Operators of Fractional Calculus and Their Applications

Abstract

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Fractional Order Runge–Kutta Methods

“…This section briefly reviews the definitions of the fractional integral and Caputo fractional derivative and explores some of their properties. A more comprehensive introduction to the fractional derivatives can be found in [1,4,33].…”

“…In this figure, we can see the interval of absolute stability for various α. Finally, we apply the IFORK method (33) to Equation (41) and get…”

“…such that the exact solution is y(t) = t 4 E α,5 (−t α ). The approximate solutions by the two-stage EFORK method (15), three-stage EFORK method (21), and IFORK methods (33) and (34) are reported in Tables 1-6 (In Appendix A some Mathematica computer programming codes are presented). The computed solutions are compared with the exact solution for different values of h, α, and T. The absolute error in time T is given by…”

Special Issue of Symmetry: “Integral Transformations, Operational Calculus and Their Applications”

“…In this connection, it seems to be worthwhile to refer the interested readers of this Special Issue to a recently-published survey-cum-expository review article (see [20]) which presented a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. Furthermore, in connection with such works as (for example) [4,7], and indeed also many papers included in the published volumes (see [15][16][17][18]), a recent survey-cum-expository review article (see [21]) will be potentially useful in order to motivate further researches and developments involving a wide variety of operators of basic (or q-) calculus and fractional q-calculus and their widespread applications in Geometric Function Theory of Complex Analysis. In the same survey-cum-expository review article (see [21]), it is also pointed out as to how known results for the q-calculus can easily (and possibly trivially) be translated into the corresponding results for the so-called (p, q)-calculus (with 0 < q < p ≤ 1) by applying some obvious parametric and argument variations, the additional parameter p being redundant (or superfluous).…”