
calvino
108
0
Why does the laplacian of u=0 when u is a solution to a certain boundary problem? Is this always the case?
 
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  • #2
after working it out, I realized I was on a totally wrong track. It's simply dependent on the stipulations on the laplacian. Perhaps one can help me with one more thing. Are there any special facts about the laplacian of a solution to a pde problem?
 
  • #3
Lu=0 is the laplace equation (L is the laplacian operator), which is itself a PDE. A Function u which satisfies the laplace equation is called an harmonic function. If the solution of a different PDE satifies also the laplace equation (it's laplacian is zero) that solution is itself an harmonic function or a sum of them or may be expresed as a series of harmonic functions, because the laplacian is a linear operator and it obbeys the superposition principle. Of course this is a very informal explanation, but i think that is what you are asking for,, maybe
 
  • #4
Calvino, spesify your question?
 

Related to PDE: If u is a solution to a certain bound problem, question about laplacian u

1. What is a PDE?

A PDE, or partial differential equation, is a mathematical equation that involves multiple variables and their partial derivatives. It is commonly used to describe physical phenomena in fields such as physics, engineering, and economics.

2. What is a bound problem?

A bound problem is a type of PDE in which the solution is subject to certain boundary conditions. These conditions specify the behavior of the solution at the boundaries of the domain in which it is defined.

3. What does it mean for u to be a solution to a bound problem?

If u is a solution to a bound problem, it means that it satisfies the PDE and the specified boundary conditions. In other words, it is a function that, when plugged into the PDE and evaluated at the boundaries, produces the desired values.

4. What is the laplacian of u?

The laplacian of u, denoted as ∇²u or Δu, is a differential operator that represents the sum of the second-order partial derivatives of u. In other words, it measures the rate of change of u in all directions at a given point.

5. How is the laplacian of u used in PDEs?

The laplacian of u is commonly used in PDEs to describe the behavior of physical systems, such as heat transfer and fluid flow. It is often used in conjunction with other terms in the PDE to represent various physical phenomena.

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