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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.
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.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"?
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.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)
ok I'll try to send you the book or in the worst case I'll take screenshotstopsquark 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
mathematix89 said:ok I'll try to send you the book or in the worst case I'll take screenshots
mathematix89 said:
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.
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.
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.
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.
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.