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