A Tricky PDE Transformation: Can You Help?

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    Pde Transformation
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Discussion Overview

The discussion revolves around the transformation of a partial differential equation (PDE) in the context of mathematical wave theory. Participants explore the manipulation of the equation through the introduction of characteristic variables and seek clarification on the underlying steps and motivations for this transformation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the transformation of the PDE and requests clarification on the missing steps involved.
  • Another participant suggests introducing new parameters, \(\xi = x - t\) and \(\zeta = x + t\), and outlines the process of using the chain rule to express derivatives \(u_t\) and \(u_x\) in terms of these new parameters.
  • A participant questions the physical motivation behind the transformation and its benefits, seeking to understand its significance beyond the mathematical manipulation.
  • In response, another participant notes that the transformation allows for easier integration of the equation, emphasizing that the function \(f\) becomes dependent solely on \(\xi\), which simplifies the integration process.
  • One participant expresses gratitude for the explanation but also raises a new question regarding the implementation of initial conditions to determine functions \(A\) and \(B\) in the context of the characteristic variables.

Areas of Agreement / Disagreement

Participants generally agree on the steps involved in transforming the PDE and the advantages of using characteristic variables, but there is uncertainty regarding the physical interpretation and implications of the transformation. The discussion remains unresolved regarding the implementation of initial conditions.

Contextual Notes

Participants note potential missing steps in the transformation process and the need for further clarification on how to relate initial conditions to the characteristic variables. There is also an acknowledgment that the integration constant in the context of PDEs is not a simple constant.

hanson
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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 someone fill in the missing links so that I could know what's going on there.
 

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New parameters are introduced such that \xi=x-t ,\hspace{1cm} \zeta=x+t. You need to work out utt and uxx in terms of the new parameters. For example, 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}, 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.
 
Last edited:
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?
 
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 \xi, so the primes denote differentiation wrt \xi. Thus you can integrate once wrt \xi and once wrt \zeta 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.
 
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?
 

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