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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 270106, 16 pages
http://dx.doi.org/10.1155/2012/270106
Research Article

On Sumudu Transform Method in Discrete Fractional Calculus

Department of Mathematics and Computer Science, Cankaya University, Eskisehir Yolu 29 km, 06810 Ankara, Turkey

Received 16 April 2012; Revised 4 June 2012; Accepted 5 June 2012

Academic Editor: Paul Eloe

Copyright © 2012 Fahd Jarad and Kenan Taş. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, starting from the definition of the Sumudu transform on a general time scale, we define the generalized discrete Sumudu transform and present some of its basic properties. We obtain the discrete Sumudu transform of Taylor monomials, fractional sums, and fractional differences. We apply this transform to solve some fractional difference initial value problems.

1. Introduction

The fractional calculus, which is as old as the usual calculus, deals with the generalization of the integration and differentiation of integer order to arbitrary order. It has recently received a lot of attention because of its interesting applications in various fields of science, such as, viscoelasticity, diffusion, neurology, control theory, and statistics, see [16].

The analogous theory for discrete fractional calculus was initiated by Miller and Ross [7], where basic approaches, definitions, and properties of the theory of fractional sums and differences were reported. Recently, a series of papers continuing this research has appeared. We refer the reader to the papers [812] and the references cited therein.

In the early 1990's, Watugala [13, 14] introduced the Sumudu transform and applied it to solve ordinary differential equations. The fundamental properties of this transform, which are thought to be an alternative to the Laplace transform were then established in many articles [1519].

The Sumudu transform is defined over the set of functions 𝐴=𝑓(𝑡)𝑀,𝜏1,𝜏2||||>0,𝑓(𝑡)<𝑀𝑒|𝑡|/𝜏𝑗,if𝑡(1)𝑗×[0,)(1.1) by 1𝐹(𝑢)=𝕊{𝑓}(𝑢)=𝑢0𝑓(𝑡)𝑒(𝑡/𝑢)𝑑𝑡,𝑢𝜏1,𝜏2.(1.2)

Although the Sumudu transform of a function has a deep connection to its Laplace transform, the main advantage of the Sumudu transform is the fact that 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 make it more advantageous than the Laplace transform. Some of these properties are(i)The Sumudu transform of a Heaviside step function is also a Heaviside step function in the transformed domain.(ii)𝕊{𝑡𝑛}(𝑢)=𝑛!𝑢𝑛.(iii)lim𝑢𝜏1𝐹(𝑢)=lim𝑡𝑓(𝑡).(iv)lim𝑢𝜏2𝐹(𝑢)=lim𝑡𝑓(𝑡).(v)lim𝑡0𝑓(𝑡)=lim𝑢0𝐹(𝑢).(vi)Foranyrealorcomplexnumber𝑐,𝕊{𝑓(𝑐𝑡)}(𝑢)=𝐹(𝑐𝑢).

In particular, since constants are fixed by the Sumudu transform, choosing 𝑐=0, it gives 𝐹(0)=𝑓(0).

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 G(u) can be treated as replicas of t and f(t), respectively [20].

Recently, an application of the Sumudu and Double Sumudu transforms to Caputo-fractional differential equations is given in [21]. In [22], the authors applied the Sumudu transform to fractional differential equations.

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 [23]. The theory of time scales was initiated by Hilger [24]. 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 dynamic process can be described with discrete or continuous models.

In this paper, starting from the definition of the Sumudu transform on a general time scale, we define the discrete Sumudu transform and present some of its basic properties.

The paper is organized as follows: in Sections 2 and 3, we introduce some basic concepts concerning the calculus of time scales and discrete fractional calculus, respectively. In Section 4, we define the discrete Sumudu transform and present some of its basic properties. Section 5 is devoted to an application.

2. Preliminaries on Time Scales

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers . The most well-known examples are 𝕋=, 𝕋=, and 𝕋=𝑞={𝑞𝑛𝑛}{0}, where 𝑞>1. The forward and backward jump operators are defined by 𝜎(𝑡)=inf{𝑠𝕋𝑠>𝑡},𝜌(𝑡)=sup{𝑠𝕋𝑠<𝑡},(2.1) respectively, where inf=sup𝕋 and sup=inf𝕋. A point 𝑡𝕋 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 𝜇𝕋[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 𝑓𝕋 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 𝑓𝕋 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 set 𝕋𝜅 is derived from the time scale 𝕋 as follows: if 𝕋 has a left-scattered maximum 𝑚, then 𝕋𝜅=𝕋{𝑚}. Otherwise, 𝕋𝜅=𝕋.

