On Sumudu Transform Method in Discrete Fractional Calculus

and Applied Analysis 3 2. Preliminaries on Time Scales A time scale T is an arbitrary nonempty closed subset of the real numbers R. The most wellknown examples are T R, T Z, and T q : {qn : n ∈ Z}⋃{0}, where q > 1. The forward and backward jump operators are defined by σ t : inf{s ∈ T : s > t}, ρ t : sup{s ∈ T : s < t}, 2.1 respectively, where inf ∅ : supT and sup ∅ : inf T. A point t ∈ T is said to be left-dense if t > infT and ρ t t, right-dense if t < supT and σ t t, left-scattered if ρ t < t, and right-scattered if σ t > t. The graininess function μ : T → 0,∞ is defined by μ t : σ t − t. For details, see the monographs 25, 26 . The following two concepts are introduced in order to describe classes of functions that are integrable. Definition 2.1 see 25 . A function f : T → R is called regulated if its right-sided limits exist at all right-dense points in T and its left-sided limits exist at all left-dense points in T. Definition 2.2 see 25 . A function f : T → R is called rd-continuous if it is continuous at right-dense points in T and its left-sided limits exist at left-dense points in T. The setT is derived from the time scaleT as follows: ifT has a left-scatteredmaximum m, then T : T − {m}. Otherwise, T : T. Definition 2.3 see 25 . A function f : T → R is said to be delta differentiable at a point t ∈ T if there exists a number fΔ t with the property that given any ε > 0, there exists a neighborhood U of t such that ∣∣[f σ t − f s ] − fΔ t σ t − s ∣∣ ≤ ε|σ t − s| ∀s ∈ U. 2.2 We will also need the following definition in order to define the exponential function on an arbitrary time scale. Definition 2.4 see 25 . A function p : T → R is called regressive provided 1 μ t p t / 0 for all t ∈ T. The set R of all regressive and rd-continuous functions forms an Abelian group under the “circle plus” addition ⊕ defined by ( p ⊕ q) t : p t q t μ t p t q t ∀t ∈ T. 2.3 The additive inverse p of p ∈ R is defined by ( p) t : − p t 1 μ t p t ∀t ∈ T. 2.4 4 Abstract and Applied Analysis Theorem 2.5 see 25 . Let p ∈ R and t0 ∈ T be a fixed point. Then the exponential function ep ·, t0 is the unique solution of the initial value problem yΔ p t y, y t0 1. 2.5 3. An Introduction to Discrete Fractional Calculus In this section, we introduce some basic definitions and a theorem concerning the discrete fractional calculus. Throughout, we consider the discrete set Na : {a, a 1, a 2, . . .}, where a ∈ R is fixed. 3.1 Definition 3.1 see 27 . Let f : Na → R and ν > 0 be given. Then the νth-order fractional sum of f is given by Δ−ν a f t : 1 Γ ν t−ν ∑ s a t − σ s ν−1f s for t ∈ Na ν. 3.2 Also, we define the trivial sum by Δ−0 a f t : f t for t ∈ Na. 3.3 Note that the fractional sum operator Δ−ν a maps functions defined on Na to functions defined on Na ν. In the above equation the term t − σ s ν−1 is the generalized falling function defined by t : Γ t 1 Γ t 1 − ν 3.4 for any t, ν ∈ R for which the right-hand side is well defined. As usual, we use the convention that division by a pole yields zero. Definition 3.2 see 27 . Let f : Na → R and ν ≥ 0 be given, and let N ∈ N be chosen such that N − 1 < ν ≤ N. Then the νth-order Riemann-Liouville fractional difference of f is given by Δaf t : Δ NΔ− N−ν a f t for t ∈ Na N−ν. 3.5 It is clear that, the fractional difference operator Δa maps functions defined on Na to functions defined on Na N−ν. As stated in the following theorem, the composition of fractional operators behaves well if the inner operator is a fractional difference. Abstract and Applied Analysis 5 Theorem 3.3 see 27 . Let f : Na → R be given and suppose ν, μ > 0 withN − 1 < ν ≤ N. Then Δa μΔ −μ a f t Δ ν−μ a f t for t ∈ Na μ N−ν. 3.6and Applied Analysis 5 Theorem 3.3 see 27 . Let f : Na → R be given and suppose ν, μ > 0 withN − 1 < ν ≤ N. Then Δa μΔ −μ a f t Δ ν−μ a f t for t ∈ Na μ N−ν. 3.6 A disadvantage of the Riemann-Liouville fractional difference operator is that when applied to a constant c, it does not yield 0. For example, for 0 < v < 1, we have Δac − c t − a −ν Γ 1 − ν . 3.7 In order to overcome this and to make the fractional difference behave like the usual difference, the Caputo fractional difference was introduced in 12 . Definition 3.4 see 12 . Let f : Na → R and ν ≥ 0 be given, and let N ∈ N be chosen such that N − 1 < ν ≤ N. Then the νth-order Caputo fractional difference of f is given by Δaf t : Δ − N−ν a Δf t for t ∈ Na N−ν. 3.8 It is clear that the Caputo fractional difference operator Δa maps functions defined onNa to functions defined onNa N−ν as well. And it follows from the definition of the Caputo fractional difference operator that Δa c 0. 3.9 4. The Discrete Sumudu Transform The following definition is a slight generalization of the one introduced by Jarad et al. 28 . Definition 4.1. The Sumudu transform of a regulated function f : Ta → R is given by