By Satwinder Jit Singh, Anindya Chatterjee (auth.), Jocelyn Sabatier, Om Prakash Agrawal, J. A. Tenreiro Machado (eds.)
In the final twenty years, fractional (or non integer) differentiation has performed a vital position in quite a few fields resembling mechanics, electrical energy, chemistry, biology, economics, keep an eye on thought and sign and photograph processing. for instance, within the final 3 fields, a few vital issues reminiscent of modelling, curve becoming, filtering, development attractiveness, part detection, identity, balance, controllability, observability and robustness are actually associated with long-range dependence phenomena. comparable development has been made in different fields in this article. The scope of the publication is hence to provide the cutting-edge within the learn of fractional platforms and the appliance of fractional differentiation.
As this quantity covers fresh purposes of fractional calculus, it is going to be of curiosity to engineers, scientists, and utilized mathematicians.
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Extra resources for Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
5 2 Fig. 1. Initialization for Caputo and LH semi-derivatives of (t 2) 2. Use of the Caputo derivative and its inferred D order history, Eq. (34), may be acceptable if i) it is found that the history acceptable to the physics defining the problem and ii) if it is acceptable to have discontinuity of the derivatives of f t of order m and greater at t 0 , where m is defined by m 1 D m. Case 2 Suppose we now consider the following fractional differential equation involving more than a single fractional derivative: C 2D 0 dt f (t ) C0 d tD f (t ) f (t ) y (t ) , 1/ 2 D 1.
14) and (15) correctly predict that ED,1(–x) will have 349 zeros. 9796276, Eqs. (14) and (15) incorrectly predict 349 zeros instead of the correct 351. 9796276, the approximations used in deriving Eqs. 9796276. 9796277 to be guaranteed that Eqs. (14) and (15) will predict the correct number of real zeros. If D is specified to a certain number of significant digits, Table 2 gives the range of D that will guarantee that the results of Eqs. (14) and (15) yield the correct number of real zeros. 24 Hanneken, Vaught, and Achar Table 2.
Html 14. Gorenflo R, Mainardi F (1996) Fractional oscillations and Mittag-Leffler functions. Fachbereich Mathematik und Informatik, A14/96, Freie Universitaet, Berlin, pp. 1–22. html 15. Ostrovskii V, Peresyokkova IN (1997) Nonasymptotic results on distribution of zeros of the function Eρ(z,µ). Anal. Math. 23(4):283–296. 16. Popov AYu (2002) The spectral values of a boundary value problem and the zeros of Mittag-Leffler functions. Differential Equations 38(5):642–653. 17. Gorenflo R, Mainardi F (1997) Fractional calculus: integral and differential equations of fractional order.
Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering by Satwinder Jit Singh, Anindya Chatterjee (auth.), Jocelyn Sabatier, Om Prakash Agrawal, J. A. Tenreiro Machado (eds.)