Generalized fractional kinetic equations involving generalized Struve function of the first kind

K.S. Nisar; S.D. Purohit; Saiful. R. Mondal

doi:10.1016/j.jksus.2015.08.005

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Original article

28 (

); 167-171

doi:

10.1016/j.jksus.2015.08.005

Generalized fractional kinetic equations involving generalized Struve function of the first kind

K.S. Nisar^{^a}, S.D. Purohit^{^b,⁎}, Saiful. R. Mondal^{^c}

a Department of Mathematics, Prince Sattam bin Abdulaziz University, 54 Wadi Aldawaser, Riyadh region, Saudi Arabia

b Department of HEAS (Mathematics), Rajasthan Technical University, Kota 324010, Rajasthan, India

c Department of Mathematics, College of Science, King Faisal University, Al-Ahsa, Saudi Arabia

⁎Corresponding author. sunil_a_purohit@yahoo.com (S.D. Purohit),

Received: 2015-5-14, Accepted: 2015-8-24, Published: 2015-04-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

In recent paper Dinesh Kumar et al. developed a generalized fractional kinetic equation involving generalized Bessel function of first kind. The object of this paper is to derive the solution of the fractional kinetic equation involving generalized Struve function of the first kind. The results obtained in terms of generalized Struve function of first kind are rather general in nature and can easily construct various known and new fractional kinetic equations.

Keywords

Fractional kinetic equations

Fractional calculus

Special functions

Laplace transforms

Generalized Struve function of the first kind

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1 Introduction

The Struve function $H_{v} (z)$ and $L_{v} (z)$ are defined as the sum of the following infinite series

(1)

H_{υ} (z) = {(\frac{z}{2})}^{υ + 1} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{Γ (k + \frac{3}{2}) Γ (k + υ + \frac{1}{2})} {(\frac{z}{2})}^{2 k}

and

(2)

L_{υ} (z) = {(\frac{z}{2})}^{υ + 1} \sum_{k = 0}^{\infty} \frac{1}{Γ (k + \frac{3}{2}) Γ (k + υ + \frac{1}{2})} {(\frac{z}{2})}^{2 k} .

The purpose of this work is to investigate the generalized form of the fractional kinetic equation involving generalized Struve function of the first kind $H_{p, b, c} (z)$ defined for complex $z \in C$ and $b, c, p \in C$ ( $R (p) > - 1$ ) by

(3)

H_{p, b, c} (z) ≔ \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} c^{k}}{Γ (p + 1 + \frac{b}{2} + k) Γ (k + 3 / 2)} {(\frac{z}{2})}^{2 k + p + 1} .

Details related to the function $H_{p, b, c} (z)$ and its particular cases can be seen in Baricz (2010, 2008), Mondal and Nisar (2014), Mondal and Swaminathan (2012) and the references therein. In this paper we consider the following transformation

(4)

ϕ_{b, c} (z) = 2^{p + 1} Γ (p + \frac{b}{2} + 1) Γ (\frac{3}{2}) z^{\frac{1}{2} - \frac{p}{2}} H_{p, b, c} (\sqrt{z}) = z + \sum_{k = 1}^{\infty} \frac{{(- c)}^{k} z^{k + 1}}{{(υ)}_{k} {(\frac{3}{2})}_{k}},

where

υ = p + \frac{b}{2} + 1 \notin Z_{0}^{-} ≔ \{0, - 1, - 2, \dots\}

and

(5)

\begin{matrix} {(a)}_{k} & = \{\begin{matrix} 1 & (k = 0) \\ a (a + 1) \dots (a + k - 1) & (k \in N ≔ {1, 2, 3, \dots}) \end{matrix} \\ = \frac{Γ (a + k)}{Γ (a)} (a \in C ⧹ Z_{0}^{-}) . \end{matrix}

The importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in physics, dynamical systems, control systems and engineering, to create the mathematical model of many physical phenomena. Especially, the kinetic equations describe the continuity of motion of substance and are the basic equations of mathematical physics and natural science. The extension and generalization of fractional kinetic equations involving many fractional operators were found (Zaslavsky, 1994; Saichev and Zaslavsky, 1997; Haubold and Mathai, 2000; Saxena et al., 2002, 2004, 2006, 2008; Chaurasia and Pandey, 2008; Gupta and Sharma, 2011; Chouhan and Sarswat, 2012; Chouhan et al., 2013; Gupta and Parihar, 2014; Kumar et al., 2015; Choi and Kumar, 2015 ). In view of the effectiveness and a great importance of the kinetic equation in certain astrophysical problems the authors develop a further generalized form of the fractional kinetic equation involving generalized Struve function of the first kind.

