Solution to system of partial fractional differential equations using the fractional exponential operators

Ansari, Alireza; Sheikhani, A. H. Refahi; Najafi, H. Saberi

doi:10.1002/mma.1545

Cited by 23 publications

(11 citation statements)

References 10 publications

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“…More recently, another fractional version of (2) has been proposed in [35] (where the exponential is replaced by the Mittag-Leffler function). An integral representation for fractional exponential operators is derived in [2] from the Bromwich integral of the inverse Mellin transform. Finally a fractional definition Downloaded by [UNSW Library] at 19:46 10 August 2015 of the shift operator is formally given in [18] as the Laplace transform of the α-stable subordinator A α (t), t ≥ 0, that is,…”

Section: Beghinmentioning

confidence: 99%

Fractional Gamma and Gamma-Subordinated Processes

Beghin

2015

Stochastic Analysis and Applications

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We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order $\nu >0$. In the case $\nu >1$, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index $1/\nu $. For $\nu $ less than one an analogous result holds, with the subordinator replaced by the inverse. In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value. Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see i.e., \cite{VAN} and the references therein).\ud As a consequence of the previous results, the fractional generalizations of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding fractional differential equations are obtained. These processes are particularly relevant for a wide range of financial and technological applications

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Section: Beghinmentioning

confidence: 99%

Fractional Gamma and Gamma-Subordinated Processes

Beghin

2015

Stochastic Analysis and Applications

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“…The field of fractional differential equations has received attention and interest only in the past 20 years or so [2,4,3]. In recent years, studies concerning the application of the fractional differential equations in science has attracted more interest among scholars [9,5]; readers can refer to [7,8] for the theory and applications of fractional calculus in this regard. For instance, [10,15] formulated the motion of a rigid plate immersing in a Newtonian fluid.…”

Section: Introductionmentioning

confidence: 99%

A Collocation Method for Solving Fractional Order Linear System

Sheikhani

Mashoof²

2017

JIMS

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Abstract. In the current paper we propose a collocation method to achieve an algorithm for numerically solving of fractional order linear systems where a fractional derivative is defined in the Caputo form. We have used the Taylor collocation method for solving fractional differential equations; a collocation method which is based on taking the truncated Taylor expansions of the vector function's solution in the fractional order linear system and substituting their matrix forms into the system. Through using collocation points we have obtained a system of linear algebraic equation. The method has been tested by some numerical examples.Key words and Phrases: Fractional system, collocation method, numerical simulation.Abstrak. In the current paper we propose a collocation method to achieve an algorithm for numerically solving of fractional order linear systems where a fractional derivative is defined in the Caputo form. We have used the Taylor collocation method for solving fractional differential equations; a collocation method which is based on taking the truncated Taylor expansions of the vector function's solution in the fractional order linear system and substituting their matrix forms into the system. Through using collocation points we have obtained a system of linear algebraic equation. The method has been tested by some numerical examples.Kata kunci: Sistem fraksional, metode kolokasi, simulasi numerik.

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“…The generalization of dynamical equations using fractional derivatives proved to be useful and more accurate in mathematical modeling related to many interdisciplinary areas. Applications of fractional order differential equations include: electrochemistry [7], porous media [8] and so on [9][10][11]. It is worth noting that recently much attention has been paid to the distributed-order differential equations and their applications in engineering fields that both integer-order systems and fractional order systems are special cases of distributed-order systems.…”

Section: Introductionmentioning

confidence: 99%