Definition 2.3 (see [25]). A function 𝑓𝕋 is said to be delta differentiable at a point 𝑡𝕋𝜅 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)𝑠𝜀𝜎(𝑡)𝑠𝑠𝑈.(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 𝑝𝕋 is called regressive provided 1+𝜇(𝑡)𝑝(𝑡)0 for all 𝑡𝕋𝜅.

The set of all regressive and rd-continuous functions forms an Abelian group under the “circle plus” addition defined by (𝑝𝑞)(𝑡)=𝑝(𝑡)+𝑞(𝑡)+𝜇(𝑡)𝑝(𝑡)𝑞(𝑡)𝑡𝕋𝜅.(2.3)

The additive inverse 𝑝 of 𝑝 is defined by (𝑝)(𝑡)=𝑝(𝑡)1+𝜇(𝑡)𝑝(𝑡)𝑡𝕋𝜅.(2.4)

Theorem 2.5 (see [25]). Let 𝑝 and 𝑡0𝕋 be a fixed point. Then the exponential function 𝑒𝑝(,𝑡0) is the unique solution of the initial value problem 𝑦Δ𝑡=𝑝(𝑡)𝑦,𝑦0=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 𝑎={𝑎,𝑎+1,𝑎+2,},where𝑎isxed.(3.1)

Definition 3.1 (see [27]). Let 𝑓𝑎 and 𝜈>0 be given. Then the 𝜈th-order fractional sum of 𝑓 is given by Δ𝑎𝜈1𝑓(𝑡)=Γ(𝜈)𝑡𝜈𝑠=𝑎(𝑡𝜎(𝑠))𝜈1𝑓(𝑠)for𝑡𝑎+𝜈.(3.2) Also, we define the trivial sum by Δ𝑎0𝑓(𝑡)=𝑓(𝑡)for𝑡𝑁𝑎.(3.3) Note that the fractional sum operator Δ𝑎𝜈 maps functions defined on 𝑎 to functions defined on 𝑎+𝜈.

In the above equation the term (𝑡𝜎(𝑠))𝜈1 is the generalized falling function defined by 𝑡𝜈=Γ(𝑡+1)Γ(𝑡+1𝜈)(3.4) for any 𝑡,𝜈 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 𝑓𝑎 and 𝜈0 be given, and let 𝑁 be chosen such that 𝑁1<𝜈𝑁. Then the 𝜈th-order Riemann-Liouville fractional difference of 𝑓 is given by Δ𝜈𝑎𝑓(𝑡)=Δ𝑁Δ𝑎(𝑁𝜈)𝑓(𝑡)for𝑡𝑎+𝑁𝜈.(3.5)

It is clear that, the fractional difference operator Δ𝜈𝑎 maps functions defined on 𝑎 to functions defined on 𝑎+𝑁𝜈.

As stated in the following theorem, the composition of fractional operators behaves well if the inner operator is a fractional difference.

Theorem 3.3 (see [27]). Let 𝑓𝑎 be given and suppose 𝜈,𝜇>0 with 𝑁1<𝜈𝑁. Then Δ𝜈𝑎+𝜇Δ𝑎𝜇𝑓(𝑡)=Δ𝑎𝜈𝜇𝑓(𝑡)for𝑡N𝑎+𝜇+𝑁𝜈.(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 Δ𝜈𝑎𝑐=𝑐(𝑡𝑎)𝜈.Γ(1𝜈)(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 𝑓𝑎 and 𝜈0 be given, and let 𝑁 be chosen such that 𝑁1<𝜈𝑁. Then the 𝜈th-order Caputo fractional difference of 𝑓 is given by 𝐶Δ𝜈𝑎𝑓(𝑡)=Δ𝑎(𝑁𝜈)Δ𝑁𝑓(𝑡)for𝑡𝑎+𝑁𝜈.(3.8)

It is clear that the Caputo fractional difference operator 𝐶Δ𝜈𝑎 maps functions defined on 𝑎 to functions defined on 𝑎+𝑁𝜈 as well. And it follows from the definition of the Caputo fractional difference operator that 𝐶Δ𝜈𝑎𝑐=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 𝑓𝕋𝑎 is given by 𝕊𝑎1{𝑓}(𝑢)=𝑢𝑎𝑒(1/𝑢)(𝜎(𝑡),𝑎)𝑓(𝑡)Δ𝑡𝑢𝒟{𝑓},(4.1) where 𝑎 is fixed, 𝕋𝑎 is an unbounded time scale with infimum 𝑎 and 𝒟{𝑓} is the set of all nonzero complex constants 𝑢 for which 1/𝑢 is regressive and the integral converges.

