How to prove that the Euler-Bernoulli equation is hyperbolic

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Discussion Overview

The discussion revolves around proving that the Euler-Bernoulli equation, represented by the partial differential equation \( u_{tt}(x,t) + u_{xxxx}(x,t) = 0 \), is hyperbolic. Participants explore the definitions and characteristics of hyperbolic partial differential equations, particularly in the context of this specific equation.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks assistance in proving the hyperbolicity of the given equation.
  • Another participant clarifies the time and space intervals, confirming that \( t \) is in the interval \([0, T]\).
  • Several participants discuss the definition of hyperbolic partial differential equations, noting that the highest derivative in time is of order 2 while in space it is of order 4.
  • A participant references a definition from Lawrence C. Evans' book on partial differential equations, suggesting it may provide clarity on the topic.
  • There is a request for a snippet or image of the relevant definition from the book to facilitate understanding.
  • Another participant mentions having read an interesting definition on Wikipedia, indicating a variety of sources being considered.

Areas of Agreement / Disagreement

Participants generally agree on the need to clarify the definition of hyperbolic partial differential equations, but there is no consensus on how the Euler-Bernoulli equation fits into this classification.

Contextual Notes

There are limitations regarding the definitions being referenced, as well as the potential need for further clarification on the implications of the order of derivatives in the equation.

mathematix89
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Hello, I would like to prove that the following partial differential equation is hyperbolic. u_{tt} (x,t)+ u_{xxxx} (x,t)= 0 with x \in \left[0 , 1\right] and x \in \left[0 , T \right ] . Can anyone help me? Thank you.
 
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I presume that the last condition is that t, not x, is in [0, T]. Okay, what is the DEFINITION of "hyperbolic partial differential equation"?
 
HallsofIvy said:
I presume that the last condition is that t, not x, is in [0, T]. Okay, what is the DEFINITION of "hyperbolic partial differential equation"?
yes indeed, t \in [0,T]
 
HallsofIvy said:
I presume that the last condition is that t, not x, is in [0, T]. Okay, what is the DEFINITION of "hyperbolic partial differential equation"?
In general when the highest derivative in space and time is of order 2 it is obvious. But here we have a derivative in time of order 2 and in space of order 4.

the definition of these equations that I have in my possession is that given by Lawrence C Evans in his book entitled Partial differential Equations volume 19.
Here is a snippet that gives a definition of these equations (page 377)
 
mathematix89 said:
In general when the highest derivative in space and time is of order 2 it is obvious. But here we have a derivative in time of order 2 and in space of order 4.

the definition of these equations that I have in my possession is that given by Lawrence C Evans in his book entitled Partial differential Equations volume 19.
Here is a snippet that gives a definition of these equations (page 377)
If we had the book that would be great. If you can't type it out you can write it down on paper (neatly please!) and upload the picture.

-Dan
 
Capture1.PNG
 
topsquark said:
If we had the book that would be great. If you can't type it out you can write it down on paper (neatly please!) and upload the picture.

-Dan
ok I'll try to send you the book or in the worst case I'll take screenshots
 
mathematix89 said:
ok I'll try to send you the book or in the worst case I'll take screenshots
Capture2.PNG
 
  • #10
Capture 4.PNG
 
  • #11
I also read a very interesting definition on wikipedia but well...
 

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