The fractional differential equation between rate of change of reaction was established by Haubold and Mathai (2000), the destruction rate and the production rate are calculated as follows:

(6)

\frac{dN}{dt} = - d (N_{t}) + p (N_{t}),

where

N = N (t)

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 (6), for spatial fluctuations or inhomogeneities in $N (t)$ the quantity are neglected, that is the equation

(7)

\frac{dN}{dt} = - c_{i} N_{i} (t)

with the initial condition that

N_{i} (t = 0) = N_{0}

is the number of density of species i at time

t = 0

and

c_{i} > 0

. If we remove the index i and integrate the standard kinetic Eq. (7), we have

(8)

N (t) - N_{0} = - c_{0} D_{t}^{- 1} N (t),

where

_{0} D_{t}^{- 1}

is the special case of the Riemann–Liouville integral operator

_{0} D_{t}^{- υ}

defined as

_{0} D_{t}^{- υ} f (t) = \frac{1}{Γ (υ)} \int_{0}^{t} {(t - s)}^{υ - 1} f (s) ds, (t > 0, R (υ) > 0) .

The fractional generalization of the standard kinetic Eq. (8) is given by Haubold and Mathai (2000) as follows:

(9)

N (t) - N_{0} = - c_{0}^{υ} D_{t}^{- 1} N (t)

and obtained the solution of (9) as follows:

(10)

N (t) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{Γ (υ k + 1)} {(ct)}^{υ k} .

Further, Saxena and Kalla (2008) considered the following fractional kinetic equation:

(11)

N (t) - N_{0} f (t) = - c_{0}^{υ} D_{t}^{- υ} N (t) (R (υ) > 0),

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 L (0, \infty

By applying the Laplace transform to (11) (see Kumar et al., 2015),

(12)

L \{N (t); p\} = N_{0} \frac{F (p)}{1 + c^{υ} p^{- υ}} = N_{0} (\sum_{n = 0}^{\infty} {(- c^{υ})}^{n} p^{- n υ}) F (p)

(n \in N_{0}, |\frac{c}{p}| < 1),

where the Laplace transform (Spiegel, 1965) is defined by

(13)

F (p) = L \{f (t); p\} = \int_{0}^{\infty} e^{- pt} f (t) dt (R (p) > 0) .

The object of this paper is to derive the solution of the fractional kinetic equation involving generalized Struve function of the first kind. The results obtained in terms of generalized Struve function of the first kind are rather general in nature and can easily construct various known and new fractional kinetic equations.

2 Solution of generalized fractional kinetic equations

In this section, we will investigate the solution of the generalized fractional kinetic equations by considering the generalized Struve function of first kind. The results are as follows.

Theorem 1

If $d > 0, υ > 0, c, b, l, t \in C$ and $R (l) > - 1$ , then the solution of the equation

(14)

N (t) - N_{0} H_{l, b, c} (t) = - d^{υ}_{0} D_{t}^{- υ} N (t)

is given by the following formula

(15)

N (t) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} Γ (2 k + l + 2)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} {(\frac{t}{2})}^{2 k + l + 1} E_{v, 2 k + l + 2} (- d^{υ} t^{υ})

where the generalized Mittag-Leffler function

E_{α, β} (x)

is given by Mittag-Leffler (1905)

E_{α, β} (x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{Γ (α n + β)} .

Proof

The Laplace transform of the Riemann–Liouville fractional integral operator is given by Erdélyi et al. (1954), Srivastava and Saxena (2001)

(16)

L \{_{0} D_{t}^{- υ} f (t); p\} = p^{- υ} F (p)

where

F (p)

is defined in (13). Now, applying the Laplace transform to the both sides of (14), gives

L \{N (t); p\} = N_{0} L \{H_{l, b, c} (t); p\} - d^{υ} L \{_{0} D_{t}^{- υ} N (t); p\}

N (p) = N_{0} (\int_{0}^{\infty} e^{- pt} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k}}{Γ (l + 1 + \frac{b}{2} + k) Γ (k + \frac{3}{2})} {(\frac{t}{2})}^{2 k + l + 1} dt) - d^{υ} p^{- υ} N (p)

\begin{matrix} N (p) + d^{υ} p^{- υ} N (p) & = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(2)}^{- (2 k + l + 1)}}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \int_{0}^{\infty} e^{- pt} t^{2 k + l + 1} dt \\ = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(2)}^{- (2 k + l + 1)}}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \frac{Γ (2 k + l + 2)}{p^{2 k + l + 2}} \end{matrix}

(17)

N (p) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(2)}^{- (2 k + l + 1)} Γ (2 k + l + 2)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \{p^{- (2 k + l + 2)} \sum_{r = 0}^{\infty} {[- {(\frac{p}{d})}^{- υ}]}^{r}\} .

