Order of Initial Conditions in PDE Solutions: How to Determine?

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Discussion Overview

The discussion revolves around the determination of the order in which to apply initial and boundary conditions when solving partial differential equations (PDEs), specifically in the context of the wave equation. Participants explore the implications of the order of conditions on the solution process.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions how to determine the order of initial and boundary conditions when solving a PDE, using the wave equation as an example.
  • Another participant suggests that the boundary value problem should be solved first to express the temporal equation's solution in terms of eigenvalues.
  • A follow-up inquiry seeks clarification on the order of the initial conditions after applying the boundary conditions.
  • One participant argues that the order of initial conditions does not affect the solution, stating that they will independently determine the constants in the wave equation.

Areas of Agreement / Disagreement

Participants express differing views on whether the order of initial conditions matters, indicating a lack of consensus on this aspect of solving PDEs.

Contextual Notes

The discussion does not resolve the implications of the order of initial conditions on the solution process, and assumptions about the nature of the PDEs and conditions are not fully articulated.

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When solving a pde using this method how do you know what ORDER to use the initial/boundary conditions given to you?

E.g. if you are asked to solve the wave equation given u(x,0), u'(x,0), u(0,t), u(l,t) how do you know what order to use these in (particularly the first two)
 
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You need to solve the boundary value problem first, so that you can write the solution to the temporal equation in terms of the eigenvalues of the boundary value problem.
 


Thanks, but after you use the two boundary conditions how do you know what order to use the two initial conditions
 
As far as situations I've seen, it doesn't matter what order you use them in. In the wave equation example you gave, the initial conditions will independently determine the constants.
 

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