We introduce nabla type Laplace transform and Sumudu transform on general time scale. We investigate the properties and the applicability of these integral transforms and their efficiency in solving fractional dynamic equations on time scales.

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 theory of time scale calculus was initiated by Hilger [

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 [

The extension of basic notions of fractional calculus to other discrete settings was performed in [

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 (

This paper is organized as follows. In Section

A time scale

The backward jump operator

For

A function

The set of ld-continuous functions

It is known from [

Let

For

A function

Let

For

We recall the notion of Taylor monomials introduced in [

For the case

For the time scale

Let

The formula (

Recently, [

Let

Further, we have the following.

Let

(i) The fractional integral of order

(ii) The Riemann-Liouville fractional derivative of order

(iii) The Caputo fractional derivative

Note that below we assume that

The following theorem is concerning the asymptotic nature of the nabla exponential function. To this end, we define the minimal graininess function

Let

The proof is similar to Theorem 3.4 of [

Let

there exists

Let

It follows that

Let

Let

The proof is similar to Theorem 5.1 in [

Let

The proof follows from the linearity property of the

Let

By using integration by parts formula (

By induction, we have the following result.

Let

For a given

In this section, we will assume that the problem (

For given functions

We state the following results without proof, since the proofs of them are similar to those in [

The convolution is associative; that is,

If

The following formula holds:

Suppose

Let

It is known [

In general, we have

For

First, we write Definition

From (

For

For the Riemann-Liouville fractional derivative derivative (

For

Write (

From (

Thus, by (

The Laplace transform (

The nabla Laplace transform of Caputo fractional derivative of order

For

Write (

By following (

Now, let us consider the generalized Mittag-Leffler function on time scales (see [

Let

In the following theorem, we give the Laplace transform of generalized Mittag-Leffler function on time scales.

For

By using Theorem

Consider the following initial value problem:

The above example coincides with the case

Now, we consider the Cauchy problem for dynamic equations with the nabla type Caputo fractional derivatives.

Consider the following initial value problem:

The last example clearly coincides with the real counter part; see [

In [

Let

Let us define the set

Let

In the special case

The following theorem states the close relationship between nabla Sumudu transform and nabla Laplace transform.

Let

The following theorem can be easily verified using induction.

Let

The following theorem presents the

Let

The proof is a direct consequence of relation (

Now, we consider the

Let

Using Theorem

In particular,

The

In the following theorem, we give the Sumudu transform of generalized Mittag-Leffler function on time scales.

For

Using the relation (

The nabla Sumudu transform of fractional integral and fractional derivatives are as follows.

(i) For

(ii) For

(iii) For

The proof to each part follows immediately after applying (

As in the case of Laplace transform (see relation (

In the following example we will illustrate the use of the

Consider the following initial value problem:

We begin by taking the

Consider the following Caputo type initial value problem:

In particular, when

Consider the following Caputo type initial value problem:

Following Theorem

if the solution of fractional dynamic equation exists by

if the solution of fractional dynamic equation exists by

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is very grateful to the referees for their helpful suggestions.