Fractional Order Calculus: Basic Concepts and Engineering Applications

Gutiérrez, Ricardo Enrique; Rosário, João Maurício; Machado, J. A. Tenreiro

doi:10.1155/2010/375858

Cited by 223 publications

(123 citation statements)

References 27 publications

(1 reference statement)

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“…whereK − α j (k j a j ) is defined by (8), and n ′ j = n j −m j . The symbol ( * ) q denotes the convolution with respect to k q that is defined by the equation…”

Section: Fractional Liouville Equation For Phase-space Continuummentioning

confidence: 99%

“…Fractional calculus [3,4,5,6,7,8,9] has a lot of applications in physics [10,11,12,13,14,15,16] and it allows us to take into account fractional power-law nonlocality of continuously distributed systems. Using the fractional calculus, we can consider fractional differential equations for conservation of probability in generalized phase spaces.…”

Section: Introductionmentioning

confidence: 99%

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Fractional Liouville equation on lattice phase-space

Tarasov

2015

Physica A: Statistical Mechanics and its Applications

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In this paper we propose a lattice analog of phase-space fractional Liouville equation. The Liouville equation for phase-space lattice with long-range jumps of power-law types is suggested. We prove that the continuum limit transforms this lattice equation into Liouville equation with conjugate Riesz fractional derivatives of non-integer orders with respect to coordinates of continuum phase-space. An application of the fractional Liouville equation with these Riesz fractional derivatives to describe properties of plasma-like nonlocal media is considered.

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“…whereK − α j (k j a j ) is defined by (8), and n ′ j = n j −m j . The symbol ( * ) q denotes the convolution with respect to k q that is defined by the equation…”

Section: Fractional Liouville Equation For Phase-space Continuummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional Liouville equation on lattice phase-space

Tarasov

2015

Physica A: Statistical Mechanics and its Applications

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“…Chang and Yortsos [5] presented a generalization of the Warren and Root [14] model to describe a double porosity fractal system (dimensionless form):…”

Section: Mathematical Modelmentioning

confidence: 99%

Analysis of diffusion process in fractured reservoirs using fractional derivative approach

Razminia

Machado

2014

Communications in Nonlinear Science and Numerical Simulation

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The fractal geometry is used to model of a naturally fractured reservoir and the concept of fractional derivative is applied to the diffusion equation to incorporate the history of fluid flow in naturally fractured reservoirs. The resulting fractally fractional diffusion (FFD) equation is solved analytically in the Laplace space for three outer boundary conditions. The analytical solutions are used to analyze the response of a naturally fractured reservoir considering the anomalous behavior of oil production. Several synthetic examples are provided to illustrate the methodology proposed in this work and to explain the diffusion process in fractally fractured systems.

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“…11 The form and properties of fractional-order derivatives as mathematical description can be found in the monographs by Podlubny, 12 Herrman 13 and Meerschaert and Sikorskii, 14 while the theoretical models and experimental applications are available and published. 15 Some applications are in¯elds that include bioengineering, [16][17][18][19] lters, 20 control theory, 21 signal processing, 22 oscillators, [23][24][25][26][27][28][29] transmission lines (TLs) [30][31][32][33] and potentially many others.…”

Section: Introductionmentioning

confidence: 99%

Application of Numerical Inverse Laplace Transform Methods for Simulation of Distributed Systems with Fractional-Order Elements

R‐Smith

Kartci

Brančík

2018

J CIRCUIT SYST COMP

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The paper presents a computationally e±cient method for modeling and simulating distributed systems with lossy transmission line (TL) including multiconductor ones, by a less conventional method. The method is devised based on 1D and 2D Laplace transforms, which facilitates the possibility of incorporating fractional-order elements and frequency-dependent parameters. This process is made possible due to the development of e®ective numerical inverse Laplace transforms (NILTs) of one and two variables, 1D NILT and 2D NILT. In the paper, it is shown that in high frequency operating systems, the frequency dependencies of the system ought to be included in the model. Additionally, it is shown that incorporating fractional-order elements in the modeling of the distributed parameter systems compensates for losses along the wires, provides higher degrees of°exibility for optimization and produces more accurate and authentic modelling of such systems. The simulations are performed in the Matlab environment and are e®ectively algorithmized.

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Fractional Order Calculus: Basic Concepts and Engineering Applications

Cited by 223 publications

References 27 publications

Fractional Liouville equation on lattice phase-space

Fractional Liouville equation on lattice phase-space

Analysis of diffusion process in fractured reservoirs using fractional derivative approach

Application of Numerical Inverse Laplace Transform Methods for Simulation of Distributed Systems with Fractional-Order Elements

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