What physics does this BVP represent?

[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.

 

[jedishrfu]

Isn't this just an oscillating system like a one dimensional spring like Hook's law:

https://courses.lumenlearning.com/boundless-physics/chapter/periodic-motion/

• The equation of motion that describes simple harmonic motion can be obtained by combining Newton’s Second Law and Hooke’s Law into a second-order linear ordinary differential equation: $F_{net}=mx''=−kx$

 

[Chandra Prayaga]

It is an oscillator, but the boundary conditions leave the amplitude arbitrary.

 

[member 428835]

Thanks for the responses! So what is a physical BC to determine the amplitude?

 

[Chandra Prayaga]

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.

 

[member 428835]

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.

 

[jedishrfu]

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)}

at both {\displaystyle t=0}

and {\displaystyle t=1}

, whereas an initial value problem would specify a value of {\displaystyle y(t)}

and {\displaystyle y'(t)}

at time {\displaystyle t=0}

.

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

 

[member 428835]

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?

 

[jedishrfu]

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.

 

[member 428835]

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).

 

[jedishrfu]

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.

 

[member 428835]

What if an engineer came to you with design constraint boundary conditions for the spring thing?

Can you elaborate please?

 

[jedishrfu]

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.

 

[member 428835]

Thanks!