achild’sgardenoffractionalderivatives-外文文獻(xiàn)(參考版)
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【正文】 Sons, 1993.10. P. A. Nekrassov, General differentiation, Mat. Sb., 14 (1888), 45168.11 K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, 1974.12. T. J. Osler, Fractional derivatives and the Leibniz rule, Amer. Math. Monthly, 78(1971), 645649.13. E. L. Post, , Generalized differentiation, Trans. Amer. Math. Soc., 32 (1930), 723781.14. B. Ross, editor, Proceedings of the International Conference on Fractional Calculusand its applications, University of New Haven, West Haven, Conn., 1974。indices quelconques, J. 201。canique, et sur unnoveau gentre pour resoudre ces questions, J. 201。s stop and think. What are the limits that will work for theexponential from (1)? Remember we want to write (11) bxax axbxaxDe edxae?==z11.What value of b will give this answer? Since the integral in (11) is reallyedxaeaeaxbxax abz=?11,we will get the form we want when 10aeab= . It will be zero when ab =?∞. So, if a ispositive, then b =?∞. This type of integral with a lower limit of ?∞ is sometimescalled the Weyl fractional derivative. In the notation from (10) we can write (1) as ?∞=De aexax axαα. Now, what limits will work for the derivative of xp in (5)? We have bxppbxppDx xdxxpbp?++==+?+z11111.10Again we want bpp++=110 . This will be the case when b = 0. We conclude that (5)should be written in the more revealing notation011Dxpxpxppααα=+?+?ΓΓ()(). So, the expression (5) for Dxpα has a builtin lower limit of 0. However,expression (1) for Deaxα has ?∞ as a lower limit. This discrepancy is why (7) and (8)do not match. In (7) we calculated ?∞D(zhuǎn)exaxα and in (8) we calculated 0Dexaxα.If the reader wishes to continue this study, we remend the very fine paper byMiller [8] as well as the excellent books by Oldham and Spanier [11] and by Miller andRoss [9]. Both books contain a short, but very good, history of the fractional calculuswith many references. The book by Miller and Ross [9] has an excellent discussion offractional differential equations. Wheeler’s notes [14] are another first rate introduction,which should be made more widely available. Wheeler gives several easily assessibleapplications, and is particularly interesting to read. Other references of historical interestare [1, 2, 4, 5, 6, 10, 13].8. Answers to questions. The following are short answers to the questions throughoutthe paper. Yes, this property does hold.: Yes, and this is easy to show from relation () Something is missing. That something is the constant of integration. We shouldhave11De edx ae c De edx ae cx cax ax ax ax ax ax??? ?==+ = =++zzz1112212 ... Let f(x) be expandable in an exponential Fourier series, fx cenninx()==?∞∞∑.Assuming we can differentiate fractionally term by term we getDfx cin enninxαα() ( )==?∞∞∑. Daxaaxαααπsin( ) sin( / )=+2 We know that Df x1() is geometrically interpreted as the slope of the curveyfx= ( ) and Dfx2( ) gives us the concavity of the curve. But the third
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