In the special case, when 𝕋𝑎=𝑎, every function 𝑓𝑎 is regulated and its discrete Sumudu transform can be written as 𝕊𝑎{1𝑓}(𝑢)=𝑢𝑘=0𝑢𝑢+1𝑘+1𝑓(𝑘+𝑎)(4.2) for each 𝑢{1,0} for which the series converges. For the convergence of the Sumudu transform, we need the following definition.

Definition 4.2 (see [27]). A function 𝑓𝑎 is of exponential order 𝑟 (𝑟>0) if there exists a constant 𝐴>0 such that ||||𝑓(𝑡)𝐴𝑟𝑡forsucientlylarge𝑡.(4.3)

The following lemma can be proved similarly as in Lemma 12 in [27].

Lemma 4.3. Suppose 𝑓𝑎 is of exponential order 𝑟>0. Then 𝕊𝑎|||{𝑓}(𝑢)existsforall𝑢{1,0}suchthat𝑢+1𝑢|||>𝑟.(4.4)

The following lemma relates the shifted Sumudu transform to the original.

Lemma 4.4. Let 𝑚0 and 𝑓𝑎𝑚 and 𝑔𝑎 are of exponential order 𝑟>0. Then for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, 𝕊𝑎𝑚𝑢{𝑓}(𝑢)=𝑢+1𝑚𝕊𝑎1{𝑓}(𝑢)+𝑢𝑚1𝑘=0𝑢𝑢+1𝑘+1𝕊𝑓(𝑘+𝑎𝑚),(4.5)𝑎+𝑚{𝑔}(𝑢)=𝑢+1𝑢𝑚𝕊𝑎1{𝑔}(𝑢)𝑢𝑚1𝑘=0𝑢+1𝑢𝑚1𝑘𝑔(𝑘+𝑎).(4.6)

Proof. For all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, we have 𝕊𝑎𝑚{1𝑓}(𝑢)=𝑢𝑘=0𝑢𝑢+1𝑘+1=1𝑓(𝑘+𝑎𝑚)𝑢𝑘=𝑚𝑢𝑢+1𝑘+11𝑓(𝑘+𝑎𝑚)+𝑢𝑚1𝑘=0𝑢𝑢+1𝑘+1=1𝑓(𝑘+𝑎𝑚)𝑢𝑘=0𝑢𝑢+1𝑘+𝑚+11𝑓(𝑘+𝑎)+𝑢𝑚1𝑘=0𝑢𝑢+1𝑘+1=𝑢𝑓(𝑘+𝑎𝑚)𝑢+1𝑚𝕊𝑎1{𝑓}(𝑢)+𝑢𝑚1𝑘=0𝑢𝑢+1𝑘+1𝕊𝑓(𝑘+𝑎𝑚),𝑎+𝑚1{𝑔}(𝑢)=𝑢𝑘=0𝑢𝑢+1𝑘+1=1𝑔(𝑘+𝑎+𝑚)𝑢𝑘=𝑚𝑢𝑢+1𝑘𝑚+1=1𝑔(𝑘+𝑎)𝑢𝑘=0𝑢𝑢+1𝑘𝑚+11𝑔(𝑘+𝑎)𝑢𝑚1𝑘=0𝑢𝑢+1𝑘𝑚+1=𝑔(𝑘+𝑎)𝑢+1𝑢𝑚𝕊𝑎1{𝑔}(𝑢)𝑢𝑚1𝑘=0𝑢+1𝑢𝑚1𝑘𝑔(𝑘+𝑎).(4.7)

Taylor monomials are very useful for applying the Sumudu transform in discrete fractional calculus.

Definition 4.5 (see [27]). For each 𝜇(), define the 𝜇th-Taylor monomial to be 𝜇(𝑡,𝑎)=(𝑡𝑎)𝜇Γ(𝜇+1)for𝑡𝑎.(4.8)

Lemma 4.6. Let 𝜇() and 𝑎,𝑏 such that 𝑏𝑎=𝜇. Then for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>1, one has 𝕊𝑏𝜇(,𝑎)(𝑢)=(𝑢+1)𝜇.(4.9)

