By Satwinder Jit Singh, Anindya Chatterjee (auth.), Jocelyn Sabatier, Om Prakash Agrawal, J. A. Tenreiro Machado (eds.)
In the final 20 years, fractional (or non integer) differentiation has performed an important function in a number of fields akin to mechanics, electrical energy, chemistry, biology, economics, regulate idea and sign and photograph processing. for instance, within the final 3 fields, a few vital concerns comparable to modelling, curve becoming, filtering, development popularity, facet detection, id, balance, controllability, observability and robustness at the moment are associated with long-range dependence phenomena. related development has been made in different fields in this article. The scope of the publication is therefore to give the state-of-the-art within the learn of fractional structures and the applying of fractional differentiation.
As this quantity covers fresh purposes of fractional calculus, it will likely be of curiosity to engineers, scientists, and utilized mathematicians.
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Extra info for Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
I − α + 1) (5) Observe that for zero ICs, the two derivatives are the same. Thus, for this condition we may switch between the two derivatives as necessary. 3 Statement of the Problem We consider the following FDEs and the ICs Dα y(t) = f (t, y(t)), (i) y (i) (0) = y0 , i = 1, · · · , n − 1. (6) (7) Observe that here we consider the FDE in terms of Caputo derivatives. This allows us to account for physical initial conditions. Equations 6 and 7 are applicable for both scalar and vector y. In the discussions to follow, we use scalar y to derive an equation.
9. 10. 11. 12. 13. Lorenzo CF, Hartley TT (2000) Initialized fractional calculus. Int. J. Appl. Math. 3(3):249–265. Lorenzo CF, Hartley TT (2001) Initialization in Fractional Order Systems, in: Proceedings of the European Control Conference, Porto, Portugal pp. 1471– 1476. Oldham KB, Spanier J (1974) The Fractional Calculus- Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York. Miller KS, Ross B (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations.
C. (2) t ³ It is assumed that the function f t is zero for all t d a the time interval between t a and t c being considered to be the “history” of the fractional integral c d t q f (t ) . Initialization consists in adding a function \ to the integral starting at time t c so that the result of fractional integration starting at time t c is equal to that of the integral starting at time t a for all t ! , q c dt Or in other words, f (t ) \ q a dt f (t ), t ! c (3) c \ 1 (t W ) q 1 f (W )dW , *(q) a ³ t !
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.)