Fractional q-Calculus Operators Pertaining to the q-Analogue of $\mathbb{M}$-Function and Its Application to Fractional q-Kinetic Equations

Abstract

In this article, the authors introduce the q-analogue of the $\mathbb{M}$-function, and establish four theorems related to the Riemann–Liouville fractional q-calculus operators pertaining to the newly defined q-analogue of $\mathbb{M}$-functions. In addition, to establish the solution of fractional q-kinetic equations involving the q-analogue of the $\mathbb{M}$-function, we apply the technique of the q-Laplace transform and the q-Sumudu transform and its inverse to obtain the solution in closed form. Due to the general nature of the q-calculus operators and defined functions, a variety of results involving special functions can only be obtained by setting the parameters appropriately.

1. Introduction

Fractional differential equations have a remarkably significant role in applied sciences, especially in dynamic systems, mathematical physics, control systems, and engineering, and they are used to build mathematical models of various physical phenomena. Solving fractional differential equations can be challenging because fractional derivatives are nonlocal operators, and analytical solutions are often not readily available. Numerical methods and special functions, like the Mittag–Leffler function, are commonly used to approximate solutions to fractional differential equations. Fractional calculus, which deals with fractional integrals and derivatives, provides a mathematical framework for studying and solving fractional differential equations. Sharma and Jain [1] recently established the generalized M-series as a function and stated it, using power series for $z,\varsigma ,\vartheta \in \mathbb{C},\Re \left(\varsigma \right)>0$, as:

$${}_r{}^{\varsigma }M_s^{\vartheta }\left(z\right)={}_r{}^{\varsigma }M_s^{\vartheta }\left(c_{1,\dots ,}{c}_r;d_{1,\dots ,}{d}_s;z\right)=\sum _{n=0}^{\infty }\frac{{\left(c_1\right)}_n\cdots {\left(c_r\right)}_n}{{\left(d_1\right)}_n\cdots {\left(d_s\right)}_n}\frac{z^n}{\mathrm{\Gamma }\left(\varsigma n+\vartheta \right)},$$

(1)

where ${\left(c_i\right)}_n$, ${\left(d_j\right)}_n$; $i=1,2,\dots ,r;j=1,2,\dots ,s$ are known as Pochhammer symbols. The $\mathbb{M}$-functions have some special cases, for example, if we set $\vartheta =1,$ the $\mathbb{M}$-function reduces in the M-series defined by Sharma [2], for $\varsigma =\vartheta =1,$ and the generalized $\mathbb{M}$-function is the well-known generalized hypergeometric function. Also, when $\varsigma =\vartheta =1,$ and $r=s=0$, then we obtain the Mittag–Leffler function [3] with one parameter, for $r=s=0$, and after that we have a Mittag–Leffler function of two indexes established by Wiman [4]. Several characteristics and applications of the M-series (1) and its particular cases have been studied by Miller [5], Saxena et al. [6], Purohit et al. [7], Mishra et al. [8], and Saigo and Kilbas [9]. The generalized $\mathbb{M}$-function is crucial in the solution of fractional order integral and differential equations.

Researchers continue to develop new techniques and tools to better understand and solve fractional differential equations due to their importance in complex physical and biological systems. More specifically, kinetic equations are the fundamental equations of natural sciences and mathematical physics because they define the continuity of the motion of the matter. They can be applied to qualitative and quantitative descriptions of biological, chemical, physical, social, and other processes. They are called “master equations” since the kinetic equation-based mathematical simulation of evolutionary processes is productive and powerful. The system of Vlasov’s equation, Liouville equation, Fokker–Planck radiation transfer equations, Vegner equations, the Boltzmann equation, and the chain of Bogoljubov’s equations for the matrix of density and diffraction are significant kinetic equations. In 2000, Haubold and Mathai [10] developed a fractional differential equation involving the reaction’s rate of change, the destruction rates, and the production rate. We discovered several papers in the literature that investigate extensions and generalizations of these equations utilizing different fractional calculus operators, for example, the work of Saxena et al. [11,12,13], Kumar et al. [14,15], Suthar et al. [16,17], Habenom et al. [18], Baleanu et al. [19], and Gupta and Parihar [20].

As a result of the applications of q-calculus in physics, statistics, and mathematics, there has recently been a significant increase in activity in the q-calculus area of research. Particularly in partition theory, exactly solvable models in statistical mechanics, combinatorics, optimal control problems, computer algebra, geometric function theory, q-transform analysis, q-integral equations, q-analysis, and q-difference have found many useful applications. Discussions of q-analysis can be found in the works of Annaby and Mansour [21], Kac and Cheung [22], and Gasper and Rahman [23]. For more information, we refer to Abdi [24] and Al-Omari et al. [25] for applications of the q-Laplace transform to solve the q-difference equations. Albayrak et al. [26] established the q-analogue of Sumudu transforms and derived some fundamental characteristics of q-Sumudu transforms such as differentiation, shifting theorems, linearity, integration, and some fascinating connection theorems that involve the q-Laplace transform.

Garg and Chanchlani [27] recently computed the following fractional q-kinetic equation in closed form using q-Laplace transforms and its inverse:

$$N_q\left(t\right)-N_0{f}_q\left(t\right)=-\omega I_q^{\tau }{N}_q\left(t\right)\left(w>0,\tau >0,0<\left|q\right|<1\right).$$

(2)

Moreover, Purohit and Ucar [28] have proposed a new solution to the q-kinetic equation with the Riemann–Liouville fractional (RLF) q-integral operator. Using the inverse q-Sumudu transform, the solution is found using the q-Mittag–Leffler function.

Motivated by the above series representation which is defined in Equation (1) and its importance and useful applications in the field of applied mathematics, we established the q-analogue of the generalized $\mathbb{M}$-functions and determined four theorems on the RLF q-calculus operators pertaining to the above mentioned function. Furthermore, we calculate the solution of the fractional q-kinetic Equation (2) involving the RLF q-integral operator by using the q-Laplace and Sumudu transform.

2. Preliminaries

For a complex or real m, the q-shifted factorial is given by in [23] as:

$${\left(m;q\right)}_0=1,{\left(m;q\right)}_n=\prod _{k=0}^{n-1}\left(1-m{q}^k\right),{\left(m;q\right)}_{\infty }=\prod _{k=0}^{\infty }\left(1-m{q}^k\right),$$

(3)

The q-beta and q-gamma functions [23] are defined as

$${\mathrm{\Gamma }}_q\left(m\right)=\frac{{\left(q;q\right)}_{\infty }}{{\left(q^m;q\right)}_{\infty }}{\left(1-q\right)}^{1-m}=\frac{{\left(m;q\right)}_{m-1}}{{\left(1-q\right)}^{m-1}},(m\in \mathbb{C},\left|q\right|<1),$$

(4)

and

$$B_q(l,m)={\int }_0^1{x}^{l-1}\frac{{\left(xq;q\right)}_{\infty }}{\left(x{q}^m;q\right)\infty }{d}_{q}x=\frac{{\mathrm{\Gamma }}_q\left(l\right){\mathrm{\Gamma }}_q\left(m\right)}{{\mathrm{\Gamma }}_q\left(l+m\right)},\left(\Re \left(l\right)>0,\Re \left(m\right)>0\right).$$

