Integral Transform Methods for Solving Fractional Dynamic Equations on Time Scales

and Applied Analysis 3 For f, g ∈ Cld(T ,C) and a, b ∈ T , the integration by parts formula is given by


Introduction
It is known that the methods connected to the employment of integral transforms are very useful in mathematical analysis.Those methods are successfully applied to solve differential and integral equations, to study special functions, and to compute integrals.One of the more widely used integral transforms is the Laplace transform defined by the following formula: The function  of a complex variable is called the Laplace transform of the function .Watugala [1] introduced a new integral transform called Sumudu transform defined by the following formula: and applied to the solution of ordinary differential equations in the control engineering problems (see also [2]).It appeared like the modification of the Laplace transform.The Sumudu transform rivals the Laplace transform in problem solving.Its main advantage is the fact that it may be used to solve problems without resorting to a new frequency domain, because it preserves scale and unit properties.The theory of time scale calculus was initiated by Hilger [3] (see also [4]).This theory is a tool that unifies the theories of continuous and discrete time system.It is a subject of recent studies in many different fields in which a dynamic process can be described with continuous and discrete models.For the detailed information on theory of time scale calculus, we refer to [5,6].The delta Laplace transform on arbitrary time scale (T) is introduced by Bohner and Peterson in [7] (see also [8]) by the following formula: L {} () := ∫ ∞  0  ()  ⊖ ( () ,  0 ) Δ,  ∈ D {} , (3) where D{} consists of all complex numbers  ∈ C for which the improper integral exists and for which 1 + () ̸ = 0 for all  ∈ T. In a similar fashion, Agwa et al. in [9] introduce the Sumudu transform on arbitrary time scale T, by the following formula: for  ∈ D{}, where D{} consists of all complex numbers  ∈ C for which the improper integral exists and for which 1 + ()/ ̸ = 0 for all  ∈ T. Note that if T = R (for real analysis), (3) ⇒ (1) and ( 4) ⇒ (2) at  0 = 0.In the case of T = Z (for discrete analysis), we have where {}() = ∑ ∞ =0 () − is the classical -transform, which will be used to solve higher order linear forward (in (, ℎ)-calculus) (see [11]).Likewise, the delta Sumudu transform on time scales not only can be applied on ordinary differential equations when T = R and on forward difference equations when T = ℎ (ℎ > 0) but also can be applied for -difference equations when T =  Z and on different types of time scales like T = ℎZ and T = T  ; for the space of the harmonic numbers, see [9].
Continuous fractional calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators.Fractional differentiation has played an important role in various areas ranging from mechanics to image processing.Their fundamental results have been surveyed, for example, in the monographs [12,13].On the other hand, discrete fractional calculus is a very new area for scientists.Foundation of this theory were formulated in pioneering works by Agarwal [14] and Díaz and Osler [15,16], where basic approach, definitions, and properties of the theory of fractional sums and differences were reported (see also [17,18]).Recently, a series of papers continuing this research has appeared (see e.g., [19][20][21][22][23][24][25][26] and the references cited therein).
The extension of basic notions of fractional calculus to other discrete settings was performed in [27,28].In these papers, the authors often preferred the power function notation based on the time scales theory, which easily exposes similarities among the results in -calculus, ℎ-calculus, (, ℎ)calculus, and the continuous case.However, this notation was employed only formally, since there was no general time scale definition of the power function and therefore the achieved results could not be generalized to other time scales.On this account, some ideas regarding fundamental properties which should be met by power functions on time scales were outlined in [29].In [30], the authors introduced fractional derivatives and integrals on time scales via the generalized Laplace transform.However, this approach suffers by some technical difficulties, connected to the inverse Laplace transform (see [8]).Recently, in [31,32] (see also [33]), the authors independently suggested an axiomatic definition of power functions on arbitrary time scale.
The aim of this paper is to introduce the nabla type Laplace transform and Sumudu transform, their properties, and applicability and its efficiency in solving fractional dynamic equations on arbitrary time scale.Of course, it is possible to consider also the delta type Laplace and Sumudu transforms (3) and (4), respectively; however, the nabla version seems be more suitable for fractional calculus as outlined, for example, in [27,28,34].
This paper is organized as follows.In Section 2, we recall basics of the time scale theory and the foundation of fractional calculus on time scales.Section 3 is devoted to nabla Laplace transform, its properties, convolution theorem, and examples of solution of fractional dynamic equations on time scales in terms of Mittag-Leffler function.Finally, in Section 4, we introduce nabla Sumudu transform and its properties on arbitrary time scales.A close relationship between nabla Sumudu transform and nabla Laplace transform and several important results were obtained.This section ended up with solving some fractional dynamic equation with nabla Sumudu transform method.The backward jump operator  : T → T is defined by
A function  : T → C is left-dense continuous or ldcontinuous provided it is continuous at left-dense points in T and its right-sided limits exist (finite) at right-dense points in T. If T = R, then  is ld-continuous if and only if  is continuous.
The set of ld-continuous functions  : T → C will be denoted by  ld (T, C) and the set of functions  : T → C that are ∇-differentiable and whose derivatives are ld-continuous is denoted by  1 ld (T, C).It is known from [5] that if  ∈  ld (T, C), then there exists a function  such that  ∇ () = ().In this case, we define the Cauchy integral by where the right-hand side integral is the Riemann integral from calculus and if T = Z, then For ,  ∈  ld (T, C) and ,  ∈ T, the integration by parts formula is given by Let  ∈  ld (T, C).The the nabla exponential function ê (⋅, ) is defined to be the unique solution of the following initial value problem: for some fixed  ∈ T. Let ℎ > 0; set and C 0 := Z 0 := C. For ℎ > 0, the Hilger real part and imaginary part of a complex number are given by respectively, where Arg denotes the principle argument function; that is, Arg : C → (−, ] R , and let R 0 () : R() and I 0 () := I().For any fixed complex number , the Hilger real part R ℎ () is a nondecreasing function of ℎ ∈ [0, ∞) (see [38]).For ℎ ≥ 0, we define the ]-cylinder transformation ξℎ : for  ∈ C ℎ .Then, the nabla exponential function can also be written in the following form: It is known that the nabla exponential function ê (⋅, ) is strictly positive on T, provided  ∈ R + ] (T, C) (see Theorem 3.18 [6]).For ,  ∈ R ] (T, C), the ]-circul plus and the ]circle minus are defined by respectively.For further details on nabla exponential function, we refer to [5].
For the time scale T =  Z for some  > 1, we have Lemma 2 (nabla Cauchy formula [37]).Let  ∈ Z + , ,  ∈ T, and let  : The formula ( 24) is a corner stone in the introduction of the nabla fractional integral  ∇ − () for  > 0. However, it requires a reasonable and natural extension of a discrete system of monomials ( ĥ ,  ∈ N 0 ) to a continuous system ( ĥ ,  ∈ R + ).However, the calculation of ĥ for  > 1 is a difficult task which seems to be answerable only in some particular cases (see Example 1).
Further, we have the following.(i) The fractional integral of order  > 0 with the lower limit  as and for  = 0 one puts (  ∇ 0 )() = ().(ii) The Riemann-Liouville fractional derivative of order  > 0 with lower limit  as where on [(), ] T is defined via the Riemann-Liouville fractional derivative by where  = [] + 1.

