# How to prove that the Euler-Bernoulli equation is hyperbolic

• MHB
• mathematix89
In summary: I'll let you read it for yourself. -DanI also read a very interesting definition on wikipedia but well...I'll let you read it for yourself. In summary, the following partial differential equation is hyperbolic.
mathematix89
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.

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

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

mathematix89 said:

I also read a very interesting definition on wikipedia but well...

## 1. What is the Euler-Bernoulli equation?

The Euler-Bernoulli equation is a mathematical equation that describes the relationship between the bending of a beam and the applied load or force. It is commonly used in engineering and physics to analyze the behavior of beams and other structural elements under different loads.

## 2. How do you prove that the Euler-Bernoulli equation is hyperbolic?

To prove that the Euler-Bernoulli equation is hyperbolic, one must show that it satisfies the conditions for a hyperbolic equation. This includes having a well-defined initial value problem, having unique solutions for all initial data, and having a finite propagation speed for disturbances in the solution.

## 3. What are the conditions for a hyperbolic equation?

The conditions for a hyperbolic equation include having a well-defined initial value problem, having unique solutions for all initial data, and having a finite propagation speed for disturbances in the solution. Additionally, the equation must have characteristic curves that intersect only once, and the eigenvalues of the associated matrix must be real and distinct.

## 4. How is the Euler-Bernoulli equation used in real-world applications?

The Euler-Bernoulli equation is used in a variety of real-world applications, including structural engineering, mechanical engineering, and aerospace engineering. It can be used to analyze the behavior of beams, plates, and other structural elements under different loads, and to design and optimize structures for maximum strength and stability.

## 5. What are some limitations of the Euler-Bernoulli equation?

While the Euler-Bernoulli equation is a useful tool for analyzing the behavior of beams and other structural elements, it has some limitations. It assumes that the beam is homogeneous, isotropic, and has a constant cross-section, which may not always be the case in real-world applications. Additionally, it does not take into account the effects of shear deformation, which can be significant in certain situations.

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