Fractional Calculus - Gauge-institute.org

Gauge Institute Journal, Volume 5, No 1, February 2009H. Vic DannonContentsIntroduction………………………………………………………………..…………41.Fractional Integral and Derivative………………………………………….61.1 Liouville’s Fractional Integral of order −1….…………………….………....621.2 Liouville’s Fractional Derivative of order 1 ………........................... ..........622. The Fundamental Theorem of the Fractional Calculus………….....82.1 Operational Derivation of the Fundamental Theorem……………...............82.2 Direct Derivation of the Fundamental Theorem…………………………….103. The Meaning of the Fractional Derivative…………..…………………..123.1 Slope of the Convolution Transform…………….……………….…………...123.2 Rate of change of the Convolution Transform………………….……………123.3 Arithmetic mean of the Convolution Transform……….............................133.4 Fractional Calculus, and Arithmetic Mean Calculus…...………………….133.5 The Meaning of Higher Order derivatives of order 1 2 …………………..….133.6 The Applicability of the Fractional Calculus………………………………...144. The Tautochrone Problem and the Fundamental Theorem………..154.1 The Tautochrone Problem………………………………………………………154.2 The Tautochrone differential Equation……………… ………………………164.3 The Tautochrone Integral Equation…………...……………………………...174.4 The Tautochrone Fractional Equation………………………………………..174.5 The Tautochrone and the Fundamental Theorem…………………………..174.6 The Cycloid Differential Equation…………………………………………….194.7 The Cycloid parametric Equation……………………………………………..192