What physics does this BVP represent?

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In summary, the conversation discusses a boundary value problem involving the equation u''(x) = -λu(x) with boundary conditions u(0) = u(1) = 0. The general solution is given as u = ∑cn sin(nπx), but since there is no way to determine cn, the problem is considered ill-posed. The conversation also touches on the physics behind the problem and how it relates to an oscillating system, specifically a one-dimensional spring described by Hook's law. The concept of boundary conditions and their distinction from initial value problems is also mentioned. Finally, the conversation ends with a discussion on how this problem could model a physical system, such as a mass-spring system or a mass
  • #1
member 428835
Hi PF!

What physics does this BVP represent:

$$u''(x) = -\lambda u(x) : u(0)=u(1) = 0$$

Also, I know a general solution is ##u = \sum_{n=1} c_n \sin(n\pi x)##. There is no way of determining ##c_n##; does that mean this problem is ill-posed? I ask about the physics because I'm wondering if there is another equation (depending on what physics this is) that could close the problem.
 
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  • #3
It is an oscillator, but the boundary conditions leave the amplitude arbitrary.
 
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  • #4
Thanks for the responses! So what is a physical BC to determine the amplitude?
 
  • #5
One physical BC is to specify the function u and its first derivative u' at a given value of x. For example,
u(0) = A, u'(0) = 0 will do the job.
 
  • #6
Chandra Prayaga said:
One physical BC is to specify the function u and its first derivative u' at a given value of x. For example,
u(0) = A, u'(0) = 0 will do the job.
But then it's an IVP and not a BVP.
 
  • #7
Here's the distinction:

Boundary value problems are similar to initial value problems. A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).

For example, if the independent variable is time over the domain [0,1], a boundary value problem would specify values for {\displaystyle y(t)}
397de1edef5bf2ee15c020f325d7d781a3aa7f50
at both {\displaystyle t=0}
43469ec032d858feae5aa87029e22eaaf0109e9c
and {\displaystyle t=1}
970dea4a5f5ec5355c4cdd62f6396fbc8b1baaa1
, whereas an initial value problem would specify a value of {\displaystyle y(t)}
397de1edef5bf2ee15c020f325d7d781a3aa7f50
and {\displaystyle y'(t)}
ac415aa71b96af9b4e78aea31eff4ba122383095
at time {\displaystyle t=0}
43469ec032d858feae5aa87029e22eaaf0109e9c
.

Finding the temperature at all points of an iron bar with one end kept at absolute zero and the other end at the freezing point of water would be a boundary value problem.

from the wikipedia article:

https://en.wikipedia.org/wiki/Boundary_value_problem
 
  • #8
jedishrfu said:
Here's the distinction:
from the wikipedia article:

https://en.wikipedia.org/wiki/Boundary_value_problem
Right, so we agree? Also, the oscillating system has time as the independent variable, so initial conditions are used. With boundary conditions, we have ##x## (space) as the independent variable. Do you know if this models any physical system?
 
  • #9
Yes, your equations model an ideal simple spring sliding back and forth (ie no friction) as long as it stay within the limits and acts according to Hooke's law otherwise all bets are off and the system will either become chaotic or damper out.
 
  • #10
jedishrfu said:
Yes, your equations model an ideal simple spring sliding back and forth (ie no friction) as long as it stay within the limits and acts according to Hooke's law otherwise all bets are off and the system will either become chaotic or damper out.
But then the boundary conditions I impose, those are actually two different moments in time, right? Isn't there a bending beam that behaves similarly, or am I remembering something wrong?

I understand that a mass-spring system is modeled by the ODE, but the BC don't make sense to me in that context (seems an IVP is more appropriate).
 
  • #11
What if an engineer came to you with design constraint boundary conditions for the spring thing?

You may be trying to pidgeon hole this problem too much. Some problem can be described as IVP and at other times as BVP depending on what info you have available to solve it.
 
  • #12
jedishrfu said:
What if an engineer came to you with design constraint boundary conditions for the spring thing?
Can you elaborate please?
 
  • #13
I need a mass damper for a building with the following constraints. I don't want it to sway more than 10 feet in any direction. I want the mass damper to respond to changes as small as X...

https://en.wikipedia.org/wiki/Tuned_mass_damper

This is a made-up example I'm sure the engineers at PF will laugh and provide a much better example.
 
  • #14
Thanks!
 

1. What is a BVP?

A BVP, or boundary value problem, is a type of mathematical problem in which the values of a function are specified at certain points, called boundary points, and the goal is to find the function that satisfies the given conditions.

2. How is a BVP different from an initial value problem?

In an initial value problem, the values of a function are specified at a single point, called the initial point. In a BVP, the values are specified at multiple boundary points, making it more complex to solve.

3. What is the role of physics in a BVP?

In a BVP, physics is used to model the real-world situation that the problem represents. The boundary conditions and equations used to solve the BVP are based on physical laws and principles.

4. How is a BVP used in physics?

A BVP is used in physics to solve problems that involve boundary conditions. It is commonly used in areas such as heat transfer, fluid dynamics, and electromagnetic theory to find solutions to physical phenomena.

5. Are there any limitations to using a BVP in physics?

While BVPs are a powerful tool in physics, they do have limitations. They may not always accurately represent real-world situations, and the solutions obtained may be subject to error and approximation. Additionally, BVPs may not be suitable for all types of physical problems.

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