Proof. By the general binomial formula (𝑥+𝑦)𝜈=𝑘=0𝑣𝑘𝑥𝑘𝑦𝑣𝑘(4.10) for 𝜈,𝑥,𝑦 such that |𝑥|<|𝑦|, where 𝑣𝑘𝜈=𝑘𝑘!,(4.11) as in [27], it follows from (4.10) and 𝑘𝑣=(1)𝑘𝑘+𝑣1𝑣1,(4.12) where 𝑘0 that 1(1𝑦)𝜈=((𝑦)+1)𝜈=𝑘=0𝑦𝑘+𝑣1𝑣1𝑘(4.13) for 𝜈 and |𝑦|<1.
And since 𝑏𝑎=𝜇, we have for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>1, (𝑢+1)𝜇=11𝑢+1(1(𝑢/(𝑢+1)))𝜇+1=1𝑢+1𝑘=0𝜇𝑢𝑘+𝜇𝑢+1𝑘=1𝑢𝑘=0𝜇𝑢𝑘+𝜇𝑢+1𝑘+1=1𝑢𝑘=0(𝑘+𝜇)𝜇𝑢Γ(𝜇+1)𝑢+1𝑘+1=1𝑢𝑘=0𝜇𝑢(𝑘+𝑏,𝑎)𝑢+1𝑘+1=𝕊𝑏𝜇(,𝑎)(𝑢).(4.14)

Definition 4.7 (see [27]). Define the convolution of two functions 𝑓,𝑔𝑎 by (𝑓𝑔)(𝑡)=𝑡𝑟=𝑎𝑓(𝑟)𝑔(𝑡𝑟+𝑎)for𝑡𝑎.(4.15)

Lemma 4.8. Let 𝑓,𝑔𝑎 be of exponential order 𝑟>0. Then for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, 𝕊𝑎{𝑓𝑔}(𝑢)=(𝑢+1)𝕊𝑎{𝑓}(𝑢)𝕊𝑎{𝑔}(𝑢).(4.16)

Proof. Since 𝕊𝑎{1𝑓𝑔}(𝑢)=𝑢𝑘=0𝑢𝑢+1𝑘+1(=1𝑓𝑔)(𝑘+𝑎)𝑢𝑘=0𝑢𝑢+1𝑘+1𝑘+𝑎𝑟=𝑎=1𝑓(𝑟)𝑔((𝑘+𝑎)𝑟+𝑎)𝑢𝑘𝑘=0𝑟=0𝑢𝑢+1𝑘+1𝑓(𝑟+𝑎)𝑔(𝑘𝑟+𝑎),(4.17) the substitution 𝜏=𝑘𝑟 yields 𝕊𝑎{1𝑓𝑔}(𝑢)=𝑢𝜏=0𝑟=0𝑢𝑢+1𝜏+𝑟+11𝑓(𝑟+𝑎)𝑔(𝜏+𝑎)=(𝑢+1)𝑢𝑟=0𝑢𝑢+1𝑟+11𝑓(𝑟+𝑎)𝑢𝜏=0𝑢𝑢+1𝜏+1𝑔(𝜏+𝑎)=(𝑢+1)𝕊𝑎{𝑓}(𝑢)𝕊𝑎{𝑔}(𝑢)(4.18) for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟.

Theorem 4.9. Suppose 𝑓𝑎 is of exponential order 𝑟1 and let 𝜈>0 with 𝑁1<𝜈𝑁. Then for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, 𝕊𝑎+𝜈Δ𝑎𝜈𝑓(𝑢)=(𝑢+1)𝜈𝕊𝑎𝕊{𝑓}(𝑢),(4.19)𝑎+𝜈𝑁Δ𝑎𝜈𝑓𝑢(𝑢)=𝑁(𝑢+1)𝑁𝜈𝕊𝑎{𝑓}(𝑢).(4.20)

Proof. First note that the shift formula (4.5) implies that for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, 𝕊𝑎+𝜈𝑁Δ𝑎𝜈𝑓(1𝑢)=𝑢𝑘=0𝑢𝑢+1𝑘+1Δ𝑎𝜈=𝑢𝑓(𝑘+𝑎+𝜈𝑁)𝑢+1𝑁𝕊𝑎+𝜈Δ𝑎𝜈𝑓1(𝑢)+𝑢𝑁1𝑘=0𝑢𝑢+1𝑘+1Δ𝑎𝜈=𝑢𝑓(𝑘+𝑎+𝜈𝑁)𝑢+1𝑁𝕊𝑎+𝜈Δ𝑎𝜈𝑓(𝑢),(4.21) taking 𝑁 zeros of Δ𝑎𝜈𝑓 into account. Furthermore, by (4.9), (4.15), and (4.16), 𝕊𝑎+𝜈Δ𝑎𝜈𝑓(1𝑢)=𝑢𝑘=0𝑢𝑢+1𝑘+1Δ𝑎𝜈=1𝑓(𝑘+𝑎+𝜈)𝑢𝑘=0𝑢𝑢+1𝑘+1𝑘+𝑎𝑟=𝑎(𝑘+𝑎+𝜈𝜎(𝑟))𝜈1=1Γ(𝜈)𝑓(𝑟)𝑢𝑘=0𝑢𝑢+1𝑘+1𝑘+𝑎𝑟=𝑎𝑓(𝑟)𝜈1=1((𝑘+𝑎)𝑟+𝑎,𝑎(𝜈1))𝑢𝑘=0𝑢𝑢+1𝑘+1𝑓𝜈1(,𝑎(𝜈1))(𝑘+𝑎)=𝕊𝑎𝑓𝜈1(,𝑎(𝜈1))(𝑢)=(𝑢+1)𝕊𝑎{𝑓}(𝑢)𝕊𝑎𝜈1(,𝑎(𝜈1))=(𝑢+1)(𝑢+1)𝜈1𝕊𝑎{𝑓}(𝑢)=(𝑢+1)𝜈𝕊𝑎{𝑓}(𝑢).(4.22) Then we obtain 𝕊𝑎+𝜈𝑁Δ𝑎𝜈𝑓𝑢(𝑢)=𝑢+1𝑁𝕊𝑎+𝜈Δ𝑎𝜈𝑓=𝑢(𝑢)𝑁(𝑢+1)𝑁𝜈𝕊𝑎{𝑓}(𝑢).(4.23)

