give in this paper a new general results in the theory of $q$-calculus. But we restrict our selves to the necessary theory to give some $q $-analogues of an integral transform named as the Natural transform and to estimate the transform relevant properties. The classical theory of the Natural transform is closely related to the classical theory of Laplace and Sumudu transforms, two of the best known of all integral transforms. The Natural transform for functions of exponential order is defined over the set $A,$$$A=\left\{ f\left( t\right) \left\vert \exists M,\tau _{1}\text{ and/or }\tau
>0\text{: }\left\vert f\left( t\right) \right\vert <Me^{\frac{\left\vert
t\right\vert }{\tau _{j}}},\text{ }t\in \left( -1\right) ^{j}\times \left[
0,\infty \right) ,\text{ }j=1,2\right. \right\}  \tag{1}$$by the integral equation$$N\left( f\right) \left( u;v\right) =\dint\nolimits_{0}^{\infty }f\left(
ut\right) \exp \left( -vt\right) dt,  \tag{2}$$where $\func{Re}v>0$ and $u\in \left( -\tau _{1},\tau _{2}\right) ,$ $u$ and $v$ being the transform variables.

The Natural transform strictly converges to the Sumudu transform (See $\left[
28,29,30\right] $ ) for $v=1$ and it strictly converges to the Laplace transform for $u=1.$ Further fundamental properties of this transform and its application to differential equations are given by $\left[ \text{14}\right] $ and $\left[ 15,16,17\right] $, respectively$.$

We organize this paper as follows. In section 2, 