(5)

An analogue of q of the power function ${(x-\tau )}^m$ is given by:

$${\left(x-\tau \right)}_q^m=x^m(\raisebox{1ex}{$\tau $}\!\left/ \!\raisebox{-1ex}{$x$}\right.;q{)}_m=x^m\frac{(\raisebox{1ex}{$\tau $}\!\left/ \!\raisebox{-1ex}{$x$}\right.;q{)}_{\infty }}{(q^m\raisebox{1ex}{$\tau $}\!\left/ \!\raisebox{-1ex}{$x$}\right.;q{)}_{\infty }},(x\ne 0, 0<|q|<1).$$

(6)

Moreover, Rajkovic et al. [29] stated the q-derivative for function $f\left(x\right)$ as

$$\left(D_{q}f\right)\left(x\right)=\frac{f\left(x\right)-f\left(qx\right)}{(1-q)x},(x\ne 0,q\ne 1).$$

(7)

and

$$\underset{q\to 1}{\lim }{D}_{q}f\left(x\right)=\frac{df\left(x\right)}{dx}$$

(8)

$$\left(D_{\frac{1}{q}}f\right)\left(x\right)=q^n\left(D_q^{n}f\right)\left(\frac{x}{q^n}\right),$$

(9)

$$D_q\left(f\left(x\right)g\left(x\right)\right)=g\left(x\right)D_{q}f\left(x\right)+f\left(qx\right)D_{q}g\left(x\right),$$

(10)

$$D_q\left(f\left(x\right)/g\left(x\right)\right)=\left(g\left(x\right)D_{q}f\left(x\right)-f\left(x\right)D_{q}g\left(x\right)\right)/g\left(x\right)g\left(qx\right).$$

(11)

A q-extension of an exponential function is stated as

$$E_q^x={}_0\varphi _0(-;-;q,-x)=\sum _{\upsilon =0}^{\infty }\frac{q^{\upsilon (\upsilon -1)/2}{x}^{\upsilon }}{{(q;q)}_{\upsilon }}={(-x;q)}_{\infty },$$

(12)

$$e_q^x={}_1\varphi _0(0;-;q,x)=\sum _{\upsilon =0}^{\infty }\frac{x^{\upsilon }}{{(q;q)}_{\upsilon }}=\frac{1}{{(-x;q)}_{\infty }},\left|x\right|<1.$$

(13)

3. q-Analogue of $\mathbb{M}$-Function

Motivated by Equation (1) and their related literature, we introduce the q-analogue of the $\mathbb{M}$-function ${}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s(q,z)$ for $z,\varsigma ,\vartheta \in \mathbb{C}$; $\Re \left(\varsigma \right)>0$ and $\left|q\right|<1$ as:

$${}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s(q,z)={}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s\left[\begin{array}{c}{q}^{c_1},\dots ,q^{c_r}\\ q^{d_1},\dots ,q^{d_s}\end{array};q,z\right]=\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{z^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)},$$

(14)

where ${\mathrm{\Gamma }}_q(.)$ is the q-gamma function and ${(q^{c_i},q)}_{\upsilon },{(q^{d_j},q)}_{\upsilon }$; $c_i{,\mathrm{d}}_j\ne 0,-1,-2\cdots (i=1,2,\cdots ,r;j=1,2,\cdots s)$ are a q-analogue of the Pochhammer symbol. If $0<\left|q\right|<1$, series (14) converges for all z if $r\le s$, and for $\left|z\right|<1$ if $r=s+1$. When $\left|q\right|>1$, the series converges for $\left|z\right|<\left|d_1\dots d_{s}q\right|/\left|c_1\dots c_r\right|.$

We describe the relationships as specific instances of ${}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s(q,z)$ with other special functions as given below:

For $r=s=0$, the $\mathbb{M}$-function becomes the q-Mittag–Leffler function defined by Mansoor [30] as:

$${}_0\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_0\left[\begin{array}{c}-;\\ -;\end{array}q,z\right]=\sum _{\upsilon =0}^{\infty }\frac{z^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}=e_{\varsigma ,\vartheta }(z,q).$$

(15)

When $\vartheta =1,$$r=s=0$, the $\mathbb{M}$-function becomes the q-Mittag–Leffler function defined by Jain [31] as:

$${}_0\stackrel{\varsigma ,1}{\mathbb{M}}_0\left[\begin{array}{c}-;\\ -;\end{array}q,z\right]=\sum _{\upsilon =0}^{\infty }\frac{z^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +1\right)}=E_{\varsigma }(z,q),(\Re \left(\varsigma \right)>0).$$

(16)

On setting $r=s=1,c_1=\delta ,d_1=1$ in (14), we obtain the generalized small q-Mittag–Leffler introduced by Purohit and Kalla [32] as follows:

$${}_1\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_1\left[\begin{array}{c}{q}^{\delta };\\ q;\end{array}q,z\right]=\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{\delta };q\right)}_{\upsilon }}{{\left(q;q\right)}_{\upsilon }}\frac{z^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}=e_{\varsigma ,\vartheta }^{\delta }(q,z).$$

(17)

On putting $r=s=1,c_1=\gamma ,d_1=\delta$ in (14), we obtain the generalized q-Mittag–Leffler introduced by Sharma and Jain [33] as follows:

$${}_1\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_1\left[\begin{array}{c}{q}^{\gamma };\\ q^{\delta };\end{array}q,z\right]=\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{\gamma };q\right)}_{\upsilon }}{{\left(q^{\delta };q\right)}_{\upsilon }}\frac{z^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}=E_{\varsigma ,\vartheta }^{\gamma ,\delta }(z,q).$$

(18)

Putting $r=1,s=0,c_1=\delta$ in (14), we obtain

$$\stackrel{1,1}{{}_1\mathbb{M}_0}\left[\begin{array}{c}{q}^{\delta };\\ -;\end{array}q,z\right]=\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{\delta };q\right)}_{\upsilon }{z}^{\upsilon }}{{\left(q;q\right)}_{\upsilon }}=\frac{{\left(q^{\delta }z;q\right)}_{\infty }}{{\left(q;q\right)}_{\infty }}={}_1\varphi _0(q^{\delta };-;q,z),$$

(19)

where the function ${}_1\varphi _0(q^{\delta };-;q,z)={(1-z)}^{-\delta }$ can be called the q-binomial function.

Finally, in view of a relations

$$\underset{q\to 1^{-}}{\lim }\frac{{\left(q^{\delta };q\right)}_{\upsilon }}{{(1-q)}^{\upsilon }}={\left(\delta \right)}_{\upsilon },$$

(20)

and

$$\underset{q\to 1^{-}}{\lim }{\mathrm{\Gamma }}_q\left(z\right)=\mathrm{\Gamma }\left(z\right).$$

(21)

Furthermore, note that

$$\underset{q\to 1^{-}}{\lim }{}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s(q,z)={}_r{}^{\varsigma }M_s^{\vartheta }\left(z\right).$$

(22)

4. Fractional q-Integration and q-Differentiation Approach

To begin, we would like to define these q-fractional operators in this section.

