a precise definition of so-called $q$-Jackson integral and developed $q$-calculus in a systematic way. It was well known that, in the literature, there are two types of $q$-analogues of integral transforms studied in detail by many authors in the recent past such as Abdi $\left[ 2\right] ,$ Hahn $\left[ 18\right] ,$ Purohit and Kalla $\left[ 3\right] ,$ Albayrak $\left[ 25\right] $, Uçar and Albayrak $\left[ 4\right] ,$ Albayrak et al. $\left[ 5\right] $ and $\left[ 6\right] ,$ Yadav and Purohit $\left[ 7\right] ,$ Fitouhi and Bettaibi $\left[ 8\right] $ and $\left[ 9\right] $ and many others, to mention but a few.

In this section of this paper we deem it proper to give the definition of the $q$-analogues of the Natural transform as in the following definition.

<span style="font-variant:small-caps;">Definition 1.</span>* Let* $\hat{A}$ *and* $\check{A}$ *be defined by*

$\begin{array}{c}
\hat{A}=\left\{ f\left( t\right) \left\vert \exists M,\tau _{1}\text{ 
\textit{and/or} }\tau >0\text{: }\left\vert f\left( t\right) \right\vert
<ME_{q}\left( \frac{\left\vert t\right\vert }{\tau _{j}}\right) ,\text{ }t\in \left( -1\right) ^{j}\times \left[ 0,\infty \right) ,\text{ }j=1,2\right. \right\} \\ 
\text{\textit{and}} \\ 
\check{A}=\left\{ f\left( t\right) \left\vert \exists M,\tau _{1}\text{ 
\textit{and/or} }\tau >0\text{: }\left\vert f\left( t\right) \right\vert
<Me_{q}\left( \tfrac{\left\vert t\right\vert }{\tau _{j}}\right) ,\text{ }t\in \left( -1\right) ^{j}\times \left[ 0,\infty \right) ,\text{ }j=1,2\right. 