Nabla Laplace Transform
Note that below we assume that ∈  R ] ; then (⊖ ] )∈  R ] and therefore ê⊖ ]  (⋅,  0 ) is well defined on T. From now on we assume that T is unbounded above.
The following theorem is concerning the asymptotic nature of the nabla exponential function.To this end, we define the minimal graininess function ] * : T → [0,∞) by and for ℎ ≥ 0 and  ∈ R, we define Theorem 5 (decay of the nabla exponential function).Let Then, for any  ∈ C ] * () (), we have the following properties: Proof.The proof is similar to Theorem 3.4 of [38].
where D ] {} consists of all complex numbers  ∈ R ] (T, C) for which the improper integral exists.
Proof.The proof is similar to Theorem 5.1 in [38].
By induction, we have the following result.
Definition 13 (see [41]).For a given  : [ 0 , ∞) T → C, the solution of the shifting problem is denoted by f and is called the shift (or delay) of .
In this section, we will assume that the problem (38) has a unique solution f for a given initial function  and that the functions , , and the complex number  are such that the operations fulfilled are valid.Definition 14 (see [41]).For given functions ,  : T → C, their convolution  *  is defined by where f is the shift of  introduced in Definition 8.
We state the following results without proof, since the proofs of them are similar to those in [6].
Let  = 0. Taking Laplace transform to the left-hand side followed by applying convolution theorem (39) yields But, from the right side of (50), we have Hence the result follow from (50) and (51).This completes the proof.
From (45), knowing we have (by taking  = ) the following result.
(56) Thus, by (37) and (56), we have The Laplace transform (54) is equivalent to the following one: The nabla Laplace transform of Caputo fractional derivative of order  is given as follows.
Proof.By using Theorem 10 and the relation (47), we obtain Example 25.Consider the following initial value problem: By taking Laplace transform of both sides of (65) and using (58), we get (68) The above example coincides with the case T = R (see [43]).Now, we consider the Cauchy problem for dynamic equations with the nabla type Caputo fractional derivatives.
Example 26.Consider the following initial value problem: By taking Laplace transform of both sides of (69) and using (59), we get (72) The last example clearly coincides with the real counter part; see [43].

Nabla Sumudu Transform
In [9], the authors introduced and studied the (delta) Sumudu transform on time scales.Many important results were produced and applied on dynamic equation on time scales.In this section, we will consider the nabla Sumudu transform.Most of the results were coated from [9,44,45] without proof since their proofs are similar.
In the special case T = N  = {,  + 1,  + 2, . ..},  ∈ R fixed (see [44]), we have The following theorem can be easily verified using induction.The following theorem presents the nabla-Sumudu transformation of convolution of two functions 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, Z, and  Z := {  :  ∈ Z} ∪ {0}, where  > 1.Let T have a right-scattered minimum  and define T  := T − {}; otherwise, set T  = T.If ,  ∈ T with  < , we denote by [, ] T the closed interval [, ] ∩ T.
on T  and positively ]-regressive if it is real valued and 1 − ] > 0 on T  .The set of ]-regressive functions and the set of positively ]-regressive functions are denoted by R ] (T, C) and R + ] (T, C), respectively, and R − ] (T, C) is defined similarly.For simplicity, we denote by  R ] (T, C) the set of complex ]-regressive constants and, similarly, we define the sets   + ] (T, C) and   − ] (T, C).