Abstract
The generalized fractional integrations of the generalized Mittag-Leffler type function (GMLTF) are established in this paper. The results derived in this paper generalize many results available in the literature and are capable of generating several applications in the theory of special functions. The solutions of a generalized fractional kinetic equation using the Sumudu transform is also derived and studied as an application of the GMLTF.
1. Introduction
The Pochhammer symbol (ϖ)n is defined by (for ϖ ∈ ℂ)[see ([1], p. 2 and p. 5)]:
The familiar generalized hypergeometric function pFq is defined as follows (see [2]):
where (ϖj)n and (χj)n given in (1) and χi can not be a negative integer or zero. Here p or q or both are permitted to be zero. For all finite x, the series (2) is absolutely convergent if p ≤ q and for |x| < 1 if p = q + 1. When p > q + 1, then the series diverge for x ≠ 0 and the series does not terminate.
In particular, if p = 2 and q = 1, (2) reduces to the Gaussian hypergeometric function
The function rΨs(z) is the generalized Wright hypergeometric series which is given by
where 𝔞i, 𝔟j ∈ ℂ, and real ϖi, χj ∈ ℝ (i = 1, 2, …, r; j = 1, 2, …, s). The asymptotic behavior of (4) for large values of argument of z ∈ ℂ were mentioned in [3, 4] (also, see [5, 6]).
To proceed our study, we need the definitions of the Mittag-Leffler functions (MLF) denoted by Eϖ(z) (see [7]) and Eϖ, χ(x) [8], respectively:
Many more generalizations and extensions of MLF widely studied recently [9, 10]. Also, the MLF performs an important role in physics and engineering problems. The derivations of physical problems of exponential nature could be governed by the physical laws through the MLF (power-law) [11–13].
Very recently, Nisar [14] defined a generalized Mittag-Leffler type function which is defined as follows
For ρ, σ, ς ∈ ℂ, ℜ(κ) > 0, δ ≠ 0, −1, −2, ⋯ , (κ)s and (ω)s denotes the Pochammer symbol.
By assuming particular values for various parameters in (7), we get many of the popular functions in the literature. For example, gives the K− function [15] and turns to [16]. Also, reduces to [17] and gives the Mittag-Leffler function Eρ, σ(z) [8]. Similar way, turns to the Mittag-Leffler functions Eρ(z) [7]. For more details one can be referred to Nisar [14].
2. Generalized Fractional Integration of GMLTF
Fractional calculus is one of the prominent branch of applied mathematics that deals with non-integer order derivatives and integrals (including complex orders), and their applications in almost all disciplines of science and engineering [18–22]. In this line, the use of special functions in connection with fractional calculus also studied widely [23–27]. For the basics of fractional calculus and its related literature, interesting readers can be referred to as Kiryakova [28], Miller and Ross [29], and Srivastava et al. [30]. In this paper, we studied the generalized fractional calculus of more generalized function given in (7). The generalized fractional integral operators (FIOs) involving the Appell functions F3 are given for ϖ, ϖ′, τ, τ′, ϵ ∈ ℂ with ℜ(ϵ) > 0 and x ∈ ℝ+as follows:
and
The integral operators of the types (8) and (9) have been introduced by Marichev [31] and later extended and studied by Saigo and Maeda [32]. Recently, many researchers (see [33–35]) have studied the image formulas for MSM FIOs involving various special functions.
The corresponding fractional differential operators (FDOs) have their respective forms:
and
Here, we recall the following results (see [32, 36]).
LEMMA 2.1. Let ϖ, ϖ′, τ, τ′, ϵ, σ ∈ ℂ be such that ℜ(ϵ) > 0 and
Then there exists the relation
LEMMA 2.2. Let ϖ, ϖ′, τ, τ′, ϵ, σ ∈ ℂ such that ℜ(ϵ) > 0 and
Then
The main aim of this paper is to apply the generalized operators of fractional calculus for the GMLTF in order to get certain new image formulas.
2.1. Sumudu Transform
The Sumudu transform is widely used to solve various type of problems in science and engineering and it is introduced by Watugala (see [37, 38]). The details of Sumudu transforms, properties, and its applications the interesting readers can be refer to Asiru [39], Belgacem et al. [40], and Bulut et al. [41].
The Sumudu transform over the set functions
is defined by
The main aim of this study is to establish the generalized fractional calculus operators and the generalized FKEs involving GMLTF.
Theorem 2.1. Let η, η′, χ, χ′, ϵ, τ, ϖ, λ, γ ∈ ℂ, ℜ(κ) > 0, δ ≠ 0, −1, −2, ⋯ , such that ℜ(τ) > max{0, ℜ(η + η′ − χ − ϵ), ℜ(η′ − χ′)}. Then
Proof. Applying the definition (7) on the left hand side (l.h.s) of Theorem 2.1,
Changing the order of integration and summation gives
Applying Lemma 2.1, we get
Using Γ(x + κ) = (x)kΓ(x), we have
In view of (4), we reached the required result. □
Theorem 2.2. Let η, η′, χ, χ′, ϵ, τ, ϖ, λ, γ ∈ ℂ, ℜ(κ) > 0, δ ≠ 0, − 1, −2, ⋯ , such that ℜ(τ) > max{ℜ(χ), ℜ(−η − η′ + ϵ), ℜ(−η − χ′ + ϵ)}. Then
Proof. Applying the definition (7) on the left hand side (l.h.s) of Theorem 2.2,
Changing the order of integration and summation gives
Applying Lemma 2.2, we get
Using Γ(x + r) = (x)rΓ(x), we have
In view of (4), we reached the required result. □
The following corollaries can derive immediately from Theorems 2.1 and 1.2 with the help of Pochhammer symbol
Corollary 2.1. Let δ = λ = 1 in Theorem 2.1, we get
Corollary 2.2. If δ = λ = 1 in Theorem 2.1, then
In the next section, we derived the generalized FKEs and we consider the Sumudu transform methodology to achieve the results.
