On the solutions of certain fractional kinetic equations involving $E^{\gamma,q}_{k,\alpha,\beta}(.)$

TL;DR

This paper introduces a generalized fractional kinetic equation involving a new k-Mittag-Leffler function, providing broad solutions that encompass known and novel results in fractional calculus.

Contribution

It develops a new generalized fractional kinetic equation with a novel k-Mittag-Leffler function, expanding the scope of solutions in fractional kinetic theory.

Findings

Derived a generalized fractional kinetic equation involving $E^{eta,q}_{k, u, ho}(.)$

Provided solutions that include known special cases as well as new results

The results are broad and applicable to various fractional kinetic models

Abstract

We develop a new generalized form of the fractional kinetic equation involving a generalized k-Bessel function. The generalized $k$-Mittag-leffler function $E^{\gamma,q}_{k,\alpha,\beta}(.)$ is discussed in terms of the solution of the fractional kinetic equation in the present paper. The results obtained here are quite general in nature and capable of yielding known and as well new results.

Figures

Equations

$$(x{)}_{n,k}=x(x+k)(x+2k)...(x+(n-1)k),$$

$$(x{)}_{(n+r)q,k}=(x{)}_{rq,k}(x+qrk{)}_{nq,k},$$

$${\Gamma }_s(\gamma )={\left(\frac{s}{k}\right)}^{\frac{\gamma }{s}-1}{\Gamma }_k\left(\frac{k\gamma }{s}\right),$$

$${\Gamma }_k(\gamma )=k^{\frac{\gamma }{k}-1}\Gamma \left(\frac{\gamma }{k}\right).$$

$$(\gamma {)}_{nq,s}={\left(\frac{s}{k}\right)}^{nq}{\left(\frac{k\gamma }{s}\right)}_{nq},$$

$$(\gamma {)}_{nq,k}=(k{)}^{nq}{\left(\frac{\gamma }{k}\right)}_{nq}.$$

$$E_{k,\alpha ,\beta }^{\gamma ,q}(z)=\sum _{n=0}^{\infty }\frac{(\gamma {)}_{nq,k}{z}^n}{{\Gamma }_k(n\alpha +\beta )n!},$$

$$(\gamma {)}_{nq}=\frac{\Gamma (\gamma +nq)}{\Gamma (\gamma )}=q^{qn}\prod _{r=1}^q{\left(\frac{\gamma +r+1}{q}\right)}_n,\mathrm{if}q\in \mathbb{N}.$$

$$E_{k,\alpha ,\beta }^{\gamma ,1}(z)=\sum _{n=0}^{\infty }\frac{(\gamma {)}_{n,k}{z}^n}{{\Gamma }_k(n\alpha +\beta )n!}=E_{k,\alpha ,\beta }^{\gamma }(z),$$

$$E_{1,\alpha ,\beta }^{\gamma ,q}(z)=\sum _{n=0}^{\infty }\frac{(\gamma {)}_{nq}{z}^n}{\Gamma (n\alpha +\beta )n!}=E_{\alpha ,\beta }^{\gamma ,q}(z),$$

$$E_{1,\alpha ,\beta }^{\gamma ,1}(z)=\sum _{n=0}^{\infty }\frac{(\gamma {)}_n{z}^n}{\Gamma (n\alpha +\beta )n!}=E_{\alpha ,\beta }^{\gamma }(z),$$

$$E_{1,\alpha ,\beta }^{1,1}(z)=\sum _{n=0}^{\infty }\frac{z^n}{\Gamma (n\alpha +\beta )n!}=E_{\alpha ,\beta }(z),$$

$$E_{1,\alpha ,1}^{1,1}(z)=\sum _{n=0}^{\infty }\frac{z^n}{\Gamma (n\alpha +1)n!}=E_{\alpha }(z).$$