Taking Laplace inverse of (17), and by using $L^{- 1} \{p^{- υ}; t\} = \frac{t^{υ - 1}}{Γ (υ)}, R (υ) > 0$ we have $L^{- 1} \{N (p)\} = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(2)}^{- (2 k + l + 1)} Γ (2 k + l + 2)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} L^{- 1} \{\sum_{r = 0}^{\infty} {(- 1)}^{r} d^{υ r} p^{- (2 k + l + 2 + υ r)}\}$ $\begin{matrix} N (t) & = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(2)}^{- (2 k + l + 1)} Γ (2 k + l + 2)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \\ \times \{\sum_{r = 0}^{\infty} {(- 1)}^{r} d^{υ r} \frac{t^{(2 k + l + 1 + υ r)}}{Γ (υ r + 2 k + l + 2)}\} \\ = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(2)}^{- (2 k + l + 1)} Γ (2 k + l + 2)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} t^{2 k + l + 1} \\ \times \{\sum_{r = 0}^{\infty} {(- 1)}^{r} d^{υ r} \frac{t^{υ r}}{Γ (υ r + 2 k + l + 2)}\} \\ = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} Γ (2 k + l + 2)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} {(\frac{t}{2})}^{2 k + l + 1} \\ \times \{\sum_{r = 0}^{\infty} {(- 1)}^{r} d^{υ r} \frac{t^{υ r}}{Γ (υ r + 2 k + l + 2)}\} \end{matrix}$ $N (t) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} Γ (2 k + l + 2)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} {(\frac{t}{2})}^{2 k + l + 1} E_{υ, 2 k + l + 2} (- d^{υ} t^{υ}) .$ □

Theorem 2

If $d > 0, υ > 0, c, b, l, t \in C$ and $R (l) > - 1$ , then the solution of the equation

(18)

N (t) - N_{0} H_{l, b, c} (d^{υ} t^{υ}) = - d_{0}^{υ} D_{t}^{- υ} N (t)

is given by the following formula

(19)

N (t) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} Γ (2 k + υ l + 2)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} {(\frac{d^{υ} t^{υ}}{2})}^{2 k + l + 1} E_{v, 2 k υ + υ l + 2} (- d^{υ} t^{υ}),

where

E_{v, 2 k υ + l + 2} (.)

is the generalized Mittag-Leffler function (Mittag-Leffler, 1905).

Proof

The Laplace transform of the Riemann–Liouville fractional integral operator is given by Erdélyi et al. (1954)

(20)

L \{_{0} D_{t}^{- υ} f (t); p\} = p^{- υ} F (p)

where

F (p)

is defined in (13). Now, applying the Laplace transform to the both sides of (18), gives

(21)

L \{N (t); p\} = N_{0} L \{H_{l, b, c} (d^{υ} t^{υ}); p\} - d^{υ} L \{_{0} D_{t}^{- υ} N (t); p\}

N (p) = N_{0} (\int_{0}^{\infty} e^{- pt} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k}}{Γ (l + 1 + \frac{b}{2} + k) Γ (k + \frac{3}{2})} {(\frac{d^{υ} t^{υ}}{2})}^{2 k + l + 1} dt) - d^{υ} p^{- υ} N (p)

\begin{matrix} N (p) + d^{υ} p^{- υ} N (p) & = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1}}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \int_{0}^{\infty} e^{- pt} t^{2 k υ + υ l + υ} dt \\ = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1}}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \frac{Γ (2 k υ + υ l + υ + 1)}{p^{2 k υ + υ l + υ + 1}} \end{matrix}

(22)

N (p) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \{p^{- (2 k υ + υ l + υ + 1)} \sum_{r = 0}^{\infty} {[- {(\frac{p}{d})}^{- υ}]}^{r}\} .

Taking Laplace inverse of (22), and by using $L^{- 1} \{p^{- υ}; t\} = \frac{t^{υ - 1}}{Γ (υ)}, R (υ) > 0$ we have $L^{- 1} \{N (p)\} = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} L^{- 1} \{\sum_{r = 0}^{\infty} {(- 1)}^{r} d^{υ r} p^{- (2 k υ + υ l + υ + υ r + 1)}\}$ $\begin{matrix} N (t) & = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \\ \times \{\sum_{r = 0}^{\infty} {(- 1)}^{r} d^{υ r} \frac{t^{υ (2 k + l + r + 1)}}{Γ (2 k υ + υ l + υ r + υ + 1)}\} \\ = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} t^{υ (2 k + l + 1)} \\ \times \{\sum_{r = 0}^{\infty} {(- 1)}^{r} d^{υ r} \frac{t^{υ r}}{Γ (2 k υ + υ l + υ r + υ + 1)}\} \\ = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} {(\frac{d^{υ}}{2})}^{2 k + l + 1} t^{υ (2 k + l + 1)} \\ \times \{\sum_{r = 0}^{\infty} {(- 1)}^{r} d^{υ r} \frac{t^{υ r}}{Γ (2 k υ + υ l + υ r + υ + 1)}\} \end{matrix}$ $N (t) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} {(\frac{d^{υ}}{2})}^{2 k + l + 1} t^{υ (2 k + l + 1)} E_{v, (2 k + l + 1) υ + 1} (- d^{υ} t^{υ})$

