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The Fourier transformations are used mainly with respect to the space variables. In certain circumstances, however, for reasons of expedience or necessity, it is desirable to eliminate time as a variable in the problem. This is achieved by means of the Laplace transformation. We specify the particular concepts of the q-Laplace transform. The convolution for these transforms is considered in some detail.

The Laplace transform provides an effective method for solving linear differential equations with constant coefficients and certain integral equations. Laplace transforms on time scales, which are intended to unify and to generalize the continuous and discrete cases, were initiated by Hilger [

Definition 2.1. A time scale T is an arbtrary nonempty closed subset of the real numbers. Thus the real numbers R, the integers Z, the natural numbers N, the nonnegative integers

Definition 2.2. Assume

We call

Definition 2.3. If

for those values of

Let us set

which is a polynomial in Z of degree

and

hold, where

where

and

so that

Lemma 2.4. For any

Therefore, for an arbitrary number

In particular,

Example 2.5. We find the q-Laplace transform of

Example 2.6. We find the q-Laplace transform of the functions

We have (see [

On the other hand, we know that

with respect to

The q-Laplace transform of the functions

and

respectively.

Theorem 2.7. If the function

where c and R are some positive constants, then the series in (1) converges uniformly with respect to z in the region

Proof. By Lemma 2.4, for the number R given in (8) we can choose an

Then for the general term of the series in (1), we have the estimate

Hence the proof is completed.

A larger class of functions for which the q-Laplace transform exists is the class

Theorem 2.8. For any

Proof. By using the reverse (5), hence

and comparison test to get the desired result.

Theorem 2.9. (Initial Value and Final Value Theorem). We have the following:

a) If

b) If

Proof. Assume

and

Hence

Multiplying

Definition 3.1. Let T be a time scale. We define the forward jump operator

Definition 3.2. For a given function

Definition 3.3. For given functions

where

Definition 3.4. For given functions

with

Theorem 3.5. (Convolution Theorem). Assume that

1) We can see from Theorem 2.9(a) that no function has its q-Laplace transform equal to the constant function 1.

2) Finally, we note that most of the results concerning the Laplace transform on

Maryam SimkhahAsil,ShahnazTaheri, (2016) q-Laplace Transform. Advances in Pure Mathematics,06,16-20. doi: 10.4236/apm.2016.61003