New identities for the Wright and the Mittag-Leffler functions using the Laplace transform

Ansari, Alireza; Sheikhani, A. H. Refahi

doi:10.1142/s1793557114500387

Cited by 21 publications

(14 citation statements)

References 7 publications

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“…Theorem 1.2 (The Titchmarsh theorem) [7,8] Let F(s) be an analytic function which has a branch cut on the negative real semiaxis, such that F(s) has the following properties:…”

Section: (The Schouten-vanderpol Theorem) [7] Let F(s) and (S) Be Anamentioning

confidence: 99%

Some results involving integral transforms of Airy functions

Ansari

Masomi

2015

Integral Transforms and Special Functions

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“…Theorem 1.2 (The Titchmarsh theorem) [7,8] Let F(s) be an analytic function which has a branch cut on the negative real semiaxis, such that F(s) has the following properties:…”

Section: (The Schouten-vanderpol Theorem) [7] Let F(s) and (S) Be Anamentioning

confidence: 99%

Some results involving integral transforms of Airy functions

Ansari

Masomi

2015

Integral Transforms and Special Functions

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“…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%

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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“…We can see the conditions of existence and uniqueness of solutions to the FDE in [9]. Moreover several numerical methods have been used to approximate the solution of fractional differential equations, such as finite difference method [11], collocation [12] method and other methods [13,5,18]. Any time function can be synthesized completely to a tolerable degree of accuracy by using set of orthogonal functions.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of fractional differential equation by wavelets and hybrid functions

Sheikhani

Mashoof

2018

B Soc Paran Mat

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In this paper, we introduce methods based on operational matrix of fractional order integration for solving a typical n-term non-homogeneous fractional differential equation (FDE). We use Block pulse wavelets matrix of fractional order integration where a fractional derivative is defined in the Caputo sense. Also we consider Hybrid of Block-pulse functions and shifted Legendre polynomials to approximate functions. By the use of these methods we translate an FDE to an algebraic linear equations which can be solved. Methods have been tested by some numerical examples.

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New identities for the Wright and the Mittag-Leffler functions using the Laplace transform

Abstract: In this paper, we state three theorems for the inverse Laplace transform and using these theorems we obtain new integral identities involving the products of the Wright and Mittag-Leffler functions. The relationships of these integral identities with the Stieltjes transform are also given.

Cited by 21 publications

References 7 publications

Some results involving integral transforms of Airy functions

Some results involving integral transforms of Airy functions

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

Numerical solution of fractional differential equation by wavelets and hybrid functions

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