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I read about fractional partial differential equation on wiki and I now know its an expansion of the usual integral power partial differential equation. I also read that it had some applications in fluid and light. etc. modeling.

However, I still have some confusions, could you explain the following questions?

  1. First, I read that for any $a>0, (P^af)=f'$ exists. However, is it unique?

  2. Further, could you give an exact form(in calculation) that what's a half-derivative? https://en.wikipedia.org/wiki/Fractional_calculus

  3. Could you give some examples of the mathematical models/application(better with link to articles) of which the fractional partial differential equation were applied?

  4. What's the limitation to the fractional partial differential equation and its recent development?

  5. Can we replace or approximate fractional partial differential equation with usual integral partial differential equation?

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    This is a lot of questions. The literature about fractional calculus is extensive. You can find much material on the web. About physical applications I suggest an old book (although outdatet) : "The Fractional Calculus", Keith B.Oldham, Jerome Spanier, Academic Press, New York, 1974. More specific in electrotechnics, see for example the curious summary of basic laws, page 11 in this paper : https://fr.scribd.com/doc/14686539/The-Fractional-Derivation-La-derivation-fractionnaire – JJacquelin Jul 08 '18 at 17:15
  • https://math.stackexchange.com/questions/573075/could-you-explain-me-the-use-of-fractional-derivatives – JJacquelin Jul 08 '18 at 17:29

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