Agarwal [34] introduced a left-sided Riemann–Liouville (R-L) q-fractional operator for $a=0$ as follows:

$$\left(I_{q,a+}^{\epsilon }f\right)\left(x\right)=\frac{x^{\epsilon -1}}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}{\int }_a^x{\left(qz/x;q\right)}_{\epsilon -1}f\left(z\right)d_{q}z.$$

(23)

Furthermore, Mansour [35] stated a right-sided R-L q-fractional operator by:

$$\left(I_{q,b-}^{\epsilon }f\right)\left(x\right)=\frac{1}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}{\int }_{qx}^b{z}^{\epsilon -1}{\left(qx/z;q\right)}_{\epsilon -1}f\left(z\right)d_{q}z.$$

(24)

From [35], for $\epsilon >0$ and $⌈\epsilon ⌉=m$, the right- and left-sided R-L fractional q-derivatives of order $\epsilon$ are given as

$$\left(D_{q,a+}^{\epsilon }f\right)\left(x\right)=\left(D_q^m{I}_{q,a+}^{m-\epsilon }f\right)\left(x\right),$$

(25)

and

$$\left(D_{q,b-}^{\epsilon }f\right)\left(x\right)={\left(\frac{-1}{q}\right)}^m\left(D_{q^{-1}}^m{I}_{q,b-}^{m-\epsilon }f\right)\left(x\right).$$

(26)

The left- and right-sided Caputo fractional q-derivatives of order $\varsigma$ are given as

$$\left({}^{c}D_{q,a+}^{\epsilon }f\right)\left(x\right)=\left(I_{q,a+}^{m-\epsilon }{D}_q^{m}f\right)\left(x\right),\left({}^{c}D_{q,b-}^{\epsilon }f\right)\left(x\right)={\left(\frac{-1}{q}\right)}^m\left(I_{q,b-}^{m-\epsilon }{D}_{q^{-1}}^{m}f\right)\left(x\right).$$

(27)

This part would determine the following interesting results in the form of theorems. Here, we present q-analogues of $\mathbb{M}$-functions in view of the fractional q-integration and q-differentiation representations.

Theorem 1. 

Let $z,\varsigma ,\vartheta \in \mathbb{C}$; $\epsilon >0,\omega \in \Re$, $\Re \left(\varsigma \right)>0$, $\left|q\right|<1$ and let it be the left-sided operator of the R-L q-fractional integration (23). Then, the following formula is true:

$$\left(I_{q,0+}^{\epsilon }\left\{t^{\vartheta -1}\stackrel{\varsigma ,\vartheta }{{}_r\mathbb{M}_s}(q,\omega t^{\varsigma })\right\}\right)\left(z\right)=z^{\epsilon +\vartheta -1}\stackrel{\varsigma ,\vartheta +\epsilon }{{}_r\mathbb{M}_s}\left(\omega z^{\varsigma };q\right).$$

(28)

Proof. 

Let ${\mathrm{\Omega }}_1$ be the left-hand side of (28), with Equation (14) and (24) on the left-hand side of (28); we have

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_1& =\frac{z^{\epsilon -1}}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}{\int }_0^z{t}^{\vartheta -1}{\left(qt/z;q\right)}_{\epsilon -1}{}_s\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_r(q,\omega t^{\varsigma })d_{q}t\hfill \\ \hfill & =\frac{z^{\epsilon -1}}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}{\int }_0^z{t}^{\vartheta -1}{\left(qt/z;q\right)}_{\epsilon -1}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }{t}^{\varsigma \upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{d}_{q}t\hfill \\ \hfill & =\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\frac{z^{\epsilon -1}}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}{\int }_0^z{t}^{\vartheta +\varsigma \upsilon -1}{\left(qt/z;q\right)}_{\epsilon -1}{d}_{q}t.\hfill \end{array}$$

(29)

Use Equation (6) in the above expression; it reduces to

$${\mathrm{\Omega }}_1=\frac{1}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }{z}^{\epsilon -1}}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{\int }_0^z\frac{{\left(qt/z;q\right)}_{\infty }}{{\left(q^{\epsilon }t/z;q\right)}_{\infty }}{t}^{\vartheta +\varsigma \upsilon -1}{d}_{q}t.$$

(30)

Let the substitution $t=\varphi z$, then $\zeta =t/z$ in (30); we obtain

$${\mathrm{\Omega }}_1=\frac{z^{\epsilon +\vartheta -1}}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{\int }_0^1\frac{{\left(q\varphi ;q\right)}_{\infty }}{{\left(q^{\epsilon }\varphi ;q\right)}_{\infty }}{\varphi }^{\vartheta +\varsigma \upsilon -1}{d}_q\varphi .$$

Using Equation (5), we obtain

$${\mathrm{\Omega }}_1=\frac{z^{\epsilon +\vartheta -1}}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\left(\omega z^{\varsigma }\right)}^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\frac{{\mathrm{\Gamma }}_q\left(\epsilon \right){\mathrm{\Gamma }}_q\left(\vartheta +\varsigma \upsilon \right)}{{\mathrm{\Gamma }}_q\left(\vartheta +\epsilon +\varsigma \upsilon \right)}.$$

(31)

We obtain (28) by analyzing the right-hand side of (31) in light of definition (14). □

Theorem 2. 

Let $z,\varsigma ,\vartheta \in \mathbb{C}$; $\epsilon >0,\omega \in \Re$,$\Re \left(\varsigma \right)>0$, $\left|q\right|<1$ and $I_{q,-}^{\epsilon }$ be the right-sided operator of R-L q-fractional integration (24). Then there holds the subsequent formula:

$$\left(I_{q,-}^{\epsilon }\left\{t^{-\epsilon -\vartheta }\stackrel{\varsigma ,\vartheta }{{}_r\mathbb{M}_s}(q,\omega t^{-\varsigma })\right\}\right)\left(z\right)=\left(z^{-\vartheta }/q\right)\stackrel{\varsigma ,\vartheta +\epsilon }{{}_r\mathbb{M}_s}(q,\omega z^{-\varsigma }).$$

(32)

Proof. 

Let ${\mathrm{\Omega }}_2$ be the left-hand side of (32), with Equations (14) and (24) on the left-hand side of (32); we obtain

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_2& =\frac{1}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}{\int }_{qz}^{\infty }{t}^{\epsilon -1}{\left(qz/t;q\right)}_{\epsilon -1}{t}^{-\epsilon -\vartheta }{}_s\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_r(q,\omega t^{-\varsigma })d_{q}t\hfill \\ \hfill & =\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\frac{1}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}{\int }_{qz}^{\infty }{\left(qz/t;q\right)}_{\epsilon -1}{t}^{-\vartheta -\varsigma \upsilon -1}{d}_{q}t.\hfill \end{array}$$

Applying Equation (6) in the above expression reduces to

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_2& =\frac{1}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\hfill \\ \hfill & \times {\int }_{qz}^{\infty }\frac{{\left(qz/t;q\right)}_{\infty }}{{\left(q^{\epsilon -1}qz/t;q\right)}_{\infty }}{t}^{-\vartheta -\varsigma \upsilon -1}{d}_{q}t.\hfill \end{array}$$