This completes the proof of Theorem 2. □

Theorem 3

If $d > 0, υ > 0, c, b, l, t \in C, a \neq d$ and $R (l) > - 1$ , then the solution of the equation

(23)

N (t) - N_{0} H_{l, b, c} (d^{υ} t^{υ}) = - a^{υ}_{0} D_{t}^{- υ} N (t)

is given by the following formula:

(24)

N (t) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} {(\frac{d^{υ}}{2})}^{2 k + l + 1} t^{υ (2 k + l + 1)} E_{v, (2 k + l + 1) υ + 1} (- a^{υ} t^{υ}) .

Proof

Applying Laplace transform to the both side of (23) we get $L_{0} \{N (t); p\} = N_{0} L \{H_{l, b, c} (d^{υ} t^{υ}); p\} - a^{υ} L \{_{0} D_{t}^{- υ} N (t); p\}$ $N (p) = N_{0} (\int_{0}^{\infty} e^{- pt} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k}}{Γ (l + 1 + \frac{b}{2} + k) Γ (k + \frac{3}{2})} {(\frac{d^{υ} t^{υ}}{2})}^{2 k + l + 1} dt) - d^{υ} p^{- υ} N (p)$ $\begin{matrix} N (p) + a^{υ} p^{- υ} N (p) & = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1}}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \int_{0}^{\infty} e^{- pt} t^{2 k υ + υ l + υ} dt \\ = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1}}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \frac{Γ (2 k υ + υ l + υ + 1)}{p^{2 k υ + υ l + υ + 1}} \end{matrix}$

(25)

N (p) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \{p^{- (2 k υ + υ l + υ + 1)} \sum_{r = 0}^{\infty} {[- {(\frac{p}{a})}^{- υ}]}^{r}\} .

Taking Laplace inverse of (25), and by using $L^{- 1} \{p^{- υ}; t\} = \frac{t^{υ - 1}}{Γ (υ)}, R (υ) > 0$ we have $L^{- 1} \{N (p)\} = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} L^{- 1} \{\sum_{r = 0}^{\infty} {(- 1)}^{r} a^{υ r} p^{- (2 k υ + υ l + υ + υ r + 1)}\}$ $\begin{matrix} N (t) & = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} \\ \times \{\sum_{r = 0}^{\infty} {(- 1)}^{r} a^{υ r} \frac{t^{υ (2 k + l + r + 1)}}{Γ (2 k υ + υ l + υ r + 1)}\} \\ = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} {(\frac{d^{υ}}{2})}^{2 k + l + 1} Γ (2 k υ + υ l + 2)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} t^{υ (2 k + l + 1)} \\ \times \{\sum_{r = 0}^{\infty} {(- 1)}^{r} a^{υ r} \frac{t^{υ r}}{Γ (2 k υ + υ l + υ r + υ + 1)}\} \\ = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} {(\frac{d^{υ}}{2})}^{2 k + l + 1} t^{υ (2 k + l + 1)} \\ \times \{\sum_{r = 0}^{\infty} {(- 1)}^{r} a^{υ r} \frac{t^{υ r}}{Γ (2 k υ + υ l + υ r + υ + 1)}\} \end{matrix}$ $N (t) = N_{0} \sum_{k = 0}^{\infty} \frac{{(- c)}^{k} Γ (2 k υ + υ l + υ + 1)}{Γ (l + k + \frac{b}{2} + 1) Γ (k + \frac{3}{2})} {(\frac{d^{υ}}{2})}^{2 k + l + 1} t^{υ (2 k + l + 1)} E_{v, (2 k + l + 1) υ + 1} (- a^{υ} t^{υ}) .$

3 Conclusion

In this work we give a new fractional generalization of the standard kinetic equation and derived solutions for the same. From the close relationship of the generalized Struve function of the first kind $H_{p} (z)$ with many special functions, we can easily construct various known and new fractional kinetic equations.

Acknowledgements

The authors would like to express their appreciation to the referees for their valuable suggestions which helped to better presentation of this paper. The author K.S. Nisar would like to thank Deanship of Scientific research, Prince Sattam bin Abdulaziz University, Saudi Arabia for financial support under the project 2014/01/2152.

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Srivastava H.M., Saxena R.K., . Operators of fractional integration and their applications. Appl. Math. Comput.. 2001;118:1-52.
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Introduction

Solution of generalized fractional kinetic equations

Conclusion

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