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

$$\begin{array}{rl}& \frac{dN}{dt}=-c_i{N}_i(t),\end{array}$$

$$\begin{array}{rl}& N(t)-N_0=-c{}_0{D}_t^{-1}N(t),\end{array}$$

$$\begin{array}{rl}& {}_0{D}_t^{-\nu }f(t)=\frac{1}{\Gamma (\nu )}{\int }_0^t{\left(t-s\right)}^{\nu -1}f(s)ds,(t>0,R(\nu )>0).\end{array}$$

$$N(t)-N_0=-c^{\nu }{}_0{D}_t^{-1}N(t),$$

$$\begin{array}{rl}& N(t)=N_0\sum _{k=0}^{\infty }\frac{(-1{)}^k}{\Gamma \left(\nu k+1\right)}{\left(ct\right)}^{\nu k}.\end{array}$$

$$\begin{array}{rl}& N(t)-N_{0}f(t)=-c^{\nu }{}_0{D}_t^{-\nu }N(t),(\Re (v)>0),\end{array}$$

$$\begin{array}{rl}& L\left\{N(t);p\right\}=N_0\frac{F(p)}{1+c^{\nu }{p}^{-\nu }}=N_0\left(\sum _{n=0}^{\infty }(-c^{\nu }{)}^n{p}^{-\nu n}\right)F(p),\\ & \left(n\in N_0,\left|\frac{c}{p}\right|<1\right)\end{array}$$

$$\begin{array}{rl}& F(p)=L\left\{N(t);p\right\}={\int }_0^{\infty }{e}^{-pt}f(t)dt,(\mathcal{R}(p)>0).\end{array}$$

$$\begin{array}{rl}& N(t)-N_0{E}_{k,\alpha ,\beta }^{\gamma ,\tau }(t)=-d^{\nu }{}_0{D}_t^{-\nu }N(t)\end{array}$$

$$\begin{array}{rl}& N(t)=N_0\sum _{n=0}^{\infty }\frac{(\gamma {)}_{n\tau ,k}}{{\Gamma }_k(n\alpha +\beta )}{t}^n{E}_{\nu ,n+1}(-d^{\nu }{t}^{\nu }),\end{array}$$

$$\begin{array}{rl}& E_{\alpha ,\beta }(x)=\sum _{n=0}^{\infty }\frac{(x{)}^n}{\Gamma \left(\alpha n+\beta \right)}.\end{array}$$

$$\begin{array}{rl}& L\left\{{}_0{D}_t^{-\nu }f(t);p\right\}=p^{-\nu }F(p),\end{array}$$

$$\begin{array}{rl}& L\left\{N(t);p\right\}=N_{0}L\left\{E_{k,\alpha ,\beta }^{\gamma ,\tau }(t);p\right\}-d^{\nu }L\left\{{}_0{D}_t^{-\nu }N(t);p\right\},\end{array}$$

$$\begin{array}{rl}& N(p)=N_0\left({\int }_0^{\infty }{e}^{-pt}\sum _{n=0}^{\infty }\frac{(\gamma {)}_{n\tau ,k}}{{\Gamma }_k(n\alpha +\beta )}\frac{t^n}{n!}dt\right)\\ & -d^{\nu }{p}^{-\nu }N(p),\end{array}$$

$$\begin{array}{rl}& N(p)+d^{\nu }{p}^{-\nu }N(p)=N_0\sum _{n=0}^{\infty }\frac{(\gamma {)}_{n\tau ,k}}{{\Gamma }_k(n\alpha +\beta )}\frac{1}{n!}\\ & \times {\int }_0^{\infty }{e}^{-pt}{t}^{n}dt\end{array}$$

$$\begin{array}{rl}& =N_0\sum _{n=0}^{\infty }\frac{(\gamma {)}_{n\tau ,k}}{{\Gamma }_k(n\alpha +\beta )}\frac{1}{n!}\frac{\Gamma (n+1)}{p^{n+1}},\end{array}$$

Full text

1. Introduction and Preliminaries

The $k$-Pochhammer symbol was introduced in [1] as follows:

[TABLE]

[TABLE]

where $x\in\mathbb{C},k\in\mathbb{R}$ and $n\in\mathbb{N}$.

Proposition 1**.**

Let $\gamma\in\mathbb{C},k,s\in\mathbb{R}$. Then the following identity holds

[TABLE]

and in particular

[TABLE]

Proposition 2**.**

Let $\gamma\in\mathbb{C},k,s\in\mathbb{R}$ and $\gamma\in\mathbb{C}$. Then the following identity holds

[TABLE]

and in particular

[TABLE]

For more details on the $k$-Pochhammer symbol, the $k$-special function and the fractional Fourier transform we refer the reader to the papers by Romero et. al. [7, 8].

Let $k\in\mathbb{R},\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re{(\gamma)}>0$ and $q\in(0,1)\cup\mathbb{N}$. Then the generalized $k$-Mittag-Leffler function, denoted by $E^{\gamma,q}_{k,\alpha,\beta}(z)$, is defined as:

[TABLE]

where $(\gamma)_{nq,k}$ denotes the $k$-Pochhammer symbol given by equation (1.6) and $\Gamma_{k}(x)$ is the $k$-gamma function given by the equation (1.4) (also see [14]).

