Do Boundary Conditions Only Require Information at a Single Point?

In summary, if you have the value of a function of many variables, and its 1st-derivatives, at a single point, and a 2nd-order partial differential equation, then you can usually use a Taylor expansion to build the entire function.
  • #1
RedX
970
3
If you have the value of a function of many variables, and its 1st-derivatives, at a single point, and a 2nd-order partial differential equation, then haven't you determined the entire function? You can use a Taylor expansion about that point to build the entire function because you have the value of the function at that point, the value of the 1st derivatives, and the value of higher derivatives (from the differential equation and differentiation of the differential equation).

Cauchy boundary conditions require the value of a function and the value of the normal derivative to an entire boundary curve of the region of interest. But don't you just need these things at a single point, and not an entire boundary curve?
 
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  • #2
RedX said:
If you have the value of a function of many variables, and its 1st-derivatives, at a single point, and a 2nd-order partial differential equation, then haven't you determined the entire function? You can use a Taylor expansion about that point to build the entire function because you have the value of the function at that point, the value of the 1st derivatives, and the value of higher derivatives (from the differential equation and differentiation of the differential equation).

Consider and arbitrary function [tex]u: \mathbb{R} ^n \rightarrow \mathbb{R}[/tex] such that [tex]u[/tex] is harmonic ie. [tex]\Delta u =0[/tex] in [tex]\mathbb{R} ^n[/tex] (actually you could do this in any domain [tex]\Omega \subset \mathbb{R} ^n[/tex] ) and suppose you want a solution of the problem:

[tex]\Delta v=0[/tex] on [tex]\mathbb{R} ^n[/tex] , [tex]v(0)=v_0[/tex] and [tex]\frac{ \partial v}{\partial x_i } (0)=v_i[/tex]

Then it's trivial to see [tex]w:= u+(v_0 -u(0))+\sum_{i=1}^{n} (v_i- \frac{ \partial u}{\partial x_i} (0))x_i[/tex] is harmonic and satisfies the given conditions, but [tex]u[/tex] was arbitrary so there are as many solutions to this as there are harmonic functions.
 
  • #3
But it ought to be true that if you were also given all the 2nd derivatives at the origin, along with the first derivatives and the function value at the origin, then the solution of the 2nd order PDE would be completely determined?

The question then becomes how do you find the 2nd derivatives? Having the value of the function at all points on a boundary curve, along with the normal first derivative on that curve, can usually give you enough information to determine the 2nd derivatives.

But what I don't get is why do you need an entire curve for your boundary? You have shown that a single point is not enough to constitute a boundary, but why isn't a tiny curve that passes through that point not enough?

Also, is it odd that a function with no singularities is completely determined by all the derivatives at a single point via a Taylor expansion? All the derivatives at a point constitute a countable infinite set, but the function itself is uncountably infinite?
 

1. What are boundary conditions in scientific experiments?

Boundary conditions refer to the specific constraints or limitations that are applied to a system or experiment in order to accurately simulate real-world conditions. They are often used to establish the starting and ending points of a system or to control the behavior of certain variables within the system.

2. Why are boundary conditions important in scientific research?

Boundary conditions are crucial in scientific research because they help to ensure that experiments and simulations accurately reflect real-world conditions. By setting specific limitations and constraints, scientists can better understand the behavior and outcomes of their experiments and draw more meaningful conclusions.

3. What are some common types of boundary conditions?

Some common types of boundary conditions include: fixed or prescribed values, symmetry or anti-symmetry, periodicity, and open or free boundaries. These conditions can be applied to different variables such as temperature, pressure, and velocity, depending on the specific experiment or system being studied.

4. How are boundary conditions determined?

The determination of boundary conditions depends on the specific experiment or system being studied. In some cases, boundary conditions may be based on known physical laws or empirical data. In other cases, they may be estimated or chosen based on the desired outcomes of the experiment. Additionally, boundary conditions may be adjusted and refined as new data is collected.

5. What happens if boundary conditions are not properly defined?

If boundary conditions are not properly defined, the results of the experiment may not accurately reflect real-world conditions. This can lead to incorrect conclusions and potentially limit the usefulness of the research. It is important for scientists to carefully consider and define boundary conditions in order to ensure the validity and reliability of their experiments.

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