(33)

Let the substitution $t=qz/\theta$, then $\theta =qz/t$ in (33); we obtain

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_2& =\frac{z^{-\vartheta }}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\left(\omega z^{-\varsigma }\right)}^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{q}^{-\vartheta -\varsigma \upsilon -1}\hfill \\ \hfill & \times {\int }_0^1\frac{{\left(\theta ;q\right)}_{\infty }}{{\left(q^{\epsilon -1}\theta ;q\right)}_{\infty }}{\theta }^{\vartheta +\varsigma \upsilon -1}{d}_q\theta .\hfill \end{array}$$

(34)

Replacing $\theta =\lambda q$ in (34), we obtain

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_2& =\frac{z^{-\vartheta }}{{\mathrm{\Gamma }}_q\left(\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\left(\omega z^{-\varsigma }\right)}^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{q}^{-1}\hfill \\ \hfill & \times {\int }_0^1\frac{{\left(\lambda q;q\right)}_{\infty }}{{\left(\lambda q^{\epsilon };q\right)}_{\infty }}{\left(\lambda \right)}^{\vartheta +\varsigma \upsilon -1}{d}_q\left(\lambda \right).\hfill \end{array}$$

(35)

Using Equation (5) in (35), we obtain

$${\mathrm{\Omega }}_2=\frac{z^{-\vartheta }}{q}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\left(\omega z^{-\varsigma }\right)}^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta +\epsilon \right)}.$$

(36)

We obtain (32) by analyzing the right-hand side of (36) in light of definition (14). □

Theorem 3. 

Let $z,\varsigma ,\vartheta \in \mathbb{C}$; $\epsilon >0,\omega \in \Re$, $\Re \left(\varsigma \right)>0$ and $\left|q\right|<1$; $D_{q,0+}^{\epsilon }$ be the left-sided operator of the R-L q-fractional differentiation (25). Then the following formula is true:

$$\left(D_{q,0+}^{\epsilon }\left\{t^{\vartheta -1}{}_s\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_r(q,\omega t^{\varsigma })\right\}\right)\left(z\right)=z^{\epsilon -\vartheta -1}\stackrel{\varsigma ,\vartheta -\epsilon }{{}_r\mathbb{M}_s}(q,\omega z^{\varsigma }).$$

(37)

Proof. 

From the left-hand side of (37), let ${\mathrm{\Omega }}_3$; using (14) and (25), we have

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_3& =D_q^m\left(I_{q,0+}^{m-\epsilon }\left\{t^{\delta -1}{}_s\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_r(q,\omega t^{\varsigma })\right\}\right)\left(z\right)\hfill \\ \hfill & =\frac{1}{{\mathrm{\Gamma }}_q\left(m-\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\hfill \\ \hfill & \times D_q^m{z}^{m-\epsilon -1}{\int }_0^z{t}^{\vartheta +\varsigma \upsilon -1}{\left(qt/z;q\right)}_{m-\epsilon -1}{d}_{q}t.\hfill \end{array}$$

Applying Equation (6) in the above expression reduces to

$${\mathrm{\Omega }}_3=\frac{1}{{\mathrm{\Gamma }}_q\left(m-\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}$$

$$\times D_q^m{z}^{m-\epsilon -1}{\int }_0^z{t}^{\vartheta +\varsigma \upsilon -1}\frac{{\left(qt/z;q\right)}_{\infty }}{{\left(q^{m-\epsilon }t/z;q\right)}_{\infty }}{d}_{q}t.$$

(38)

Let the substitution $t=\varphi z$, then $\zeta =t/z$ in (38), and we obtain

$${\mathrm{\Omega }}_3=\frac{1}{{\mathrm{\Gamma }}_q\left(m-\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}$$

$$\times D_q^m{z}^{m-\epsilon +\vartheta +\varsigma \upsilon -1}{\int }_0^1\frac{{\left(\varphi q;q\right)}_{\infty }}{{\left(\varphi q^{m-\epsilon };q\right)}_{\infty }}{\varphi }^{\vartheta +\varsigma \upsilon -1}{d}_q\varphi .$$

Applying Equation (5) in the above expression, we obtain the following

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_3& =\frac{1}{{\mathrm{\Gamma }}_q\left(m-\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\hfill \\ \hfill & \times D_q^m{z}^{m-\epsilon +\vartheta +\varsigma \upsilon -1}{B}_q(\vartheta +\varsigma \upsilon ,m-\epsilon ).\hfill \end{array}$$

(39)

Implementing Equation (5) and making a simple calculation, we obtain our result (37). □

Theorem 4. 

Let $z,\varsigma ,\vartheta \in \mathbb{C}$; $\epsilon >0,\vartheta -\epsilon +\left\{\epsilon \right\}>1,\omega \in \Re$, $\Re \left(\varsigma \right)>0$, $\left|q\right|<1$ and $D_{q,-}^{\epsilon }$ be the right-sided operator of the R-L q-fractional differentiation (26). Then there holds the subsequent formula:

$$\left(D_{q,-}^{\epsilon }\left\{t^{\epsilon -\vartheta }\stackrel{\varsigma ,\vartheta }{{}_r\mathbb{M}_s}(q,\omega t^{-\varsigma })\right\}\right)\left(z\right)=\frac{z^{-\vartheta }}{q^{m+1}}\stackrel{\varsigma ,\vartheta -\epsilon }{{}_r\mathbb{M}_s}(q,\omega z^{-\varsigma }).$$

(40)

Proof. 

Let ${\mathrm{\Omega }}_4$ be the left-hand side of (40), with Equations (14) and (24); then we have

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_4& ={\left(\frac{-1}{q}\right)}^m{D}_{q^{-1}}^m\left(I_{q,-}^{m-\epsilon }\left\{t^{\epsilon -\vartheta }{}_s\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_r(q,\omega t^{-\varsigma })\right\}\right)\left(x\right)\hfill \\ \hfill & =\frac{1}{{\mathrm{\Gamma }}_q\left(m-\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{\left(\frac{-1}{q}\right)}^m\hfill \\ \hfill & \times D_{q^{-1}}^m{\int }_{qz}^{\infty }{\left(qz/t;q\right)}_{m-\epsilon -1}{t}^{m-\vartheta -\varsigma \upsilon -1}{d}_{q}t.\hfill \end{array}$$

(41)

Applying Equation (6) in the above expression reduces to

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_4& =\frac{1}{{\mathrm{\Gamma }}_q\left(m-\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\hfill \\ \hfill & \times {\left(\frac{-1}{q}\right)}^m{D}_{q^{-1}}^m{\int }_{qz}^{\infty }\frac{{\left(qz/t;q\right)}_{\infty }}{{\left(q^{m-\epsilon }z/t;q\right)}_{\infty }}{t}^{m-\vartheta -\varsigma \upsilon -1}{d}_{q}t.\hfill \end{array}$$

(42)

