The physical meaning of the PDE?

In summary, this equation is an ODE that describes the motion of a fluid. The units of x and u are unknown, but they could be measured in terms of distance or time. The equation also has a magical constant, A, which represents the change of speed according to distance.
  • #1
thepioneerm
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the physical meaning of the PDE?!

Homework Statement



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Homework Equations



How can I know the physical meaning of the following partial differential equation?!

 
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  • #2


You cannot, It can refer to anything.
In physics the first thing is the units. What is the unit of x and u?
And what is the background of the equation? Why the boundary conditions have been choosen like this?
After you have the answer to these questions, you can start wondering what would the equation means.
 
  • #3


You can't. Mathematics is not physics and equations may have physical applications but separate from the application, there is no "physical" meaning. The crucial point is the meaning or interpretation of u and x themselves. As magwas suggested, you can learn something about the that by looking at the units x and u have.

By the way, this is NOT a "partial" differential equation. Since there is only one independent variable, it is an ordinary differential equation.
 
  • #4


magwas said:
You cannot, It can refer to anything.
In physics the first thing is the units. What is the unit of x and u?
And what is the background of the equation? Why the boundary conditions have been choosen like this?
After you have the answer to these questions, you can start wondering what would the equation means.


thank you for the hint.

u: velocity of a fluid m/sec
x: Distance m
[-1,1] : the Domain.

Can you help me now wondering what would the equation means?!
 
  • #5


HallsofIvy said:
You can't. Mathematics is not physics and equations may have physical applications but separate from the application, there is no "physical" meaning. The crucial point is the meaning or interpretation of u and x themselves. As magwas suggested, you can learn something about the that by looking at the units x and u have.

By the way, this is NOT a "partial" differential equation. Since there is only one independent variable, it is an ordinary differential equation.



I'm sorry :blushing: ...that's right... it is an ODE.
 
  • #6


Well, let's see. We have a fluid. It moves relative to our 2m wide/long something.
It would be helpful to know the direction of the speed relative to the something.
If the sopeed is parallel to the something (I could think of a pipe which have the same cross area at -1 and 1 (but fluid dynamics is much more complicated than that)), then we have u'(x) with the units of 1/s, some kind of frequency.
If it is not paralell, then u'(x) have units of [tex]\frac{m_{y}}{s m_{x}}[/tex] (just not to confuse length in one direction to length in other direction.
Similarly u''(x) is either 1/ms or [tex]\frac{m_{y}}{s m_{x}^2}[/tex].
Anyway, we should have a magical constant A of units [tex]m^2[/tex], thus the diff equation is really
[tex] A u''(x) = u(x)[/tex]

Now we should try to figure out the physical meaning of u''(x). It is the change of change of speed according to distance, which I honestly don't know what could mean. Maybe looking up equations of fluid dynamics or knowing more about the reasoning which led to this diff equation would help to understand more.
 
  • #7


magwas said:
Well, let's see. We have a fluid. It moves relative to our 2m wide/long something.
It would be helpful to know the direction of the speed relative to the something.
If the sopeed is parallel to the something (I could think of a pipe which have the same cross area at -1 and 1 (but fluid dynamics is much more complicated than that)), then we have u'(x) with the units of 1/s, some kind of frequency.
If it is not paralell, then u'(x) have units of [tex]\frac{m_{y}}{s m_{x}}[/tex] (just not to confuse length in one direction to length in other direction.
Similarly u''(x) is either 1/ms or [tex]\frac{m_{y}}{s m_{x}^2}[/tex].
Anyway, we should have a magical constant A of units [tex]m^2[/tex], thus the diff equation is really
[tex] A u''(x) = u(x)[/tex]

Now we should try to figure out the physical meaning of u''(x). It is the change of change of speed according to distance, which I honestly don't know what could mean. Maybe looking up equations of fluid dynamics or knowing more about the reasoning which led to this diff equation would help to understand more.



Thank you for this explanation
and I will try to find more about "change of change of speed according to distance".
 

1. What is a PDE?

A PDE, or partial differential equation, is a mathematical equation that involves partial derivatives of an unknown function of several independent variables. It is used to model and describe physical phenomena in various fields such as physics, engineering, and economics.

2. What is the physical meaning of a PDE?

The physical meaning of a PDE is that it describes the relationship between the dependent variable and its independent variables in a physical system. It represents the underlying laws and principles that govern the behavior of the system.

3. How are PDEs used in science?

PDEs are used in science to model and analyze physical systems, such as fluid flow, heat transfer, and electromagnetic fields. They also play a crucial role in developing mathematical models and simulations for predicting and understanding real-world phenomena.

4. What are the types of PDEs?

There are several types of PDEs, including elliptic, parabolic, and hyperbolic. Elliptic PDEs are used to describe steady-state phenomena, while parabolic PDEs are used for problems involving heat and diffusion. Hyperbolic PDEs are used for problems involving waves and vibrations.

5. What is the difference between a PDE and an ODE?

A PDE is a differential equation that involves partial derivatives, while an ODE is a differential equation that involves ordinary derivatives. PDEs are used to model systems with multiple independent variables, while ODEs are used to model systems with a single independent variable.

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