By Richard Herrmann
The e-book offers a concise advent to the fundamental tools and methods in fractional calculus and allows the reader to meet up with the state-of-the-art during this box in addition to to take part and give a contribution within the improvement of this interesting examine region. The contents are dedicated to the appliance of fractional calculus to actual difficulties. The fractional thought is utilized to topics in classical mechanics, team concept, quantum mechanics, nuclear physics, hadron spectroscopy and quantum box thought and it'll shock the reader with new fascinating insights. This new, prolonged variation now additionally covers extra chapters approximately photo processing, folded potentials in cluster physics, infrared spectroscopy and native points of fractional calculus. a brand new characteristic is workouts with elaborated suggestions, which considerably helps a deeper figuring out of common facets of the idea. therefore, this e-book also needs to be important as a aiding medium for academics and classes dedicated to this topic.
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Additional resources for Fractional calculus : an introduction for physicists
Example text
29) it may be interpreted as a regularized Liouville derivative. 7. Some functions and the corresponding Feller derivative for θ = 1, or F D1α f (x) dα f (x) dxα sin(kx) cos(kx) sign(k)|k|α cos(kx) −sign(k)|k|α sin(kx) 1+α 2 − k π2 x cos(πα/2)Γ( 1−α )Γ(1 + α)1 F1 ( 2+α ; 32 ; −kx2 ), 2 2 divergent e−kx p F q ({ai }; {bj }; kx) k≥0 with coefficients 1 sin(θπ/2) A1 (θ, α) = − (c+ (θ, α) − c− (θ, α)) = − 2 2 sin(απ/2) cos(θπ/2) 1 A2 (θ, α) = − (c+ (θ, α) + c− (θ, α)) = − 2 2 cos(απ/2) Hence we obtain: F Dθα = sin(θπ/2)F D1α + cos(θπ/2)RZ Dα or more accurately sin(θπ/2)F D1α + cos(θπ/2)RZ Dα sin(θπ/2) d F D1α−1 + cos(θπ/2)RZ Dα dx α F Dθ = α−2 d2 d2 α−2 sin(θπ/2) + cos(θπ/2) dx 2 RZ D dx2 F D1 .
3), which determines the dynamical behaviour of a classical point mass object to a fractional friction force. 22) m is determined. The trivial solution ω = 0 for α > 0 describes a constant, time independent function x(t) = const and is a consequence of the fact, that the differential equation is built from derivatives of x(t) only. 23) m The actual number of different solutions depends on the numerical value of α. January 12, 2011 11:38 World Scientific Book - 9in x 6in HerrmannFC˙main 27 Friction Forces If α is in the range of 0 < α < 1 two different conjugate complex solutions exist.
Some special functions and the corresponding fractional derivatives according to 3 D α . 9. Some special functions and the corresponding fractional derivatives according to 4 D α . 102) These definitions are valid for 0 ≤ α < k. 103) kD dxnk and choosing n so that 0 ≤ α − nk < k the definitions given are valid for all α > 0. kD α = January 12, 2011 11:38 World Scientific Book - 9in x 6in HerrmannFC˙main 51 Fractional Calculus In the same manner the Feller fractional derivative definition may be extended to fractional derivatives of higher order.