Uniqueness of the solution with certain boundary conditions

Dirichlet or Neumann boundary conditions. Chet asks for an explanation and the other person suggests proving it mathematically by considering the linearity of the equations. In summary, Chet is questioning the uniqueness of solutions in electrodynamics and the other person suggests proving it mathematically by considering the linearity of the equations.
  • #1
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Hey!


Speaking electrodynamics, I can't seem to get mathematically or even physically convinced that the solution with Dirichlet or Neumann boundary conditions is UNIQUE.

Can someone explain it?

Thanks.
 
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  • #2
What equation are you solving? Sometimes you have uniqueness, sometimes you might not depending on the problem and the boundary conditions.
 
  • #3
M. next said:
Hey!


Speaking electrodynamics, I can't seem to get mathematically or even physically convinced that the solution with Dirichlet or Neumann boundary conditions is UNIQUE.

Can someone explain it?

Thanks.
Are the equations linear? If so, are you aware that the solution has to be unique? See if you can prove this mathematically by making use of the characteristics of linear equations.

Chet
 

1. What is the uniqueness of a solution with certain boundary conditions?

The uniqueness of a solution with certain boundary conditions refers to the fact that there is only one possible solution that satisfies the given conditions. This means that there are no other solutions that could potentially fulfill the same conditions.

2. How do boundary conditions affect the uniqueness of a solution?

Boundary conditions play a crucial role in determining the uniqueness of a solution. They provide specific constraints that the solution must meet, and these constraints often help to narrow down the possible solutions to just one unique solution.

3. Can a problem have multiple solutions with the same boundary conditions?

No, a problem cannot have multiple solutions with the same boundary conditions. If two solutions satisfy the same boundary conditions, then they are essentially the same solution and are not considered unique.

4. Are there any exceptions to the uniqueness of a solution with certain boundary conditions?

Yes, there can be exceptions to the uniqueness of a solution with certain boundary conditions. For example, in some cases, there may be multiple solutions that satisfy the given boundary conditions, especially in more complex or nonlinear systems.

5. How does the uniqueness of a solution with certain boundary conditions impact real-world applications?

The uniqueness of a solution with certain boundary conditions is crucial in real-world applications as it ensures that the solution is reliable and accurate. In fields such as engineering, physics, and mathematics, having a unique solution is essential for making informed and precise decisions and predictions.

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