fractional calculus

plural fractional calculi

1. uncountablemathematical analysisThe branch of mathematics that studies generalisations of calculus to allow noninteger (i.e., real or complex) powers of the differentiation operator D and the integration operator J; (countable) any one of said generalisations of calculus.

   This chapter presents a number of functions that have been found to be useful in providing solutions to the problems of fractional calculus. […]The Mittag-Leffler function is the basis function to functional calculus, as the exponential function is to the integer-order calculus.

   2008, Shantanu Das, Functional Fractional Calculus for System Identification and Controls, Springer, page 19

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   2000, Carl F. Lorenzo, Tom T. Hartley, Initialized Fractional Calculus, NASA, NASA/TP—2000-209943, page 12, The paper presents the definition sets required for initialized fractional calculi. Two underlying bases have been used, the Riemann-Liouville based fractional calculus and the Grünwald based functional calculus (by reference).

   One of the major advantages of fractional calculus is that it can be considered as a superset of integer-order calculus. Thus fractional calculus has the potential to accomplish what integer-order calculus cannot. We believe that many of the great future developments will come from the applications of fractional calculus to different fields.

   2007, J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado, editors, Advances in Fractional Calculus, Springer, page xii