achild’sgardenoffractionalderivatives-外文文獻-閱讀頁
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【正文】 and diagram in Figure 1 shows that Dfx ftdtdttxx?=zz210211bg bg .Since ft1b g is not a function of t2, it can be moved outside the inner integral, soDfx ft dtdt ft xtdtxtxx?==?zz z21021 101bg bg bgb gorDfx ftxtdtx?=?z20bg bgb g .Using the same procedure we can show thatDfx ftxtdtx?=?z30212bg bgb g , Dfx ftxtdtx?=??z403123bg bgb g ,8and in general, Dfxnft x t dtnxn??=??zbg bgb g1101()!. Now, as we have previously done, let us replace the ?n with arbitrary α and thefactorial with the gamma function to get(9) Dfxftdtxtxαααbgb gb gbg=??+z110Γ.This is a general expression (using an integral) for fractional derivatives that has thepotential of being used as a definition. But, there is a problem . If α 1, the integral isimproper. This occurs because as tx→ , xt?→0 . This integral diverges for everyα≥0. When ? 10α the improper integral converges. Thus if α is negative there isno problem. Since (9) converges only for negative α , it is truly a fractional integral. Before we leave this section we want to mention that the choice of zero for the lowerlimit was arbitrary. The lower limit could just as easily have been b . However, theresulting expression will be different. Because of this, many people who work in thisfield use the notation bxDfxαb g indicating limits of integration going from b to x . Thuswe have from (9)(10) bxbxDfxftdtxtαααbgb gb gbg=??+z11Γ .Question What lower limit of fractional differentiation “b” will give us the resultbxppDxcppxcααα()()()()?=+?+??ΓΓ11?97. The mystery solved. Now you may begin to see what went wrong in Section 5. Weare not surprised that fractional integrals involve limits, because integrals involve limits.Since ordinary derivatives do not involve limits of integration, no one expects fractionalderivatives to involve such limits. We think of derivatives as local properties offunctions. The fractional derivative symbol Dα incorporates both derivatives (positiveα ) and integrals ( negative α ). Integrals are between limits. It turns out that fractionalderivatives are between limits also. The reason for the contradiction in Section 5 is thatthere were two different limits of integration being used. Now we can resolve themystery of the contradiction. What is the secret? Let39。ometrie et de m233。cole Polytech., 13(1832), 169.126. J. Liouville, Memoire: sur le calcul des differentielles 224。colePolytech., 13 (1832), 71162.7. A. C. McBride and G. F. Roach, Fractional Calculus, Pitman Publishing, 1985.8. K. S. Miller, Derivatives of noninteger order, Math. Mag., 68 (1995), 183192.9. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and FractionalDifferential Equations, John Wiley amp。 SpringerVerlag, New York, 1975.15. N. Wheeler, Construction and Physical Application of the fractional Calculus, notesfor a Reed College Physics Seminar, 1997.
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