Is Separation of Variables Valid for Solving Partial Differential Equations?

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Discussion Overview

The discussion focuses on the validity of the separation of variables technique for solving partial differential equations (PDEs). Participants explore the conditions under which this method can be applied, particularly in relation to linearity and the nature of the equations involved.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant questions the validity of separation of variables, suggesting it provides a particular general solution depending on the coordinate system used.
  • Another participant asserts that separation of variables is valid for linear equations, emphasizing the ability to "disassemble" and reassemble the equation.
  • A different participant challenges the assertion that all linear PDEs are separable, indicating that this may be too strong a claim.
  • Further inquiry is made into the conditions for separability, specifically whether all first-order linear PDEs would be separable and how forcing functions might affect the approach.

Areas of Agreement / Disagreement

Participants express differing views on the conditions for the validity of separation of variables, with some asserting it applies to linear equations while others contend that not all linear PDEs are separable. The discussion remains unresolved regarding the specific conditions that govern separability.

Contextual Notes

Limitations include a lack of consensus on the conditions under which separation of variables is valid, as well as the potential influence of forcing functions on the separability of first-order linear PDEs.

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when is the separation of variables technique for partial differential equations valid? it seems to give a particular general solution (such as a general Fourier series, or series of legendre polynomials) to a problem depending which coordinate system that you are in?
 
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Any time your equation is linear. Separation of variables works as long as it is possible to "disassemble" your equation, solve each part, then put them back together into a solution of the entire equation. That is basically what "linear" allows us to do.
 
HallsofIvy said:
Any time your equation is linear.

I think that's a little strong since not every linear partial differential equation is separable.
 
dhris said:
I think that's a little strong since not every linear partial differential equation is separable.

I wonder what the conditions are. Would all first order linear PDEs be separable? If there was a forcing function could we just deal with the homogeneous part like we can for ODEs?
 

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