Substituting $t=qz/\zeta$, then $\zeta =qz/t$ in (42), we obtain

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_4& =\frac{1}{{\mathrm{\Gamma }}_q\left(m-\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{\left(\frac{-1}{q}\right)}^m\hfill \\ \hfill & \times D_{q^{-1}}^m{z}^{m-\vartheta -\varsigma \upsilon }{q}^{m-\vartheta -\varsigma \upsilon -1}{\int }_0^1\frac{{\left(\zeta ;q\right)}_{\infty }}{{\left(q^{m-\epsilon -1}\zeta ;q\right)}_{\infty }}{\zeta }^{-m+\vartheta +\varsigma \upsilon -1}{d}_q\zeta .\hfill \end{array}$$

(43)

Replacing $\zeta =\lambda q$ in (43), we obtain

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_4& =\frac{1}{{\mathrm{\Gamma }}_q\left(m-\epsilon \right)}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{\left(\frac{-1}{q}\right)}^m\hfill \\ \hfill & \times D_{q^{-1}}^m{z}^{m-\vartheta -\varsigma \upsilon }{q}^{-1}{\int }_0^1{\lambda }^{\vartheta -m+\varsigma \upsilon -1}\frac{{\left(\lambda q;q\right)}_{\infty }}{{\left(q^{m-\epsilon }\lambda ;q\right)}_{\infty }}{d}_q\left(\lambda \right).\hfill \end{array}$$

(44)

Using Equation (5) in (35), we have

$${\mathrm{\Omega }}_4=\frac{1}{q}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{\left(\frac{-1}{q}\right)}^m$$

$$\times D_{q^{-1}}^m{z}^{m-\vartheta -\varsigma \upsilon }\frac{{\mathrm{\Gamma }}_q\left(\vartheta -m+\varsigma \upsilon \right)}{{\mathrm{\Gamma }}_q\left(\vartheta -\epsilon +\varsigma \upsilon \right)}.$$

(45)

Moreover, using Equation (9), we have

$${\mathrm{\Omega }}_4=\frac{1}{q}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\omega }^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{\left(\frac{-1}{q}\right)}^m$$

$$\times q^m{D}_q^m\left(\frac{z^{m-\vartheta -\varsigma \upsilon }}{q^m}\right)\frac{{\mathrm{\Gamma }}_q\left(\vartheta -m+\varsigma \upsilon \right)}{{\mathrm{\Gamma }}_q\left(\vartheta -\epsilon +\varsigma \upsilon \right)}.$$

(46)

We obtain (40) by analyzing the right-hand side of (46) in light of definition (14). □

5. Generalized Fractional Kinetic Equation

The fractional order of differential equations have recently proven to be useful modeling instruments in a wide range of physical phenomena. This is due to the fact that the feasible modeling of a physical phenomenon depends on the instant time, which is additionally achieved effectively by using fractional calculus. Their treatment from the point of view of the q-calculus by using fractional q-calculus operators of the R-L type can also open up new avenues.

If p is the production rate on N, let $N\left(t\right)$ be an arbitrary time-dependent reaction, d be the destruction rate, and then a mathematical description of the three quantities is as follows:

$$\frac{dN}{dt}=-d+p.$$

(47)

A differential fractional equation for the quantities $N\left(t\right)$, p, and d, as introduced by Haubold and Mathai [10], is as follows:

$$\frac{dN}{dt}=-d\left(N_t\right)+p\left(N_t\right),$$

(48)

where $N_t\left(t^{\ast }\right)=N\left(t-t^{\ast }\right)$ for $t^{\ast }>0.$ In addition, Haubold and Mathai [10] discovered that when the spatial shifts or inhomogeneities in quantities $N\left(t\right)$ are ignored, Equation (48) becomes:

$$\frac{d{N}_i}{dt}=-c_i{N}_i\left(t\right).$$

(49)

For the starting situation, let $N_i\left(t=0\right)=N_0$ be a number density of species i at a time $t=0$; the constant $c_i>0,$ is called the standard kinetic equation. So, the solution of Equation (49) is provided by:

$$N_i\left(t\right)=N_0{e}^{-c_{i}t}.$$

(50)

Or else, when the index i is removed and the standard kinetic equation is integrated (49), we obtain:

$$N\left(t\right)-N_0=c_0{D}_t^{-1}N\left(t\right),$$

(51)

where ${}_{0}D_t^{-1}$ is the standard operator of the integral.

Haubold and Mathai [10] stated the fractional generalization of Equation (51) by:

$$N\left(t\right)-N_0=c^v{}_{0}D_t^{-v}N\left(t\right),$$

(52)

where ${}_{0}D_t^{-v}$ is the well-known standard R-L fractional integral operator, and this is stated by:

$${}_{0}D_t^{-v}f\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(v\right)}{\int }_0^t{\left(t-u\right)}^{v-1}f\left(u\right)du,\Re \left(v\right)>0$$

(53)

By computing the fractional kinetic Equation (52), we have the following:

$$N\left(t\right)=N_0\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k}{\mathrm{\Gamma }\left(vk+1\right)}{\left(ct\right)}^{vk}.$$

(54)

5.1. Solution of Generalized Fractional q-Kinetic Equations by Using q-Laplace Transform

At this phase, we use a q-Laplace transform technique to evaluate the generalized q-kinetic equation involving q-analogues of the $\mathbb{M}$-functions (14).

Hahn ([36], see also [24]) stated the q-analogues of the well-known classical Laplace transforms using the following q-integral as:

$${}_{q}L_{\kappa }\left\{f\left(t\right)\right\}=\frac{1}{1-q}{\int }_0^{{\kappa }^{-1}}{E}_q^{-q\kappa t}f\left(t\right)d_{q}t,$$

(55)

when $E_q^z$ is the q-exponential which is defined in (12).

Further, Garg and Chanchlani [27] gave the q-Laplace convolution theorem as follows:

$${}_{q}L_{\kappa }\left\{f{\ast }_{q}g\right\}\left(s\right)={}_{q}L_{\kappa }{\left\{f\left(t\right)\right\}}_q{L}_{\kappa }\left\{g\left(t\right)\right\},$$

when $f{\ast }_{q}g$ is the q-convolution of two analytic functions $f\left(t\right)$ and $g\left(t\right)$ is given as

$$f\left(t\right){\ast }_{q}g\left(t\right)=\frac{1}{1-q}{\int }_0^{t}f\left(u\right)g\left[t-qu\right]d_{q}t,$$

(56)

where the function $g\left(t\right)={\sum }_{n=0}^{\infty }{a}_n{t}^n,$

$$g\left[t-qu\right]=\sum _{n=0}^{\infty }{a}_n(t-qu)(t-q^{2}u)\dots (t-q^{n}u).$$

(57)

Theorem 5. 

Let $\omega >0,\varsigma >0,0<\left|q\right|<1$; then the following fractional q-kinetic equation

$$N_q\left(t\right)-N_0{t}^{\vartheta -1}{}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s(q,-\omega t^{\varsigma })=-\omega I_q^{\varsigma }{N}_q\left(t\right),$$

(58)

has the following solution

$$N^q\left(t\right)=N_0{t}^{\vartheta -1}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}{(-\omega t^{\varsigma })}^{\upsilon }{e}_{\varsigma ,\varsigma \upsilon +\vartheta }(-\omega t^{\varsigma };q).$$

(59)

Proof. 

