Initial value problems in discrete fractional calculus

By Ferhan Merdivenci Atici, Paul W. Eloe. 2008

662 citations recorded by the source corpus.

This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a v-th (0 < v ≤ 1) order fractional difference equation is defined. A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example. Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/d8d6e4a4dc6b572ff36ee4cbe6f320cc568a1110

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In Numerical Analysis

Citations in the corpus

The Theory of Discrete Fractional Calculus: Development and Application (2011, background): “Moreover, Atici and Eloe pay little attention in [2] and [3] to function domains or to lower limits of summation and di¤erentiation, two details vital for a rigorous and correct treatment of the power rule and the fractional composition rules.”
Discrete chaos in fractional delayed logistic maps (2015, background): “After that, several authors developed the theory of the fractional difference equations on time scales [9], such as the initial value problems [10], the discrete calculus of variations [11], the Laplace transform [12,13], the properties of the Caputo and the Riemann-Liouville difference [14], and so”
Existence and Ulam Stability of Solutions for Discrete Fractional Boundary Value Problem (2013, background): “[3] F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981–989, 2009.”
On Delta and Nabla Caputo Fractional Differences and Dual Identities (2011, background): “After then, several authors started to deal with discrete fractional calculus [1, 3, 4, 5, 6, 14, 12, 20, 21, 22, 27, 28], benefiting from time scales calculus originated in 1988 (see [15]).”
NECESSARY OPTIMALITY CONDITIONS FOR FRACTIONAL DIFFERENCE PROBLEMS OF THE CALCULUS OF VARIATIONS (2010, background): “−σ(s))(ν−1)f(s), t ∈ T, ν ≥ 0, (6) 4 N. R. O. BASTOS, R. A. C. FERREIRA AND D. F. M. TORRES t∆ −ν b f(t) = f(t)+ ν Γ(ν +1) Xb s=σ(t) (s+ν −σ(t))(ν−1)f(s), t ∈ T, ν ≥ 0. The next theorem was proved in [10]. Theorem 2.2 ([10]). Let f ∈ F and ν > 0. Then, the equality a∆ −ν t ∆f(t) = ∆(a∆ −ν t f(t))”
lp ‐maximal regularity for fractional difference equations on UMD spaces (2015, neutral): “To mention a few references, see [1], [2], [8], [9], [28], [30], [31], [35], [40] and [43].”
Fixed points and asymptotic stability of nonlinear fractional difference equations (2011, background): “[4] F. M. Atici, P. W. Eloe, A transform method in discrete fractional calculus, Intern.”
Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one (2013, background): “It took almost 20 years for researchers to start developing this theory [2,3] but now authors are expanding this field of science in various directions [6,8,14,15].”
FRACTIONAL h-DIFFERENCE EQUATIONS ARISING FROM THE CALCULUS OF VARIATIONS (2011, methodology): “Liouville fractional derivative and proved some properties of the fractional diﬀerence operator [24]. More results on the theory introduced by Miller and Ross are given in the works of Atici and Eloe [5, 6]. See also [10, 11, 18], and [7] for applications to the Gompertz fractional diﬀerence equatio”
Caputo q-fractional initial value problems and a q-analogue Mittag–Leffler function (2011, background): “ Very recently and after the appearance of time scale calculus (see for example [6]), some authors started to pay attention and apply the techniques of time scale to discrete fractional calculus ([3],[4],[5], [1]) beneﬁtting from the results announced before in [7]. All of these results are mainly a”