A new transform method in nabla discrete fractional calculus

Cite this article as: Jarad et al.: A new transform method in nabla discrete fractional calculus. Advances in Difference Equations A new transform method in nabla discrete fractional calculus Fahd Jarad Billur Kaymakçalan Kenan Tas¸ Starting from the definition of the Sumudu transform on a general nabla time scale, we define the generalized nabla discrete Sumudu transform. We obtain the nabla discrete Sumudu transform of Taylor monomials, fractional sums, and differences. We apply this transform to solve some fractional difference equations with initial value problems. MSC: 44A15; 44A55 discrete Sumudu transform; fractional sums; fractional differences; convolution; time scale - F(u) := S{f }(u) := Without doubt, the Sumudu transform of a function has a deep connection to its Laplace transform. However, the Sumudu transform is a bit superior as it may be used to solve problems without resorting to a new frequency domain because it preserves scales and unit properties. By these properties, the Sumudu transform may be used to solve intricate problems in engineering and applied sciences that can hardly be solved when the Laplace transform is used. Moreover, some properties of the Sumudu transform makes it more advantageous than the Laplace transform. Some of these properties are as follows. () S{tn}(u) = n!un. () limu→–τ F(u) = limt→–∞ f (t). () limu→τ F(u) = limt→∞ f (t). () limt→∓ f (t) = limu→∓ F(u). () For any real or complex number c, S{f (ct)}(u) = F(cu). Particularly, since constants are fixed by the Sumudu transform, choosing c = , it gives F() = f (). In dealing with physical applications, this aspect becomes a major advantage, especially in instances where keeping track of units and dimensional factor groups of constants is relevant. This means that in problem solving, u and F(u) can be treated as replicas of t and f (t), respectively []. Recently, an application of the Sumudu and double Sumudu transforms to Caputo fractional differential equations is given in []. In [], the authors applied the Sumudu transform to fractional differential equations. In [], the authors obtained the discrete Sumudu transform of Taylor monomials, fractional sums, and fractional differences and applied this transform to solve a fractional difference initial value problem. Starting with a general definition of the Laplace transform on an arbitrary time scale, the concepts of the h-Laplace and consequently the discrete Laplace transform were specified in []. The theory of time scales was initiated by Stefan Hilger []. This theory is a tool that unifies the theories of continuous and discrete time systems. It is a subject of recent studies in many different fields in which a dynamic process can be described with discrete or continuous models. In this paper, starting from the definition of the Sumudu transform on a general nabla time scale, we define the nabla discrete Sumudu transform and present some of its basic properties. The paper is organized as follows. In Sections  and , we introduce some basic concepts concerning the calculus of time scales and discrete fractional calculus, respectively. In Section , we define the nabla discrete Sumudu transform and present some of its basic properties. Section  is devoted to some applications. 2 Preliminaries on time scales A time scale T is an arbitrary nonempty closed subset of the real numbers R. The most well-known examples are T = R, T = Z, and T = qZ := {qn : n ∈ Z} ∪ {}, where q > . The forward and backward jump operators are defined by σ (t) := inf{s ∈ T : s > t} and ρ(t) := sup{s ∈ T : s < t}, respectively, where inf ∅ := sup T and sup ∅ := inf T. A point t ∈ T is said to be left-dense if t > inf T and ρ(t) = t, right-dense if t < sup T and σ (t) = t, left-scattered if ρ(t) < t, and right-scattered if σ (t) > t. The backwards graininess function ν : T → [, ∞) is defined by ν(t) := t – ρ(t). For details, see the monographs [, ]. The following two concepts are introduced in order to describe the classes of functions that are integrable. Definition . [] 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 . [] A function f : T → R is called ld-continuous if it is continuous at left-dense points in T and its right-sided limits exist at right-dense points in T. The set Tκ is derived from the time scale T as follows: If T has a right-scattered minimum m, then Tκ := T – {m}. Otherwise, Tκ := T. Definition . [] A function f : T → R is said to be nabla differentiable at a point t ∈ Tκ if there exists a number f ∇ (t) with the property that given any ε > , there exists a neighborhood U of t such that f ρ(t) – f (s) – f ∇ (t) ρ(t) – s ≤ ε ρ(t) – s for all s ∈ U. We shall also need the following definition in order to define the nabla exponential function on an arbitrary time scale. The set Rν (...truncated)