3. Generalized Fractional Kinetic Equations Involving GMLTF
The generalized fractional kinetic equations (FKEs) involving the GMLTF with the Sumudu transform is derived in this section. The FKEs are studied widely in many papers [42–45].
Let 𝔎 = (𝔎t) be the arbitrary reaction defined by a time-dependent quantity. The destruction 𝔡 and production 𝔭 depend on the quantity 𝔎 itself: 𝔡 = 𝔡(𝔎) or 𝔭 = 𝔭(𝔎) [see [42]]. The fractional differential equation can be expressed by
where 𝔎t described by (see, [42]). A special case of (15) is
with 𝔎i(t = 0) = 𝔎0, 𝔠i > 0 and the solution of (16) is
Performing the integration of (16) leads to
where is the particular case of Riemann–Liouville (R-L) integral operator and 𝔠 is a constant. The fractional form of (18) is (see [42])
where is given by
Theorem 3.1. For ϖ, λ, γ ∈ ℂ, δ ≠ 0, −1, −2, ⋯ , d > 0, ϵ > 0 then the solution of
is given by
Proof. The Sumudu transform (ST) of the R-L fractional operator is
where G(u) is defined in (14). Now, applying the ST on the both sides of (21) and using (7) and (23), we have
which gives
which implies that
After some simple calculation, we get
The inverse ST of (27) and using the formula , ℜ(ϵ) > 0 gives
In view of the Mittag-Leffler function definition, we arrived the needful result.
Theorem 3.2. For ϖ, λ, γ ∈ ℂ, δ ≠ 0, −1, −2, ⋯ , d > 0, ϵ > 0 then the equation
have the following solution
Proof. Applying the Sumudu transform on the both sides of (29)
and using (7) and (23), we get
which gives
which can be simplified as
Taking the Sumudu inverse of (34) and using , we get
In view of the definition of the Mittag-Leffler function, we get the required result. □
Theorem 3.3. For ϖ, λ, γ ∈ ℂ, δ ≠ 0, −1, −2, ⋯ , d > 0, ϵ > 0 and a ≠ d then the equation
have the following solution
Proof. Applying the Sumudu transform on the both sides of (36)
and using (7) and (23), we get
which gives
which can be simplified as
Taking the Sumudu inverse of (41) and using , we get
In view of the definition of the Mittag-Leffler function, we get the required result. □
If we take δ = 1 in Theorem 1.3, we get the generalized FKE involving K − function as follows:
Corollary 3.1. For ϖ, λ, γ ∈ ℂ, δ ≠ 0, −1, −2, ⋯ , d > 0, ϵ > 0 then
is given by
If we take δ = 1, p = q = 0 in Theorem 3.1, we have the generalized FKE involving the Prabhakar function:
Corollary 3.2. For ϖ, λ, γ ∈ ℂ, δ ≠ 0, −1, −2, ⋯ , d > 0, ϵ > 0 then
is given by
If we choose δ = 1, p = q = 0 and γ = 1 in Theorem 3.1, then the generalized FKE involving the Wiman function:
Corollary 3.3. For ϖ, λ ∈ ℂ, δ ≠ 0, −1, −2, ⋯ , d > 0, ϵ > 0 then
is given by
If we choose δ = 1, p = q = 0 and γ = 1 in Theorem 3.1, then we get the generalized FKE involving the Mittag-Leffler function:
Corollary 3.4. For ϖ ∈ ℂ, δ ≠ 0, −1, −2, ⋯ , d > 0, ϵ > 0 then
is given by
Remark 3.1. By choosing the suitable parameters in Theorems 3.2 and 3.3, one can derive the generalized FKEs of GMLTF as similar as above corollaries.
4. Conclusion
The generalized fractional integrations of the generalized Mittag-Leffler type function is studied in this paper. The obtained results are expressed in terms of the generalized Wright hypergeometric function and generalized hypergeometric functions. To show the potential application of GMLTF, the solutions of fractional kinetic equations are derived with the help of Sumudu transform. The results obtained in this study have significant importance as the solution of the equations are general and can derive many new and known solutions of FKEs involving various type of special functions.
Statements
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Author contributions
The author confirms being the sole contributor of this work and has approved it for publication.
Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
The Mittag-Leffler type function, fractional calculus, fractional kinetic equations; MSC [2000]: 33C05; 33C20; 33E12; 26A33; 44A20; 35Qxx, riemam-liouville derivative, special function
Citation
Nisar KS (2020) Generalized Mittag-Leffler Type Function: Fractional Integrations and Application to Fractional Kinetic Equations. Front. Phys. 8:33. doi: 10.3389/fphy.2020.00033
Received
21 December 2019
Accepted
05 February 2020
Published
28 February 2020
Volume
8 - 2020
Edited by
Devendra Kumar, University of Rajasthan, India
Reviewed by
Haci Mehmet Baskonus, Harran University, Turkey; Yudhveer Singh, Amity University Jaipur, India
Updates
Copyright
© 2020 Nisar.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Kottakkaran Sooppy Nisar n.sooppy@psau.edu.sa
This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics
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