Theorem 4.10. Suppose 𝑓𝑎 is of exponential order 𝑟1 and let 𝜈>0 with 𝑁1<𝜈𝑁. Then for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, 𝕊𝑎+𝑁𝜈Δ𝜈𝑎𝑓(𝑢)=(𝑢+1)𝑁𝜈𝑢𝑁𝕊𝑎{𝑓}(𝑢)𝑁1𝑘=0𝑢𝑘𝑁Δ𝑎𝜈𝑁+𝑘𝑓(𝑎+𝑁𝜈).(4.24)

Proof. Let 𝑓,𝑟,𝜈, and 𝑁 be as in the statement of the theorem. We already know from Theorem 3.8 in [28] that (4.24) holds when 𝜈=𝑁, that is, 𝕊𝑎Δ𝑁𝑓1(𝑢)=𝑢𝑁𝕊𝑎{𝑓}(𝑢)𝑁1𝑘=0𝑢𝑘𝑁Δ𝑘𝑓(𝑎).(4.25) If 𝑁1<𝜈<𝑁, then 0<𝑁𝜈<1 and hence it follows from (3.6), (4.19), and (4.25) that 𝕊𝑎+𝑁𝜈Δ𝜈𝑎𝑓(𝑢)=𝕊𝑎+𝑁𝜈Δ𝑁Δ𝑎(𝑁𝜈)𝑓=1(𝑢)𝑢𝑁𝕊𝑎+𝑁𝜈Δ𝑎(𝑁𝜈)𝑓(𝑢)𝑁1𝑘=0𝑢𝑘𝑁Δ𝑘Δ𝑎(𝑁𝜈)=𝑓(𝑎+𝑁𝜈)(𝑢+1)𝑁𝜈𝑢𝑁𝕊𝑎{𝑓}(𝑢)𝑁1𝑘=0𝑢𝑘𝑁Δ𝑎𝜈𝑁+𝑘𝑓(𝑎+𝑁𝜈).(4.26)

In the following theorem the Sumudu transform of the Caputo fractional difference operator is presented.

Theorem 4.11. Suppose 𝑓𝑎 is of exponential order 𝑟1 and let 𝜈>0 with 𝑁1<𝜈𝑁. Then for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, 𝕊𝑎+𝑁𝜈𝐶Δ𝜈𝑎𝑓(𝑢)=(𝑢+1)𝑁𝜈𝑢𝑁𝕊𝑎{𝑓}(𝑢)𝑁1𝑘=0𝑢𝑘Δ𝑘.𝑓(𝑎)(4.27)

Proof. Let 𝑓,𝑟,𝜈, and 𝑁 be as in the statement of the theorem. We already know from (4.25) that 𝑣=𝑁, (4.27) holds. If 𝑁1<𝜈<𝑁, then 0<𝑁𝜈<1 and hence it follows from (4.19) and (4.25) that 𝕊𝑎+𝑁𝜈𝐶Δ𝜈𝑎𝑓(𝑢)=𝕊𝑎+𝑁𝜈Δ𝑎(𝑁𝜈)Δ𝑁𝑓=(𝑢)(𝑢+1)𝑁𝜈𝕊𝑎Δ𝑁𝑓=(𝑢)(𝑢+1)𝑁𝜈𝑢𝑁𝕊𝑎{𝑓}(𝑢)𝑁1𝑘=0𝑢𝑘Δ𝑘.𝑓(𝑎)(4.28)

