Fractional Calculus - Gauge-institute.org

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Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic DannonThe meaning of order n +1Fractional Derivative follows from21 1 12 2 + 1 2n + n n ( − )D f( x) = D D f( x) = D F ( x).3.6 The Applicability of the Fractional CalculusThe meaning of the Fractional Derivative enables us to evaluatethe applicability of the Fractional Calculus in simple terms:The Fractional Derivative of order 1 2may be useful in a modelthat involves the Convolution Transform12( − )F ( x). That is,If you can see the Convolution Transform underlying your model,you may be well served by the Fractional Derivative.The solution of the Tautochrone Problem in the next section withthe aid of the Fundamental Theorem, shows thatthe required Convolution Transform modeling and identification isfar from self-evident.14
Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic Dannon4.The Tautochrone Problem and theFundamental TheoremThe Tautochrone Problem is associated with Abel [Abel]. Itssolution by Abel involves the use of the Fundamental Theorem.4.1 The Tautochrone ProblemA bead of mass m slips along a frictionless wire,from heightto heightη = y , at time t = 0η = 0, at time t = T .We want to find the smooth curve -if there is such curve- alongwhich the bead’s falling time is the same for any y . To that end,15
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Fractional Calculus - Gauge-institute.org

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