On Fractional Mittag–Leffler Operators

Ansari, Alireza; Darani, Mohammadreza Ahmadi; Moradi, Mohammad

doi:10.1016/s0034-4877(13)60017-8

Cited by 11 publications

(11 citation statements)

References 7 publications

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“…used to describe F (− ln s) remains valid for s ∈ C. Based on this argument, solution of (41) can be obtained by inverting F (G(s)) back to time domain using Theorem 1. To this aim firstly, Faa di Bruno's formula (7) may be used to obtain the Maclaurin expansion coefficients of F (s) as follows:…”

Section: Fractional Distributed Order Relaxation Processesmentioning

confidence: 99%

“…In case r = 1, it is obviously followed that lim t→0 þ F lnt ð Þ ¼ 1. Otherwise, Faa di Bruno's formula (7) helps in calculation of the limit by providing the following expression…”

Section: Fractional Distributed Order Relaxation Processesmentioning

confidence: 99%

“…The above‐mentioned problem was visited by Duffy leading to the Schouten‐Vanderpol Theorem asserting the following relation

L_{s \to t}^{- 1} ({}, F ((), G ((), s))) = \int_{0}^{+ \infty} f ((), τ) L_{s \to t}^{- 1} ({}, exp ((), - G ((), s) τ)) italicdτ

where

L_{s \to t}^{- 1} ({}, .)

denotes inverse Laplace operator, provided that F ( s ) and G ( s ) are analytic in the half plane Re{ s } > s 0 . Schouten‐Vanderpol Theorem for Laplace and Mellin transforms appeared in different problems contexts (for some samples, see Ansari and Masomi, Ansari and Sheikhani, Ansari et al, and Ansari) indicating the existing demand for a reliable technique to treat the aforementioned problem. In the present paper, this problem is tackled by means of what is called in this paper Bell polynomial series.…”

Section: Introductionmentioning

confidence: 99%

“…Proof. Substituting the higher order derivative terms F (r − 1) (ln t) in (44) with their expressions given by Faa di Bruno's formula (7) yields…”

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confidence: 99%

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The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

Taghavian

2019

Math Methods in App Sciences

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This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact solution of fractional distributed order relaxation processes as well as time‐domain impulse response of fractional distributed order operators in new series forms. Evaluation of the obtained series expansions through computer simulations is also given. The results are then used to present novel series expansions for some special functions, including the one‐parameter Mittag‐Leffler function. It is shown that truncating these series expansions when combined with using potential partition polynomials provides efficient approximations for these functions. At the end, the results are shown to be also useful in studying asymptotical behavior of partial Bell polynomials. Numerical simulations as well as analytical examples are provided to verify the results of this paper.

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Section: Fractional Distributed Order Relaxation Processesmentioning

confidence: 99%

“…In case r = 1, it is obviously followed that lim t→0 þ F lnt ð Þ ¼ 1. Otherwise, Faa di Bruno's formula (7) helps in calculation of the limit by providing the following expression…”

Section: Fractional Distributed Order Relaxation Processesmentioning

confidence: 99%

“…The above‐mentioned problem was visited by Duffy leading to the Schouten‐Vanderpol Theorem asserting the following relation

L_{s \to t}^{- 1} ({}, F ((), G ((), s))) = \int_{0}^{+ \infty} f ((), τ) L_{s \to t}^{- 1} ({}, exp ((), - G ((), s) τ)) italicdτ

where

L_{s \to t}^{- 1} ({}, .)

Section: Introductionmentioning

confidence: 99%

“…Proof. Substituting the higher order derivative terms F (r − 1) (ln t) in (44) with their expressions given by Faa di Bruno's formula (7) yields…”

mentioning

confidence: 99%

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The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

Taghavian

2019

Math Methods in App Sciences

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show abstract

“…Most of these works have been surveyed using the operational calculus of this function such as the Laplace and Mellin transforms. See for example [3,4,5,6], [9], [11,12] and [15]. Therefore for the importance and significance of this function, in this paper, we get the Fourier transform of the M-Wright and its associate functions.…”

mentioning

confidence: 99%