Fractional Integrations of a Generalized Mittag-Leffler Type Function and Its Application

Nisar, Kottakkaran Sooppy

doi:10.3390/math7121230

Cited by 4 publications

(12 citation statements)

References 43 publications

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“…Similar way, 0 E ρ,1;1 0;1 (z) turns to the Mittag-Leffler functions E ρ (z) [7]. For more details one can be referred to Nisar [14].…”

Section: Introductionmentioning

confidence: 90%

“…where G (u) is defined in (14). Now, applying the ST on the both sides of (21) and using (7) and (23), we have…”

Section: Generalized Fractional Kinetic Equations Involving Gmltfmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Mittag-Leffler Type Function: Fractional Integrations and Application to Fractional Kinetic Equations

Nisar

2020

Front. Phys.

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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.

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“…Similar way, 0 E ρ,1;1 0;1 (z) turns to the Mittag-Leffler functions E ρ (z) [7]. For more details one can be referred to Nisar [14].…”

Section: Introductionmentioning

confidence: 90%

“…where G (u) is defined in (14). Now, applying the ST on the both sides of (21) and using (7) and (23), we have…”

Section: Generalized Fractional Kinetic Equations Involving Gmltfmentioning

confidence: 99%

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Generalized Mittag-Leffler Type Function: Fractional Integrations and Application to Fractional Kinetic Equations

Nisar

2020

Front. Phys.

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“…

normalΩ ((), t) - Ω_{0} = - c_{ν} I_{0 +}^{ν} normalΩ ((), t),

where

I_{0 +}^{ν}, ν > 0

is known as the Riemann–Liouville fractional operator. For the interest of readers, more comprehensive discussions and developments on the fractional kinetic equation can also be found in other studies 4–17 . Throughout in the article,

ℜ

denotes the real part of a complex number,

ℝ

and

ℂ

are used to symbolize the set of real and complex numbers respectively.…”

Section: Introductionmentioning

confidence: 99%

A new representation of the extended k‐gamma function with applications

Tassaddiq

2021

Math Methods in App Sciences

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In this study, a new series representation of the extended k‐gamma function is investigated. A new representation expresses this function as an infinite sum of delta functions. Several researchers have examined this family of functions, but no study has been conducted dealing with the fractional kinetic equation. A new representation proves useful to solve the fractional kinetic equation involving an extended k‐gamma function. Particular cases involving the original gamma function are discussed as corollaries. Such an application of the gamma function is not possible by using known representations; nevertheless, using this representation, new fractional transform formulae are evaluated. A new representation is also worthwhile to compute new integrals of products of the extended k‐gamma functions, proving to be reliable with known identities. Several distributional properties of the extended k‐gamma function are also discussed using Fourier transform.

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“…Furthermore, derivations of physical phenomena of exponential nature could be determined by the physical laws via the M-LTF (power-law), (Bhatter, Mathur, Kumar, Nisar, & Singh, 2020;Djida, Mophou, &Area, 2019 andSaqib, Khan, &Shafie, 2019). Due to successful diverse applications for M-LTFs, correlated with FC, in physics and mathematic allied problems, several researchers prompted a lot of attention to the behaviour of M-LTFs and extended their outcomes to the complex domain, (for instance, see Al-Janaby, 2018;Al-Janaby & Ahmad, 2018;Al-Janaby &Darus, 2019 andNisar, 2019).…”

Section: Introductionmentioning

confidence: 99%

Some analytical merits of Kummer-Type function associated with Mittag-Leffler parameters

Ghanim

Al‐Janaby

2021

Arab Journal of Basic and Applied Sciences

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As of late, the study of Fractional Calculus (FC) and Special Functions (SFs) has been interestingly prompted in various realms of mathematics, engineering and sciences. This is due to the considerable demonstrated potential of their applications. Among these SFs, Gamma function and Mittag-Leffler functions are the most renowned and distinguished. Numerous authors continue to study this line. The current analysis attempts to introduce and further examine new modifications of Gamma and Kummer function in terms of Mittag-Leffler functions, respectively. Several attributes and formulations of this new Kummer-type function that include integral representations, Beta transform, Laplace transform, derivative formulas, and recurrence relation are investigated. Furthermore, outcomes of Riemann-Liouville fractional integral and fractional derivative in relation to this newly established Kummer function are also investigated.

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Fractional Integrations of a Generalized Mittag-Leffler Type Function and Its Application

Cited by 4 publications

References 43 publications

Generalized Mittag-Leffler Type Function: Fractional Integrations and Application to Fractional Kinetic Equations

Generalized Mittag-Leffler Type Function: Fractional Integrations and Application to Fractional Kinetic Equations

A new representation of the extended k‐gamma function with applications

Some analytical merits of Kummer-Type function associated with Mittag-Leffler parameters

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