Lemma 4.12. Let 𝑓𝑎 be given. For any 𝑝0 and 𝜈>0 with 𝑁1<𝜈𝑁, one has 𝐶Δ𝑎𝜈+𝑝𝑓(𝑡)=𝐶Δ𝜈𝑎Δ𝑝𝑓(𝑡)𝑓𝑜𝑟𝑡𝑎+𝑁𝑣.(4.29)

Proof. Let f, v, N, and p be given as in the statement of the lemma. Then 𝐶Δ𝑎𝜈+𝑝𝑓(𝑡)=Δ𝑎(𝑁+𝑝𝜈𝑝)Δ𝑁+𝑝𝑓(𝑡)=Δ𝑎(𝑁𝜈)Δ𝑁Δ𝑝=𝑓(𝑡)𝐶Δ𝜈𝑎Δ𝑝𝑓(𝑡).(4.30)

Corollary 4.13. Suppose 𝑓𝑎 is of exponential order 𝑟1, 𝜈>0 with 𝑁1<𝜈𝑁 and 𝑝0. Then for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, 𝕊𝑎+𝑁𝜈𝐶Δ𝑎𝜈+𝑝𝑓(𝑢)=(𝑢+1)𝑁𝜈𝑢𝑁+𝑝𝕊𝑎{𝑓}(𝑢)𝑁+𝑝1𝑘=0𝑢𝑘Δ𝑘.𝑓(𝑎)(4.31)

Proof. The proof follows from (4.25), (4.27), and (4.29).

5. Applications

In this section, we will illustrate the possible use of the discrete Sumudu transform by applying it to solve some initial value problems. The following initial value problem was solved in Theorem 23 in [27] by using the Laplace transforms.

Example 5.1. Suppose 𝑓𝑎 is of exponential order 𝑟1 and let 𝜈>0 with 𝑁1<𝜈𝑁. The unique solution to the fractional initial value problem Δ𝜈𝑎+𝜈𝑁𝑦(𝑡)=𝑓(𝑡),𝑡𝑎Δ𝑘𝑦(𝑎+𝜈𝑁)=𝐴𝑘,𝑘{0,1,,𝑁1},𝐴𝑘(5.1) is given by 𝑦(𝑡)=𝑁1𝑘=0𝛼𝑘(𝑡𝑎)𝜈+𝑘𝑁+Δ𝑎𝜈𝑓(𝑡),𝑡𝑎+𝜈𝑁,(5.2) where 𝛼𝑘=Δ𝜈𝑁+𝑘𝑎+𝜈𝑁𝑦(𝑎)=Γ(𝜈+𝑘𝑁+1)𝑘𝑝=0𝑘𝑝𝑗=0(1)𝑗𝑘!(𝑘𝑗)𝑁𝜈𝑘𝑝𝑗𝐴𝑘𝑝𝑝(5.3) for 𝑘{0,1,,𝑁1}.

