Talk:Cauchy boundary condition

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I stumbled upon this article because a student found it and got a completely wrong notion of boundary conditions. After a short reading, I consider this article dangerous and detrimental. First of all, it should NOT be called Cauchy boundary condition, because it is not a boundary condition. It is more of the kind of an initial condition. The part on partial differential equations was plain wrong and I will apply a minimal fix. I am not sure though how to handle the overall problem with this article. Section 4.6 in Part I of Evans' book gives more insight to this. Guidokanschat (talk) 12:35, 7 September 2016 (UTC)
 * The Cauchy problem is not the same as Cauchy boundary conditions. Stop editing Wikipedia articles on things you don't understand. Mathbb (talk) 11:55, 13 September 2016 (UTC)

The heat equation example does not appear to be an example of a Cauchy boundary condition, in fact it seems to lack enough boundary conditions to allow for a unique solution. AxelBoldt 03:02, 21 March 2007 (UTC)

great example of separation of variables, however, it stops right before application of the Cauchy conditions and shows us no solution. —Preceding unsigned comment added by 24.98.18.211 (talk) 15:20, 3 October 2010 (UTC)

DRH (11/9/2010): Probably because there is no solution, as far as I can tell. Perhaps a solution would exist if the boundary were rectangular. Otherwise, it appears that polar coordinates might be required to handle the curved boundary.

This is not a Cauchy boundary condition, because the conditions it describes are on non-overlapping sections of the boundary - temperature held at zero on the curved portion of the boundary, insulation on the straight portion. Cauchy boundary conditions should describe both a Dirichlet and a Neumann condition on an exact same portion of a given boundary.130.216.209.81 (talk) 08:04, 21 March 2013 (UTC)

weighted mean
The description of a 'weighted mean' on boundary conditions is confusing -- it applies better to a Robin boundary condition Chris2crawford (talk) 19:43, 12 December 2011 (UTC)

function notation
The notation for the PDE
 * $$\psi_{xx} + \psi_{yy}= \psi(x,y) \ $$

is awkward to say the least. Could the writer clarify what exactly he intends there? Is it same as
 * $$\Delta_2\psi - \psi = 0 \ $$ ?

Also agree that the example is not helpful in illustrating Cauchy B.C. Eureka_sat (talk)

hyperbolic equations
It would be good to add a section on which types of equations are well-defined Cauchy boundary values. As I understand it, the 2nd order differential equation has to be hyperbolic, not elliptic or parabolic. In that case, your example $$\psi_{xx} + \psi_{yy}= \psi(x,y) \ $$ does not have a solution! Chris2crawford (talk) 14:55, 10 November 2012 (UTC)

Several problems
This entry has several weak points, as also suggested by the previous comments. I have raised a number of "caution" flags and done a few simple edits, but it would be great if somebody could actually rewrite the whole thing!

I removed the following example, due to problems mentioned by previous comments (most importantly, it is actually not a Cauchy probelm). It would be good to have a good example on a real Cauchy problem.

Example (removed from actual entry by Uffe Thygesen)
Let us define the heat equation in two spatial dimensions as follows
 * $$u_t = \alpha \nabla^2 u \ $$

where $$\alpha \ $$ is a material-specific constant called thermal diffusivity

and suppose that such equation is applied over the region $$G \ $$, which is the upper semidisk of radius $$a \ $$ centered at the origin. Suppose that the temperature is held at zero on the curved portion of the boundary, while the straight portion of the boundary is insulated, i.e., we define the Cauchy boundary conditions as
 * $$u=0 \ \forall (x,y) \in r=a, 0\leq \theta \leq \pi \ $$

and
 * $$u_y = 0, y = 0 \ $$

We can use separation of variables by considering the function as composed by the product of the spatial and the temporal part
 * $$u(x,y,t)= \phi (x,y) \psi (t)\ $$

applying such product to the original equation we obtain
 * $$\phi (x,y) \psi ' (t)= \alpha \phi '' (x,y) \psi (t) \ $$

whence


 * $$ \frac{\psi '(t)}{\alpha \psi (t)} = \frac{\phi '' (x,y)}{\phi (x,y)} $$

Since the left hand side (l.h.s.) depends only on $$t \ $$, and the right hand side (r.h.s.) depends only on $$(x,y) \ $$, we conclude that both should be equal to the same constant


 * $$\frac {\psi '(t)}{\alpha \psi (t)}= - \lambda = \frac {\phi '' (x,y)}{\phi (x,y)} $$

Thus we are led to two equations: the first in the spatial variables


 * $$\phi_{xx}+\phi_{yy}+\lambda \phi (x,y)=0 \ $$

and a second equation in the $$t \ $$ variable,


 * $$\psi '(t) +\lambda \alpha \psi (t)=0 \ $$

Once we impose the boundary conditions, the solution of the temporal ODE is


 * $$\psi (t) =A e^{-\lambda \alpha t}\ $$

where A is a constant which could be defined upon the initial conditions. The spatial part can be solved again by separation of variables, substituting $$\phi (x,y) = X(x)Y(y) \ $$ into the PDE and dividing by $$ X(x) Y(y) \ $$ from which we obtain (after reorganizing terms)


 * $$\frac {Y}{Y}+\lambda =-\frac {X}{X} $$

since the l.h.s. depends only on y and r.h.s. only depends on $$x \ $$, both sides must equal a constant, say $$\mu \ $$,


 * $$\frac {Y}{Y}+ \lambda =- \frac {X}{X} = \mu $$

so we obtain a pair of ODE's upon which we can impose the boundary conditions that we defined

— Preceding unsigned comment added by UffeHThygesen (talk • contribs) 10:02, 5 December 2014 (UTC)