Differential equations without Green functions

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Discussion Overview

The discussion revolves around the existence of Green functions for differential equations (DEs), specifically questioning whether certain DEs lack Green functions and under what conditions a DE can possess one. The scope includes both linear and non-linear DEs.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant inquires about the existence of differential equations that do not have a Green function and seeks conditions that would allow a DE to possess one.
  • Another participant asserts that the concept of Green functions is useless for non-linear DEs.
  • A different participant agrees that the concept is not applicable to non-linear DEs but suggests it can be used for all linear DEs.
  • Another participant challenges the previous assertion by stating that while the concept may seem useless for non-linear DEs, it can still be useful for solving them perturbatively.

Areas of Agreement / Disagreement

Participants express disagreement regarding the utility of Green functions for non-linear DEs, with some asserting their uselessness while others propose that they can still be beneficial in certain contexts.

Contextual Notes

There are unresolved assumptions regarding the definitions and conditions under which Green functions apply, particularly in the context of non-linear DEs.

ShayanJ
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Are there differential equations that, for some reason, don't have a Green function? Are there conditions for a DE to satisfy so that it can have a Green function?
Thanks
 
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The entire concept is useless for non-linear DEs.
 
pasmith said:
The entire concept is useless for non-linear DEs.
But can be used for all linear DEs?
 
pasmith said:
The entire concept is useless for non-linear DEs.

It is useful for solving non-linear DEs perturbatively so I don't know if I would call it useless for non-linear DEs.
 

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