Proof. Since 𝑓 is of exponential order 𝑟, then 𝕊𝑎{𝑓}(𝑢) exists for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟. So, applying the Sumudu transform to both sides of the fractional difference equation in (5.1), we have for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, 𝕊𝑎Δ𝜈𝑎+𝜈𝑁𝑦(𝑢)=𝕊𝑎{𝑓}(𝑢).(5.4) Then from (4.24), it follows (𝑢+1)𝑁𝜈𝑢𝑁𝕊𝑎+𝜈𝑁{𝑦}(𝑢)𝑁1𝑘=0𝑢𝑘𝑁Δ𝜈𝑁+𝑘𝑎+𝜈𝑁𝑦(𝑎)=𝕊𝑎{𝑓}(𝑢)(5.5) and hence 𝕊𝑎+𝜈𝑁𝑢{𝑦}(𝑢)=𝑁(𝑢+1)𝑁𝜈𝕊𝑎{𝑓}(𝑢)+𝑁1𝑘=0𝑢𝑘(𝑢+1)𝑁𝜈Δ𝜈𝑁+𝑘𝑎+𝜈𝑁𝑦(𝑎).(5.6) By (4.20), we have 𝑢𝑁(𝑢+1)𝑁𝜈𝕊𝑎{𝑓}(𝑢)=𝕊𝑎+𝜈𝑁Δ𝑎𝜈𝑓(𝑢).(5.7) Considering the terms in the summation, by using the shifting formula (4.5), we see that for each 𝑘{0,1,,𝑁1}, 𝑢𝑘(𝑢+1)𝑁𝜈=𝑢𝑢+1𝑘(𝑢+1)𝜈+𝑘𝑁=𝑢𝑢+1𝑘𝕊𝑎+𝜈+𝑘𝑁𝜈+𝑘𝑁(,𝑎)(𝑢)=𝕊𝑎+𝜈𝑁𝜈+𝑘𝑁((1,𝑎)𝑢)𝑢𝑘1𝑖=0𝑢𝑢+1𝑖+1𝜈+𝑘𝑁(𝑖+𝑎+𝜈𝑁,𝑎)=𝕊𝑎+𝜈𝑁𝜈+𝑘𝑁(,𝑎)(𝑢)(5.8) since 𝜈+𝑘𝑁(𝑖+𝑎+𝜈𝑁,𝑎)=(𝑖+𝜈𝑁)𝜈+𝑘𝑁Γ=(𝜈+𝑘𝑁+1)Γ(𝑖+𝜈𝑁+1)Γ(𝑖𝑘+1)Γ(𝜈+𝑘𝑁+1)=0(5.9) for 𝑖{0,𝑘1}.
Consequently, we have 𝕊𝑎+𝜈𝑁{𝑦}(𝑢)=𝕊𝑎+𝜈𝑁Δ𝑎𝜈𝑓(𝑢)+𝑁1𝑘=0Δ𝜈𝑁+𝑘𝑎+𝜈𝑁𝑦(𝑎)𝕊𝑎+𝜈𝑁𝜈+𝑘𝑁(,𝑎)(𝑢)=𝕊𝑎+𝜈𝑁𝑁1𝑘=0Δ𝜈𝑁+𝑘𝑎+𝜈𝑁𝑦(𝑎)𝜈+𝑘𝑁(,𝑎)+Δ𝑎𝜈𝑓(𝑢).(5.10) Since Sumudu transform is a one-to-one operator (see [28, Theorem 3.6]), we conclude that for 𝑡𝑎+𝜈𝑁, 𝑦(𝑡)=𝑁1𝑘=0Δ𝜈𝑁+𝑘𝑎+𝜈𝑁𝑦(𝑎)𝜈+𝑘𝑁(𝑡,𝑎)+Δ𝑎𝜈=𝑓(𝑡)𝑁1𝑘=0Δ𝜈𝑁+𝑘𝑎+𝜈𝑁𝑦(𝑎)Γ(𝜈+𝑘𝑁+1)(𝑡𝑎)𝜈+𝑘𝑁+Δ𝑎𝜈𝑓(𝑡),(5.11) where Δ𝜈𝑁+𝑘𝑎+𝜈𝑁𝑦(𝑎)=Γ(𝜈+𝑘𝑁+1)𝑘𝑝=0𝑘𝑝𝑗=0(1)𝑗𝑘!(𝑘𝑗)𝑁𝜈𝑘𝑝𝑗Δ𝑘𝑝𝑘𝑦(𝑎+𝜈𝑁),(5.12) (see [27, Theorem 11]).

Example 5.2. Consider the initial value problem (5.1) with the Riemann-Liouville fractional difference replaced by the Caputo fractional difference. 𝐶Δ𝜈𝑎+𝜈𝑁𝑦(𝑡)=𝑓(𝑡),𝑡𝑎,Δ𝑘𝑦(𝑎+𝜈𝑁)=𝐴𝑘,𝑘{0,1,,𝑁1},𝐴𝑘.(5.13)
Applying the Sumudu transform to both sides of the difference equation, we get for all 𝑢{1,0} such that |(𝑢+1)/𝑢|>𝑟, 𝕊𝑎𝐶Δ𝜈𝑎+𝜈𝑁𝑦(𝑢)=𝕊𝑎{𝑓}(𝑢).(5.14)
Then from (4.27), it follows (𝑢+1)𝑁𝜈𝑢𝑁𝕊𝑎+𝜈𝑁{𝑦}(𝑢)𝑁1𝑘=0𝑢𝑘𝐴𝑘=𝕊𝑎{𝑓}(𝑢).(5.15) By (4.20), we have 𝕊𝑎+𝜈𝑁{𝑦}(𝑢)=𝑁1𝑘=0𝑢𝑘𝐴𝑘+𝑢𝑁(𝑢+1)𝑁𝜈𝕊𝑎={𝑓}(𝑢)𝑁1𝑘=0𝑢𝑘𝐴𝑘+𝕊𝑎+𝜈𝑁Δ𝑎𝜈𝑓(𝑢).(5.16) Since from [28], we have 𝕊0{𝑡𝑛}(𝑢)=𝑛!𝑢𝑛,𝑛0,(5.17) hence 𝑦(𝑡)=𝑁1𝑘=0𝐴𝑘(𝑡𝑎𝜈+𝑁)𝑘𝑘!+Δ𝑎𝜈𝑓(𝑡).(5.18)

Remark 5.3. The initial value problem (5.1) can also be solved by using Proposition 15 in [12].