Now, using the q-Laplace transform on both sides of Equation (58), it becomes

$${}_{q}L_{\kappa }\left\{N_q\left(t\right)\right\}-{}_{q}L_{\kappa }\left\{N_0{t}^{\vartheta -1}{}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s(q,-\omega t^{\varsigma })\right\}={}_{q}L_{\kappa }\left\{-\omega I_q^{\varsigma }{N}_q\left(t\right)\right\}.$$

(60)

Now, using Equation (14) in Equation (60), we have

$$\begin{array}{cc}\hfill {}_{q}L_{\kappa }\left\{N_q\left(t\right)\right\}+& {}_{q}L_{\kappa }\left\{\omega I_q^{\varsigma }{N}_q\left(t\right)\right\}\hfill \\ \hfill & ={}_{q}L^{\kappa }\left\{N_0{t}^{\vartheta -1}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\left(-\omega t^{\varsigma }\right)}^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\right\}\hfill \\ \hfill & =N_0\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{(-\omega )}^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{}_{q}L_{\kappa }\left\{t^{\vartheta +\varsigma \upsilon -1}\right\}.\hfill \end{array}$$

(61)

The authors’ q-Laplace transform of the R-L fractional q-integral operator in [27] is as follows:

$${}_{q}L_{\kappa }\left\{I_q^{\varsigma }f\left(t\right)\right\}={\left\{\frac{1-q}{\kappa }\right\}}^{\varsigma }{}_{q}L_{\kappa }f\left(t\right),\Re \left(\varsigma \right)>0,$$

(62)

where the R-L fractional q-integral operator [37] is provided as

$$I_q^{\varsigma }f\left(t\right)=\frac{1}{{\mathrm{\Gamma }}_q\left(\varsigma \right)}{\int }_o^t{(t-zq)}_q^{\varsigma -1}f\left(z\right)d_{q}z,\Re \left(\varsigma \right)>0.$$

(63)

Using Equation (62), the power function of q-Laplace transform

$${}_{q}L_{\kappa }\left\{t^{\varsigma }\right\}=\frac{{(1-q)}^{\varsigma }{\mathrm{\Gamma }}_q(\varsigma +1)}{{\kappa }^{\varsigma +1}},\Re \left(\varsigma \right)>0,$$

and ${}_{q}L_{\kappa }\left\{N_q\left(t\right)\right\}=N_q^{\ast }\left(\kappa \right)$ in Equation (61), above, we obtain

$$N_q^{\ast }\left(\kappa \right)\left(1+\frac{\omega {(1-q)}^{\varsigma }}{{\kappa }^{\varsigma }}\right)=N_0\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}$$

$$\times \frac{{(-\omega )}^{\upsilon }{(1-q)}^{\vartheta +\varsigma \upsilon -1}{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right){\kappa }^{\varsigma \upsilon +\vartheta }}.$$

After rearrangement of the terms, we use the geometric series expansion formula, and we obtain

$$\begin{array}{cc}\hfill N_q^{\ast }\left(\kappa \right)=N_0& \sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{(-\omega )}^{\upsilon }{(1-q)}^{\vartheta +\varsigma \upsilon -1}}{{\kappa }^{\varsigma \upsilon +\vartheta }}\hfill \\ \hfill & \times \sum _{r=0}^{\infty }\frac{{(-1)}^r{(-\omega )}^r{(1-q)}^{\varsigma r}}{{\kappa }^{\varsigma r}}.\hfill \end{array}$$

(64)

Using the inverse q-Laplace transform on both sides of (64), we have

$$\begin{array}{cc}\hfill {}_{q}L_{\kappa }^{-1}\left\{N_q^{\ast }\left(\kappa \right)\right\}=N_0& \sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}{(-\omega )}^{\upsilon }{(1-q)}^{\vartheta +\varsigma \upsilon -1}\hfill \\ \hfill & \times \sum _{r=0}^{\infty }{(-1)}^r{(-\omega )}^r{(1-q)}^{\varsigma r}{}_{q}L_{\kappa }^{-1}\left\{{\kappa }^{-\left(\varsigma r+\varsigma \upsilon +\vartheta \right)}\right\}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill N_q\left(t\right)& =N_0\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}{(-\omega )}^{\upsilon }{(1-q)}^{\vartheta +\varsigma \upsilon -1}\hfill \\ \hfill & \times \sum _{r=0}^{\infty }\frac{{\left(-\omega \right)}^r{(1-q)}^{\varsigma r}{t}^{\left(\varsigma r+\varsigma \upsilon +\vartheta \right)}}{{(1-q)}^{\varsigma r+\varsigma \upsilon +\vartheta -1}{\mathrm{\Gamma }}_q(\varsigma r+\varsigma \upsilon +\vartheta )}\hfill \\ \hfill & =N_0{t}^{\vartheta -1}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}{(-\omega t^{\varsigma })}^{\upsilon }\sum _{r=0}^{\infty }\frac{{\left(-\omega t^{\varsigma }\right)}^r}{{\mathrm{\Gamma }}_q(\varsigma r+\varsigma \upsilon +\vartheta )}.\hfill \end{array}$$

(65)

We obtain the required result by interpreting the result in (65) in light of (15), that is,

$$N_q\left(t\right)=N_0{t}^{\vartheta -1}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}{(-\omega t^{\varsigma })}^{\upsilon }{e}_{\varsigma ,\varsigma \upsilon +\vartheta }(-\omega t^{\varsigma };q).$$

□

5.2. Alternative Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform

In this section, we look at how a q-Sumudu transform can be used to solve a generalized fractional q-kinetic equation involving the q-analogue of the $\mathbb{M}$-functions (14).

Albayrak et al. [26] established and studied a q-Sumudu transform to aid in the process of solving integral and differential equations in the domain time, as well as for use in different applications of systems in applied physics and engineering. The q-Sumudu transform has very unique and helpful characteristics, and it can be used to solve problems governing kinetic equations in science and engineering. The q-Sumudu transforms are the theoretical counterpart of the q-Laplace transforms.

The primary type of q-analogue of the Sumudu transform defined and represented by Albayrak et al. [26] (see also [28]) is as follows:

$$S_q\left\{f\left(x\right);\sigma \right\}=\frac{1}{\left(1-q\right)\sigma }{\int }_0^{\sigma }{E}_q\left(\frac{q}{\sigma }t\right)d_{q}x,\sigma \in \left(-{\tau }_1,{\tau }_2\right),$$

(66)

$$={\left(q;q\right)}_{\infty }\sum _{k=0}^{\infty }\frac{q^{k}f\left(\sigma q^k\right)}{{\left(q;q\right)}_k}.$$

(67)

When applied cross the set of functions, we have

$$A=\left\{f\left(x\right)\exists M,{\tau }_1{\tau }_2>0,\left|f\left(x\right)<M{E}_q\left(\left|t\right|/{\tau }_j\right),x\in {\left(-1\right)}^j\times \left[0,\infty )\right.\right|\right\}.$$

The following are the q-Sumudu transforms of some unique functions used in the sequel:

$$S_q\left\{x^{\varsigma -1};\sigma \right\}={\sigma }^{\varsigma -1}{\left(1-q\right)}^{\varsigma -1}{\mathrm{\Gamma }}_q\left(\varsigma \right),\left(\Re \left(\varsigma \right)>0\right).$$