The generalized Pochhammer symbol is defined as (see [9, page 22])

[TABLE]

We consider particular cases of $E^{\gamma,q}_{k,\alpha,\beta}(z)$:

(i) For $q=1$, equation(1.7) yields the $k$-Mittag-Leffler function (see [2]), defined as:

[TABLE]

(ii) For $k=1$, n(1.7) yields the Mittag-Leffler function, defined in [12]:

[TABLE]

(iii) For $q=1$ and $k=1$, (1.7) gives the Mittag-Leffler function, defined in Dorrego and Cerutti [2]:

[TABLE]

(iv) For $q=1,k=1$ and $\gamma=1$, (1.7) gives the Mittag-Leffler function (see [15]), defined as:

[TABLE]

(v) For $q=1,k=1,\gamma=1$ and $\beta=1$, (1.7) gives the Mittag-Leffler function (see [6]), defined as:

[TABLE]

A detailed account of the Mittag-Leffler function and their applications can be found in the survey paper by Saxena et. al. [11].

2. Fractional differential equations

In physics, dynamical systems, control systems and engineering, fractional differential equations model many physical phenomena and kinetic equations describe the continuity of motion of substances. The extension and generalization of fractional kinetic equations involving fractional operators can be found in [4, 5, 10].

The fractional differential equation between the rate of change of the reaction, the destruction rate and the production rate was established by Haubold and Mathai[4] and is given as follows:

[TABLE]

where $N=N(t)$ denotes the rate of reaction, $d=d(N)$ the rate of destruction, $p=p(N)$ the rate of production and $N_{t}$ denotes the function defined by $N_{t}(t^{*})=N(t-t^{*}),t^{*}>0.$

The special case of (2.1) for spatial fluctuations and inhomogeneities in $N(t)$ the quantities are neglected , that is the equation

[TABLE]

with the initial condition that $N_{i}(t=0)=N_{0}$ is the number density of the species $i$ at time $t=0$ and $c_{i}>0$. If we remove the index $i$ and integrate the standard kinetic equation (2.2), we have

[TABLE]

where ${}_{0}D^{-1}_{t}$ is the special case of the Riemann-Liouville integral operator ${}_{0}D^{-\nu}_{t}$ defined as

[TABLE]

The fractional generalization of the standard kinetic equation(2.3) is given by Haubold and Mathai [4] as follows:

[TABLE]

and they obtained the solution of (2.5) as

[TABLE]

In [10] the authors considered the following fractional kinetic equation:

[TABLE]

where $N(t)$ denotes the number density of a given species at time $t$, $N_{0}=N(0)$ is the number density of that species at time $t=0$, $c$ is a constant and $f\in\mathcal{L}(0,\infty)$. Apply the Laplace transform to (2.7) (see [5]), so

[TABLE]

where the Laplace transform [13] is given by

[TABLE]

The objective of this paper is to derive the solution of the fractional kinetic equation involving the generalized $k$-Mittag-Leffler function. The results obtained in terms of the Mittag-Leffler function are rather general in nature and we can easily construct various known and new fractional kinetic equations.

3. Solution of generalized fractional kinetic equations

In this section, we investigate the solution of the generalized fractional kinetic equations by considering the generalized $k$-Bessel function.

Theorem 1**.**

If $d>0,\nu>0,$ $k\in\mathbb{R},\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re{(\gamma)}>0$ and $q\in(0,1)\cup\mathbb{N}$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where the generalized Mittag-Leffler function $E_{\alpha,\beta}(x)$ is given by [6]

[TABLE]

Proof: The Laplace transform of the Riemann-Liouville fractional integral operator is given by [3]

[TABLE]

where $F(p)$ is defined in (2.9). Now, applying the Laplace transform to both sides of (3.1) gives

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Taking the Laplace inverse of (3.9), and using

[TABLE]

we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Theorem 2**.**

If $d>0,\nu>0,$ $k\in\mathbb{R},\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re{(\gamma)}>0$ and $q\in(0,1)\cup\mathbb{N}$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Theorem 3**.**

If $a>0,d>0,\nu>0,$ $k\in\mathbb{R},\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re{(\gamma)}>0$ and $q\in(0,1)\cup\mathbb{N}$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Proof: The proof of (3.16) and (3.18) is similar to that given to prove (3.2).