Example 5.4. Consider the initial value problem 𝐶Δ𝜈+1𝑎+𝜈1𝑦(𝑡)𝐶Δ𝜈𝑎+𝜈1𝑦(𝑡)=0,𝑡𝑎,Δ𝑘𝑦(𝑎+𝑣𝑁)=𝐴𝑘,𝑘{0.1},𝐴𝑘,(5.19) where 0<𝜈1. Applying the Sumudu transform to both sides of the equation and using (4.31) and (4.27), respectively, we get (𝑢+1)1𝜈𝑢2𝕊𝑎+𝜈1{𝑦}(𝑢)𝐴0𝑢𝐴1(𝑢+1)1𝜈𝑢𝕊𝑎+𝜈1{𝑦}(𝑢)𝐴0=0.(5.20) Hence we get 𝕊𝑎+𝜈1𝐴{𝑦}(𝑢)=0𝐴1+𝐴1.1𝑢(5.21) Since from [28], we have 𝕊0(1+𝜆)𝑡1(𝑢)=|||1𝜆𝑢for(1+𝜆)𝑢|||𝑢+1<1,(5.22) then 𝑦𝐴(𝑡)=0𝐴1+𝐴12𝑡𝑎𝜈+1.(5.23)

References

  1. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
  2. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives—Theory and Applications, Gordon and Breach Science Publishers, Linghorne, Pa, USA, 1993.
  3. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  4. R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Redding, Conn, USA, 2006.
  5. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar
  6. N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, pp. 765–771, 2006.
  7. K. S. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the Univalent Functions, Fractional Calculus, and Their Applications, pp. 139–152, Nihon University, 1989.
  8. F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165–176, 2007.
  9. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. F. M. Atıcı and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–12, 2009. View at Zentralblatt MATH
  11. T. Abdeljawad and D. Baleanu, “Fractional differences and integration by parts,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 574–582, 2011. View at Zentralblatt MATH
  12. T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. G. K. Watugala, “Sumudu transform: a new integral transform to solve differential equations and control engineering problems,” International Journal of Mathematical Education in Science and Technology, vol. 24, no. 1, pp. 35–43, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. G. K. Watugala, “The Sumudu transform for functions of two variables,” Mathematical Engineering in Industry, vol. 8, no. 4, pp. 293–302, 2002. View at Zentralblatt MATH
  15. M. A. Asiru, “Sumudu transform and the solution of integral equations of convolution type,” International Journal of Mathematical Education in Science and Technology, vol. 32, no. 6, pp. 906–910, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. M. A. Aşiru, “Further properties of the Sumudu transform and its applications,” International Journal of Mathematical Education in Science and Technology, vol. 33, no. 3, pp. 441–449, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. F. B. M. Belgacem, A. A. Karaballi, and S. L. Kalla, “Analytical investigations of the Sumudu transform and applications to integral production equations,” Mathematical Problems in Engineering, no. 3-4, pp. 103–118, 2003. View at Publisher · View at Google Scholar
  18. F. B. M. Belgacem and A. A. Karballi, “Sumudu transform fundemantal properties investigations and applications,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2006, Article ID 91083, 23 pages, 2006. View at Publisher · View at Google Scholar
  19. A. Kılıçman and H. Eltayeb, “On the applications of Laplace and Sumudu transforms,” Journal of the Franklin Institute, vol. 347, no. 5, pp. 848–862, 2010. View at Publisher · View at Google Scholar
  20. F. B. M. Belgacem, “Introducing and analysing deeper Sumudu properties,” Nonlinear Studies, vol. 13, no. 1, pp. 23–41, 2006. View at Zentralblatt MATH
  21. F. Jarad and K. Tas, “Application of Sumudu and double Sumudu transforms to Caputo-Fractional dierential equations,” Journal of Computational Analysis and Applications, vol. 14, no. 3, pp. 475–483, 2012.
  22. Q. D. Katatbeh and F. B. M. Belgacem, “Applications of the Sumudu transform to fractional differential equations,” Nonlinear Studies, vol. 18, no. 1, pp. 99–112, 2011. View at Zentralblatt MATH
  23. M. Bohner and G. Sh. Guseinov, “The h-Laplace and q-Laplace transforms,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 75–92, 2010. View at Publisher · View at Google Scholar
  24. S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at Zentralblatt MATH
  25. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001. View at Publisher · View at Google Scholar
  26. M. Bohner and A. Peterson, Advances in Dynamic equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. View at Publisher · View at Google Scholar
  27. M. T. Holm, The theory of discrete fractional calculus: development and application [Ph.D. thesis], 2011.
  28. F. Jarad, K. Bayram, T. Abdeljawad, and D. Baleanu, “On the discrete sumudu transform,” Romanian Reports in Physics. In press.