Cauchy vs. Dirichelt/Neumann Condition for PDE

In summary, the difference between a Cauchy Boundary condition and a combined Dirichlet/Neumann Boundary Condition for PDEs is that Cauchy conditions involve both u and du/dn being given on the boundary, while Dirichlet conditions only have u given and Neumann conditions only have du/dn given. A combination of Dirichlet and Neumann conditions does not necessarily result in a Cauchy condition. Additionally, Cauchy conditions can be used to solve Open Hyperbolic PDEs, whereas Dirichlet/Neumann conditions are only applicable to Elliptic and Parabolic PDEs.
  • #1
ginarific
4
0
Hi,

Can anybody tell me the difference between a Cauchy Boundary condition and a combined Dirichlet/Neumann Boundary Condition for PDEs?

The reason why I'm asking is because Cauchy boundary conditions can be used to solve Open Hyperbolic PDEs, whereas Dirichlet/Neumann can only be used to solve Elliptic and Parabolic PDEs.

My textbook says:

Cauchy Conditions: have u and du/dn given on C

Dirichlet Conditions: have u given on C

Neumann Conditions: du/dn given on C

So if you have a combination of Dirichlet and Neumann conditions, is that a Cauchy condition?

Any help would be much appreciated!

Thanks,
Gina
 
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  • #2
If it helps at all, I think I've seen some discussion on these BC's inhttp://youtu.be/-BleG7PBwEA" .
 
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FAQ: Cauchy vs. Dirichelt/Neumann Condition for PDE

What is the difference between Cauchy and Dirichlet/Neumann conditions for PDE?

The Cauchy condition for a partial differential equation (PDE) specifies both the value of the solution and its normal derivative on a given surface. This is known as an initial-boundary value problem. On the other hand, the Dirichlet condition specifies the value of the solution on a given boundary, while the Neumann condition specifies the normal derivative of the solution on a given boundary. Both of these are known as boundary value problems.

When should Cauchy conditions be used instead of Dirichlet/Neumann conditions?

Cauchy conditions are typically used when there is a surface in the problem domain that is not a boundary. In this case, the solution and its normal derivative are both needed to uniquely determine the solution.

Are Cauchy conditions more difficult to solve than Dirichlet/Neumann conditions?

In general, Cauchy conditions are more difficult to solve than Dirichlet/Neumann conditions because they require information about the solution and its normal derivative. This means that more data is needed to uniquely determine the solution, making the problem more complex.

Can Cauchy conditions be used for all types of PDEs?

No, Cauchy conditions are typically only used for linear PDEs with constant coefficients. For nonlinear or variable coefficient PDEs, Cauchy conditions may not be applicable or may lead to a non-unique solution.

How do Cauchy and Dirichlet/Neumann conditions affect the behavior of the solution?

Cauchy conditions will typically lead to a smoother solution compared to Dirichlet/Neumann conditions, as they provide more information about the solution. However, Dirichlet/Neumann conditions are often used to model physical boundaries, such as walls or boundaries of a domain, and can accurately describe specific behaviors at these boundaries.

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