(68)

$$S_q\left\{\frac{\left(1-q\right)}{x}{E}_{\varsigma ,0}\left(-\omega x^{\varsigma };q\right);\sigma \right\}=\frac{1}{\sigma \left(1+\omega {\sigma }^{\varsigma }{\left(1-q\right)}^{\varsigma }\right)},\left(\Re \left(\varsigma \right)>0\right).$$

(69)

Albayrak et al. [26] provided a q-Sumudu convolution theorem as follows:

$$S_q\left\{\left(f{\ast }_{q}g\right)\left(x\right);\sigma \right\}=\sigma S_q\left\{f\left(x\right);\sigma \right\}{S}_q\left\{g\left(x\right);\sigma \right\},$$

(70)

when $f{\ast }_{q}g$ are the q-convolutions of two analytical functions $f\left(t\right)$ and $g\left(t\right)$, provided by

$$\left(f{\ast }_{q}g\right)\left(z\right)=\frac{z}{1-q}{\int }_0^{1}f\left(zx\right)g\left[z\left(1-qx\right)\right]d_{q}x,$$

(71)

where $g\left(z\right)={\sum }_{k=0}^{\infty }{a}_n{z}^n$ and $g\left[z-x\right]={\sum }_{k=0}^{\infty }{a}_n{\left(z-x\right)}_q^n.$

Discussion 1. 

Let $0<\left|q\right|<1,\omega >0,\varsigma >0$; the solution to the generalized fractional q-kinetic Equation (58) is provided by (59).

Using the q-Sumudu applied to both sides of Equation (58), it becomes

$$S_q\left\{N_q\left(t\right);\sigma \right\}-S_q\left\{N_0{t}^{\vartheta -1}{}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s(q,-\omega t^{\varsigma });\sigma \right\}=S_q\left\{-\omega I_q^{\varsigma }{N}_q\left(t\right);\sigma \right\}.$$

(72)

The q-Sumudu transform of the R-L fractional q-integral operator is defined by Purohit and Ucar [28] as:

$$S_q\left\{I_q^{\varsigma }f\left(t\right);s\right\}=s^{\varsigma }{\left(1-q\right)}^{\varsigma }{S}_q\left\{f\left(t\right);s\right\}$$

(73)

where the R-L fractional q-integral operator was established by Al-Salam [37] and was provided independently by Agarwal [34] as:

$$I_q^{\varsigma }f\left(t\right)=\frac{t^{\varsigma -1}}{{\mathrm{\Gamma }}_q\left(\varsigma \right)}{\int }_0^t{\left(qz/t;q\right)}_{\varsigma -1}{d}_{q}x,\Re \left(\varsigma \right)>0,\varsigma \notin \left\{-1,-2,\dots \right\}.$$

Now, using Equations (14) and (76) in Equation (72), we have the following:

$$S_q\left\{N_q\left(t\right);\sigma \right\}+\omega {\sigma }^{\varsigma }{\left(1-q\right)}^{\varsigma }{S}_q\left\{N_q\left(t\right);\sigma \right\}$$

$$=S_q\left\{N_0{t}^{\vartheta -1}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\left(-\omega t^{\varsigma }\right)}^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\right\}.$$

Substituting $S_q\left\{N_q\left(t\right);\sigma \right\}=N\left(\sigma \right)$, we have

$$N\left(\sigma \right)+\omega {\sigma }^{\varsigma }{\left(1-q\right)}^{\varsigma }N\left(\sigma \right)=N_0\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\left(-\omega \right)}^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}{S}_q\left\{t^{\vartheta +\varsigma \upsilon -1}\right\}.$$

(74)

Using Equation (68), we obtain

$$\begin{array}{cc}\hfill N\left(\sigma \right)\left\{1+\omega {\sigma }^{\varsigma }{\left(1-q\right)}^{\varsigma }\right\}=N_0& \sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}\frac{{\left(-\omega \right)}^{\upsilon }}{{\mathrm{\Gamma }}_q\left(\varsigma \upsilon +\vartheta \right)}\hfill \\ \hfill & \times {\sigma }^{\vartheta +\varsigma \upsilon -1}{\left(1-q\right)}^{\vartheta +\varsigma \upsilon -1}{\mathrm{\Gamma }}_q\left(\vartheta +\varsigma \upsilon \right).\hfill \end{array}$$

(75)

After rearrangement of the terms, we use a geometric series expansion formula and obtain

$$\begin{array}{cc}\hfill N\left(\sigma \right)=N_0& \sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}{\left(-\omega \right)}^{\upsilon }{\sigma }^{\vartheta +\varsigma \upsilon -1}{\left(1-q\right)}^{\vartheta +\varsigma \upsilon -1}\hfill \\ \hfill & \times \sum _{r=0}^{\infty }{(-1)}^r{(-\omega {\sigma }^{\varsigma })}^r{(1-q)}^{\varsigma r}.\hfill \end{array}$$

(76)

Using the inverse q-Sumudu transform with both sides of (73), we have

$$N_q\left(t\right)=N_0{t}^{\vartheta -1}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{c_1};q\right)}_{\upsilon }\cdots {\left(q^{c_r};q\right)}_{\upsilon }}{{\left(q^{d_1};q\right)}_{\upsilon }\cdots {\left(q^{d_s};q\right)}_{\upsilon }}{(-\omega t^{\varsigma })}^{\upsilon }\sum _{r=0}^{\infty }\frac{{\left(-\omega t^{\varsigma }\right)}^r}{{\mathrm{\Gamma }}_q(\varsigma r+\varsigma \upsilon +\vartheta )}.$$

(77)

We easily arrive at the desired result (59) by interpreting the result (77) in the context of (15).

6. Concluding Observations

This portion will produce the following amazing results in the form of corollaries using the results produced in the previous segment. For $q\to 1^{-}$, Theorems 1–4 generate the following equality relations:

Corollary 1. 

Let $z,\varsigma ,\vartheta \in \mathbb{C}$; $\epsilon >0,\omega \in \Re$, $\Re \left(\varsigma \right)>0$; then the formula is

$$\left(I_{0+}^{\epsilon }\left\{t^{\vartheta -1}{}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s\left(\omega t^{\varsigma }\right)\right\}\right)\left(z\right)=z^{\epsilon +\vartheta -1}\stackrel{\varsigma ,\vartheta +\epsilon }{{}_r\mathbb{M}_s}\left(\omega z^{\varsigma }\right).$$

(78)

Corollary 2. 

Let $z,\varsigma ,\vartheta \in \mathbb{C}$; $\epsilon >0,\omega \in \Re$, $\Re \left(\varsigma \right)>0$; then the formula is

$$\left(I_{-}^{\epsilon }\left\{t^{-\epsilon -\vartheta }{}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s\left(\omega t^{-\varsigma }\right)\right\}\right)\left(z\right)=z^{-\vartheta }\stackrel{\varsigma ,\vartheta +\epsilon }{{}_r\mathbb{M}_s}\left(\omega z^{-\varsigma }\right).$$

(79)

Corollary 3. 