4. Special Cases

If we choose $q=1$, then (3.2), (3.16) and (3.18) reduces to the following:

Corollary 1**.**

If $d>0,\nu>0,$ $k\in\mathbb{R},\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0$ and $\Re{(\gamma)}>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\alpha,\beta}(x)$ is the Mittag-Leffler function defined in equation (3.3).

Corollary 2**.**

If $d>0,\nu>0,$ $k\in\mathbb{R},\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0$ and $\Re{(\gamma)}>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the Mittag-Leffler function defined in equation (3.3).

Corollary 3**.**

If $a>0,d>0,\nu>0,$ $k\in\mathbb{R},\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0$ and $\Re{(\gamma)}>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

If we choose $k=1$, then (3.2), (3.16) and (3.18) reduces to the following:

Corollary 4**.**

If $d>0,\nu>0,$ $\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re{(\gamma)}>0$ and $q\in(0,1)\cup\mathbb{N}$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 5**.**

If $d>0,\nu>0,$ $\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re{(\gamma)}>0$ and $q\in(0,1)\cup\mathbb{N}$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 6**.**

If $a>0,d>0,\nu>0,$ $\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re{(\gamma)}>0$ and $q\in(0,1)\cup\mathbb{N}$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

If we choose $q=1$ and $k=1$, then (3.2), (3.16) and (3.18) reduces to the following:

Corollary 7**.**

If $d>0,\nu>0,$ $\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0$ and$\Re{(\gamma)}>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 8**.**

If $d>0,\nu>0,$ $\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0$ and$\Re{(\gamma)}>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 9**.**

If $a>0,d>0,\nu>0,$ $\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0$ and$\Re{(\gamma)}>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

If we choose $q=1,k=1$ and $\gamma=1$, then (3.2), (3.16) and (3.18) reduces to the following:

Corollary 10**.**

If $d>0,\nu>0;\alpha,\beta\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 11**.**

If $d>0,\nu>0;\alpha,\beta\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 12**.**

If $a>0,d>0,\nu>0;\alpha,\beta\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

If we choose $q=1,k=1,\gamma=1$ and $\beta=1$, then (3.2), (3.16) and (3.18) reduces to the following:

Corollary 13**.**

If $d>0,\nu>0;\alpha\in\mathbb{C};\Re(\alpha)>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 14**.**

If $d>0,\nu>0;\alpha\in\mathbb{C};\Re(\alpha)>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 15**.**

If $a>0,d>0,\nu>0;\alpha\in\mathbb{C};\Re(\alpha)>0$, then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

If we choose $q=1,k=1,\gamma=1,\alpha=0$ and $\beta=1$, then (3.2), (3.16) and (3.18) reduces to the following:

Corollary 16**.**

If $d>0,\nu>0$ then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 17**.**

If $d>0,\nu>0$ then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

Corollary 18**.**

If $a>0,d>0,\nu>0$ then the solution of the equation

[TABLE]

is given by

[TABLE]

where $E_{\nu,\nu n+1}(.)$ is the generalized Mittag-Leffler function defined in equation (3.3).

5. Database and Graphical Interpretation

In this section we establish database for numerical solutions of the kinetic equations (3.2), (3.16) and (3.18) for particular values of the parameters and their graphs and Mesh-plot are plotted in figure 1 and 2 respectively. Here we denote the solutions of equations (3.2) as $N(t)=N(N_{0},\gamma,\tau,k,\alpha,\beta,d,\nu,t)$; where $N_{0}=.05,\gamma=2,\tau=1,k=2,\alpha=6,\beta=7,d=3,\nu=1$; those for equation (3.16) as $N(t)=N(N_{0},\gamma,\tau,k,\alpha,\beta,d,\nu,t)$; where $N_{0}=.05,\gamma=2,\tau=1,k=2,\alpha=6,\beta=7,d=3,\nu=5$ and those for equation (3.18) as $N(t)=N(N_{0},\gamma,\tau,k,\alpha,\beta,d,a,\nu,t)$; where $N_{0}=.05,\gamma=2,\tau=1,k=2,\alpha=6,\beta=7,d=3,a=3,\nu=7$. We obtained the database (see Table 1) by using these values. On the basis of the same values assigned to the parameters in equations (3.2), (3.16) and (3.18), we have the graphs in figure 1 and 2.