A Tricky PDE Transformation: Can You Help?

In summary, the PDE is transformed by introducing the characteristic variables which changes the equation from a difficult to solve to an integrable equation. The transformation has a physical motivation in that you can integrate the equation.
  • #1
hanson
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0
A trick on PDE??

Hi all.
I am reading a text in mathematical wave theory.
I saw and am confused by a manipulation of a PDE, as shown in the attached figure.

I don't really undertand how the equation (1.9) is transformed by "introducing the charcteristic variables). (as indicated by the red arrow and the question mark)

I guess there are some missing steps? Could somone fill in the missing links so that I could know what's going on there.
 

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  • #2
New parameters are introduced such that [tex]\xi=x-t ,\hspace{1cm} \zeta=x+t[/tex]. You need to work out utt and uxx in terms of the new parameters. For example, [tex]u_t=\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial t}+\frac{\partial u}{\partial \zeta}\frac{\partial \zeta}{\partial t}=-u_{\xi}+u_{\zeta}[/tex], using the chain rule. Use a similar method to find ux, and then to find utt and uxx.

When you've worked out utt and uxx sub them into the LHS of (1.9) and you should obtain the result.
 
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  • #3
Thanks! I think I could work it out.
But what is the motivation of doing this?
I mean, is there any phyical meaning attached? Or What benefits can be obtained after this transformation?
 
  • #4
hanson said:
Thanks! I think I could work it out.
But what is the motivation of doing this?
I mean, is there any phyical meaning attached? Or What benefits can be obtained after this transformation?

I'm not sure of the physical meaning, but the advantage of performing the transformation is that you can integrate the equation! Note that f is now a function of only [itex]\xi[/itex], so the primes denote differentiation wrt [itex]\xi[/itex]. Thus you can integrate once wrt [itex]\xi[/itex] and once wrt [itex]\zeta[/itex] and obtain the result given (noting that this is a PDE and so the constant of integration is not just a constant in the normal sense, but is a function of the coordinate not integrated with repect to.)

IF you don't make the substitution, then the equation is a fair bit harder to solve.
 
  • #6
Oops...
I still find it difficult to implement the two initial conditions to determine functions A and B...
how to relate the conditions in terms of the characteristic variables?
 

1. What is the purpose of this PDE transformation?

The purpose of this PDE transformation is to simplify a given partial differential equation (PDE) into a more manageable form, making it easier to solve and analyze. It involves manipulating the PDE using various techniques and transformations, such as change of variables, substitution, and integration, to transform it into an equivalent form that is easier to work with.

2. What are the benefits of using a PDE transformation?

There are several benefits to using a PDE transformation. Firstly, it can help to reduce the complexity of the PDE, making it easier to solve and interpret. Secondly, it can reveal hidden relationships and patterns within the PDE, providing a deeper understanding of the underlying physical or mathematical principles. Lastly, it can also lead to more efficient and accurate numerical methods for solving the PDE.

3. What are some commonly used techniques for PDE transformations?

Some commonly used techniques for PDE transformations include change of variables, substitution, integration, and symmetry methods. Change of variables involves replacing certain variables in the PDE with new ones to simplify the equation. Substitution involves replacing the PDE with an equivalent one that is easier to solve. Integration involves integrating the PDE with respect to certain variables to obtain a new form. Symmetry methods involve using the symmetries of the PDE to transform it into a simpler form.

4. Are PDE transformations always successful in simplifying the equation?

No, PDE transformations are not always successful in simplifying the equation. Some PDEs may be too complex or do not have any known transformations that can simplify them. In these cases, other methods, such as numerical methods, may be required to solve the PDE.

5. How can PDE transformations be applied in real-world problems?

PDE transformations can be applied in various fields such as physics, engineering, and finance, to name a few. They can be used to model and analyze complex physical systems, such as fluid dynamics, heat transfer, and quantum mechanics. In engineering, PDE transformations can be used to optimize designs and improve efficiency. In finance, they can be used to model and predict market behavior. Overall, PDE transformations have a wide range of applications in solving real-world problems.

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