Laplace’s transform of fractional order via the Mittag–Leffler function and modified Riemann–Liouville derivative

Jumarie, Guy

doi:10.1016/j.aml.2009.05.011

Cited by 155 publications

(117 citation statements)

References 16 publications

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“…In the present article, we have recalled some properties of the Mittag-Leffler function and Mittag-Leffler logarithm function as described in [20]. Then we have presented fractional trigonometric function and the fractional Floquet system based on the modified Riemann-Liouville derivation.…”

Section: Resultsmentioning

confidence: 99%

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Analytical studies for linear periodic systems of fractional order

2016

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Section: Resultsmentioning

confidence: 99%

“…The idea of the fractional trigonometric functions has been stated by Jumarie [20] asserting that these functions are not periodic. Now, we introduce new fractional trigonometric functions which are periodic with the period 2p a % 2p.…”

Section: Fractional Trigonometric Functions and Mittagleffler Logaritmentioning

confidence: 99%

Analytical studies for linear periodic systems of fractional order

2016

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“…This fractional derivative was successfully implemented to fractional Laplace problems [35], fractional variational calculus [36], and probability calculus [37]. Jumarie's modified Riemann-Liouville derivative has many interesting properties: the˛order derivative of a constant is zero and it can be applied to both differentiable and nondifferentiable functions.…”

Section: Preliminariesmentioning

confidence: 99%

An analytic approach to a class of fractional differential‐difference equations of rational type via symbolic computation

Aslan

2013

Math Methods in App Sciences

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Communicated by S. G. GeorgievFractional derivatives are powerful tools in solving the problems of science and engineering. In this paper, an analytical algorithm for solving fractional differential-difference equations in the sense of Jumarie's modified Riemann-Liouville derivative has been described and demonstrated. The algorithm has been tested against time-fractional differentialdifference equations of rational type via symbolic computation. Three examples are given to elucidate the solution procedure. Our analyses lead to closed form exact solutions in terms of hyperbolic, trigonometric, and rational functions, which might be subject to some adequate physical interpretations in the future.

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“…Amongst many extensions of this formalism the ones which appear the most promising use the generalizations of the integral transforms to their fractional integral analogs. The clear exposition of this approach can be found in [19], while the applications are developed in [39] and [18].…”

Section: The Borel Transformmentioning

confidence: 99%

Operational Versus Umbral Methods and the Borel Transform

Dattoli

Palma

Sabia

et al. 2017

Int. J. Appl. Comput. Math

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Abstract. Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.

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Laplace’s transform of fractional order via the Mittag–Leffler function and modified Riemann–Liouville derivative

Cited by 155 publications

References 16 publications

Analytical studies for linear periodic systems of fractional order

Analytical studies for linear periodic systems of fractional order

An analytic approach to a class of fractional differential‐difference equations of rational type via symbolic computation

Operational Versus Umbral Methods and the Borel Transform

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