Solution of Fractional Differential Equation Systems and Computation of Matrix Mittag–Leffler Functions

Duan, Jun‐Sheng; Lian, Chen

doi:10.3390/sym10100503

Cited by 27 publications

(14 citation statements)

References 36 publications

(49 reference statements)

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“…1. The first formalism presented in this review was related to the fractional walker, this method has a great focus of investigation in the present day, because daily new techniques, theorems and theories continue to be developed by the mathematicians who investigate the fractional calculus [112,113,114,115,116,117,118,119,120,121,122,123,124]. With all this progress of the fractional calculus, physics advances together, since it allows the modelling of a series of interesting problems in physics, such as diffusion equation with tempered derivatives [125,126,127,128,129,130], memory systems [131,132,133,134,135], non-homogeneous systems [60,136,137,138], etc [139,140,141].…”

Section: Brief Discussion and Some Considerationsmentioning

confidence: 99%

Analytic approaches of the anomalous diffusion: A review

Santos

2019

Chaos, Solitons & Fractals

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This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken nowadays. The discussion is started by a brief historical report that starts with the studies of thermal machines and combines in theories such as the statistical mechanics of Boltzmann-Gibbs and the Brownian Movement. In this scenario, in the twentieth century, a series of experiments were reported that were not described by the usual model of diffusion. Such experiments paved the way for deeper investigation into anomalous diffusion, i.e. (∆x) 2 ∼ t α to α = 1. These processes are very abundant in physics, and the mechanisms for them to occur are diverse. For this reason, there are many possible ways of modelling the diffusive processes. This article discusses three analytic approaches to investigate anomalous diffusion: fractional diffusion equation, nonlinear diffusion equation and Langevin equation in the presence of fractional, coloured or multiplicative noises. All these formalisms presented different degrees of complexity and for this reason, they have succeeded in describing anomalous diffusion phenomena.

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Section: Brief Discussion and Some Considerationsmentioning

confidence: 99%

Analytic approaches of the anomalous diffusion: A review

Santos

2019

Chaos, Solitons & Fractals

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“…where 𝐸 2,2 (−𝜔 0 2 𝑡 2 ) is the Mittag-Leffler function which has the general form (Duan and Chen, 2018):…”

Section: The Helical Trajectory Equationsmentioning

confidence: 99%

Fractional solution of helical motion of a charged particle under the influence of Lorentz force

Altarawneh¹

2022

Int. j. adv. appl. sci.

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In this study, a generalized solution for the helical motion of a charged particle in uniform electric and magnetic fields is obtained using a powerful fractional derivative approach. Using this approach, the differential equations that describe the helical motion of a charged particle in the fields were obtained. The solution for the fractional differential equations is presented in great detail in terms of a series solution using the Mittag-Leffler function. The Laplace transform technique was used to solve the differential equations in the regular form and in the fractional form (with fractional parameter γ). Two and three-dimensional plots were presented for the trajectory of the particle before and after introducing the fractional operator for different values of γ. Features of delay in the motion and dissipation in the medium have been observed in the fractional solution too. The importance of our work stems from the two- and three-dimensional visualization of the obtained generalized helical trajectories that can be applied to similar types of motions in nature and the universe.

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“…Garrappa and N. Popolizio [11] investigated the computation of the Mittag-Leffler function and evaluated a three-parameter Mittag-Leffler function using the inverse Laplace transform method; the results have been presented in [12]. Duan and his colleague [13,14] employed different methods, such as the inverse Laplace transform method, Jordan canonical matrix method, and minimal polynomial method, for solving a system of FDEs, wherein the solutions were expressed in terms of the matrix Mittag-Leffler function.…”

Section: Introductionmentioning

confidence: 99%

Solutions of a System of Fuzzy Fractional Differential Equations in Terms of the Matrix Mittag-Leffler Functions

Padmapriya¹,

Kaliyappan²

2022

IJFIS

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The matrix Mittag-Leffler functions play a crucial role in several applications related to systems with fractional dynamics. These functions represent a generalization for fractional-order systems of the matrix exponential function involved in integer-order systems. Computational techniques for evaluating the matrix Mittag-Leffler functions are therefore of particular importance. In this study, a fuzzy system of differential equations with fractional derivatives was solved in terms of the matrix Mittag-Leffler functions. The matrix Mittag-Leffler function was evaluated based on the Jordan canonical form and the minimal polynomial of the matrix.

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Solution of Fractional Differential Equation Systems and Computation of Matrix Mittag–Leffler Functions

Cited by 27 publications

References 36 publications

Analytic approaches of the anomalous diffusion: A review

Analytic approaches of the anomalous diffusion: A review

Fractional solution of helical motion of a charged particle under the influence of Lorentz force

Solutions of a System of Fuzzy Fractional Differential Equations in Terms of the Matrix Mittag-Leffler Functions

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