Is Separation of Variables Valid for Solving Partial Differential Equations?

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SUMMARY

The separation of variables technique is valid for solving linear partial differential equations (PDEs) when it allows for the disassembly of the equation into solvable parts. This method yields general solutions such as Fourier series or Legendre polynomials, depending on the coordinate system used. However, not all linear PDEs are separable, indicating that specific conditions must be met for this technique to apply effectively. Understanding these conditions is crucial for accurately applying separation of variables in various scenarios.

PREREQUISITES
  • Linear Partial Differential Equations (PDEs)
  • Separation of Variables Technique
  • Fourier Series and Legendre Polynomials
  • Homogeneous and Non-Homogeneous PDEs
NEXT STEPS
  • Research the conditions under which linear PDEs are separable
  • Study the application of Fourier series in solving PDEs
  • Explore the role of forcing functions in linear PDEs
  • Learn about the classification of first-order linear PDEs
USEFUL FOR

Mathematicians, physicists, and engineers working with partial differential equations, particularly those interested in the application of separation of variables in various contexts.

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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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