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The Wikipedia article on fundamental solution says

In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function, which normally further addresses boundary conditions.

However, everything that the rest of the article says about fundamental solutions seems to apply equally well to Green's functions, and boundary conditions are not mentioned again. At a level that a typical physicist could understand, what is the difference between a Green's function and a fundamental solution, and why do mathematicians prefer the latter?

(Clearly the answer to this question will require more advanced math than physicists usually use - otherwise they would use fundamental solutions instead of Green's functions - but I would appreciate the simplest possible answer that captures the difference with the bare minimum of advanced mathematics.)

AccidentalFourierTransform
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tparker
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    Related terminology question here, though of course this question is different. – knzhou Jan 03 '18 at 07:56
  • I believe this question belongs in the mathematical section. The "Green function" concept used by physicists is either in the known electromagnetism set-up (see the book by J.D. Jackson) and in QFT where they bear the name propagators. Here I like the rigorous treatment by Urs Schreiber: https://www.physicsforums.com/insights/newideaofquantumfieldtheory-propagators/ – DanielC Jan 03 '18 at 09:25
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    Green function $\equiv$ fundamental solution. – AccidentalFourierTransform Jan 03 '18 at 09:42
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    The question has been asked in mathSE and answered there: The Green fctn includes smooth fctns to comport with BCs. Physicists need not agonize on this. – Cosmas Zachos Jan 03 '18 at 16:32
  • @CosmasZachos Ah, thank you very much. I was led astray by the Wiki article's confusing phrasing - I though the word "which" after the second comma modified the phrase "fundamental solution" rather than "Green's function". I wonder why mathematicians prefer fundamental solutions, as Green's functions are more general and therefore more powerful. – tparker Jan 03 '18 at 18:02
  • @AccidentalFourierTransform The notes linked above indicate that is not entirely true; a fundamental solution always has BC's that the field goes to zero at spatial infinity, while a Green's function can add a solution to the homogeneous equation which satisfies boundary conditions at finite distance. – tparker Jan 03 '18 at 18:06

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