Let $z,\varsigma ,\vartheta \in \mathbb{C}$; $\epsilon >0,\omega \in \Re$, $\Re \left(\varsigma \right)>0$; then the formula is

$$\left(D_{0+}^{\epsilon }\left\{t^{\vartheta -1}{}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s\left(\omega t^{\varsigma }\right)\right\}\right)\left(z\right)=z^{\epsilon -\vartheta -1}\stackrel{\varsigma ,\vartheta +\epsilon }{{}_r\mathbb{M}_s}\left(\omega z^{\varsigma }\right).$$

(80)

Corollary 4. 

Let $z,\varsigma ,\vartheta \in \mathbb{C}$; $\epsilon >0,\omega \in \Re$, $\Re \left(\varsigma \right)>0$; then the formula is

$$\left(D_{-}^{\epsilon }\left\{t^{\epsilon -\vartheta }{}_r\stackrel{\varsigma ,\vartheta }{\mathbb{M}}_s\left(\omega t^{-\varsigma }\right)\right\}\right)\left(z\right)=z^{-\vartheta }\stackrel{\varsigma ,\vartheta +\epsilon }{{}_r\mathbb{M}_s}\left(\omega z^{-\varsigma }\right).$$

(81)

Now, these results are immediate consequences of Equations (15) and (17), and Theorems 1–4; therefore, they are given with no corollaries as evidence here.

Let $r=s=0$, in Theorems 1–4; we obtain some amazing results in the form of Corollaries 5–8, respectively.

Corollary 5. 

$$\left(I_{q,0+}^{\epsilon }\left\{t^{\vartheta -1}{e}_{\varsigma ,\vartheta }(q,\omega t^{\varsigma })\right\}\right)\left(z\right)=z^{\epsilon +\vartheta -1}{e}_{\varsigma ,\vartheta +\epsilon }\left(\omega z^{\varsigma };q\right).$$

(82)

Corollary 6. 

$$\left(I_{q,-}^{\epsilon }\left\{z^{-\epsilon -\vartheta }{e}_{\varsigma ,\vartheta }(q,\omega t^{-\varsigma })\right\}\right)\left(x\right)=\left(z^{-\vartheta }/q\right)e_{\varsigma ,\vartheta +\epsilon }(q,\omega z^{-\varsigma }).$$

(83)

Corollary 7. 

$$\left(D_{q,0+}^{\epsilon }\left\{t^{\vartheta -1}{e}_{\varsigma ,\vartheta }(q,\omega t^{\varsigma })\right\}\right)\left(z\right)=z^{\epsilon -\vartheta -1}{e}_{\varsigma ,\vartheta -\epsilon }(q,\omega z^{\varsigma }).$$

(84)

Corollary 8. 

$$\left(D_{q,-}^{\epsilon }\left\{t^{\epsilon -\vartheta }{e}_{\varsigma ,\vartheta }(q,\omega t^{-\varsigma })\right\}\right)\left(z\right)=\left(z^{-\vartheta }/q^{m+1}\right)e_{\varsigma ,\vartheta -\epsilon }(q,\omega z^{-\varsigma }).$$

(85)

If $q\to 1^{-}$ and we use a limit formula of (20)–(22), we see the outcomes as shown in Corollaries 5–8, providing the q-extensions of the known outcomes found by Saxena and Saigo [[38], Corollary ((1.1), (1.3), (1.5), (1.7)), respectively].

If we put $r=s=1,c_1=\delta ,d_1=1$ in Theorems 1–4, we obtain some fascinating outcomes in the form of Corollaries 9–12, respectively.

Corollary 9. 

$$\left(I_{q,0+}^{\epsilon }\left\{t^{\vartheta -1}{e}_{\varsigma ,\vartheta }^{\delta }(q,\omega t^{\varsigma })\right\}\right)\left(z\right)=z^{\epsilon +\vartheta -1}{e}_{\varsigma ,\vartheta +\epsilon }^{\delta }\left(\omega z^{\varsigma };q\right).$$

(86)

Corollary 10. 

$$\left(I_{q,-}^{\epsilon }\left\{z^{-\epsilon -\vartheta }{e}_{\varsigma ,\vartheta }^{\delta }(q,\omega t^{-\varsigma })\right\}\right)\left(x\right)=\left(z^{-\vartheta }/q\right)e_{\varsigma ,\vartheta +\epsilon }^{\delta }(q,\omega z^{-\varsigma }).$$

(87)

Corollary 11. 

$$\left(D_{q,0+}^{\epsilon }\left\{t^{\vartheta -1}{e}_{\varsigma ,\vartheta }^{\delta }(q,\omega t^{\varsigma })\right\}\right)\left(z\right)=z^{\epsilon -\vartheta -1}{e}_{\varsigma ,\vartheta -\epsilon }^{\delta }(q,\omega z^{\varsigma }).$$

(88)

Corollary 12. 

$$\left(D_{q,-}^{\epsilon }\left\{t^{\epsilon -\vartheta }{e}_{\varsigma ,\vartheta }^{\delta }(q,\omega t^{-\varsigma })\right\}\right)\left(z\right)=\left(z^{-\vartheta }/q^{m+1}\right)e_{\varsigma ,\vartheta -\epsilon }^{\delta }(q,\omega z^{-\varsigma }).$$

(89)

Similarly, letting $q\to 1^{-}$, Corollaries 9–12 provide the q-extensions of the known outcomes of Saxena and Saigo [[38], Theorems 1, 2, 3 and 4, respectively].

Furthermore, by focusing the variables involved in the main theorem, we can solve some other fractional q-kinetic equations and also the ordinary q-kinetic equations.

If we set $r=s=1,c_1=\delta ,d_1=1$ in Theorem 5 and by using definition (17), we have the following outcomes, that is, a q-analogue of the results obtained by Garg and Chanchlani [[27], p.39, eq. 30].

Corollary 13. 

Let $\omega >0,\varsigma >0,0<\left|q\right|<1$; then the following fractional q-kinetic equation

$$N_q\left(t\right)-N_0{t}^{\vartheta -1}{E}_{\varsigma ,\vartheta }^{\delta }(q,-\omega t^{\varsigma })=-\omega I_q^{\varsigma }{N}_q\left(t\right),$$

(90)

has the following solution

$$N_q\left(t\right)=N_0{t}^{\vartheta -1}\sum _{\upsilon =0}^{\infty }\frac{{\left(q^{\delta };q\right)}_{\upsilon }}{{\left(q;q\right)}_{\upsilon }}{(-\omega t^{\varsigma })}^{\upsilon }{e}_{\varsigma ,\varsigma \upsilon +\vartheta }(-\omega t^{\varsigma };q).$$

(91)

If we take $q\to 1$ in our Theorem 5 and using Equations (15)–(17), we also determine that the findings presented here are general enough to produce solutions to a lot of known or novel fractional kinetic equations involving such other unique functions as (for example) those examined by Saxena et al. [12], as well as solutions to the problems examined by Saxena et al. [11,13] and Haubold and Mathai [10]. Also, we conclude by noting that the q-Sumudu transform method is an extremely successful and practical alternative method for solving fractional q-differential equations. The extended kinetic equations are expected to have potential applications in nuclear power, nuclear physics, astrophysics, and other related fields.