Communications in nonlinear science & numerical simulation, ISSN 1007-5704, 2013, Volume 18, Issue 11, pp. 2945 - 2948

•A violation of the Leibniz rule is a basic property of fractional derivatives.•Any fractional derivative, which satisfy Leibniz rule, has order equal to...

Fractional derivative | Leibniz rule | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | Computer simulation | Mathematics - Classical Analysis and ODEs

Fractional derivative | Leibniz rule | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | Computer simulation | Mathematics - Classical Analysis and ODEs

Journal Article

International journal of modern physics. B, Condensed matter physics, statistical physics, applied physics, ISSN 1793-6578, 2013, Volume 27, Issue 9, pp. 1330005 - 1330032

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law...

fractional calculus | fractional models | Fractional dynamics | open quantum systems | systems with memory | long-range interaction | fractal media | PHYSICS, CONDENSED MATTER | PHYSICS, APPLIED | STATIONARY STATES | FIELD | POWERS | PHYSICS, MATHEMATICAL | CONTINUUM-MECHANICS | NUMERICAL-SOLUTIONS | FRACTAL DISTRIBUTION | CANONICAL QUANTIZATION | DYNAMICS | ACCRETION | SYSTEMS | Dynamics | Mathematical analysis | Fractal analysis | Fractals | Calculus | Nanostructure | Differentiation | Dynamical systems | Physics - General Physics

fractional calculus | fractional models | Fractional dynamics | open quantum systems | systems with memory | long-range interaction | fractal media | PHYSICS, CONDENSED MATTER | PHYSICS, APPLIED | STATIONARY STATES | FIELD | POWERS | PHYSICS, MATHEMATICAL | CONTINUUM-MECHANICS | NUMERICAL-SOLUTIONS | FRACTAL DISTRIBUTION | CANONICAL QUANTIZATION | DYNAMICS | ACCRETION | SYSTEMS | Dynamics | Mathematical analysis | Fractal analysis | Fractals | Calculus | Nanostructure | Differentiation | Dynamical systems | Physics - General Physics

Journal Article

Journal of Mathematical Physics, ISSN 0022-2488, 08/2014, Volume 55, Issue 8, p. 83510

A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and...

FOKKER-PLANCK EQUATION | FRACTIONAL SPACE | CONTINUUM-MECHANICS | LIOUVILLE EQUATIONS | SOLIDS | ELECTROMAGNETIC-FIELD | WAVE-EQUATION | POROUS-MEDIA | CONTINUOUS MEDIUM MODEL | PHYSICS, MATHEMATICAL | QUANTUM-FIELD-THEORY | Operators | Timoshenko's equation | Continuum modeling | Anisotropy | Differential equations | Anisotropic media | Calculus | Fractals | Timoshenko beams | MATHEMATICAL SPACE | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | BEAMS | POISSON EQUATION | ANISOTROPY | VECTORS | VECTOR FIELDS

FOKKER-PLANCK EQUATION | FRACTIONAL SPACE | CONTINUUM-MECHANICS | LIOUVILLE EQUATIONS | SOLIDS | ELECTROMAGNETIC-FIELD | WAVE-EQUATION | POROUS-MEDIA | CONTINUOUS MEDIUM MODEL | PHYSICS, MATHEMATICAL | QUANTUM-FIELD-THEORY | Operators | Timoshenko's equation | Continuum modeling | Anisotropy | Differential equations | Anisotropic media | Calculus | Fractals | Timoshenko beams | MATHEMATICAL SPACE | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | BEAMS | POISSON EQUATION | ANISOTROPY | VECTORS | VECTOR FIELDS

Journal Article

Annals of Physics, ISSN 0003-4916, 2008, Volume 323, Issue 11, pp. 2756 - 2778

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector...

Fractal media | Derivatives and integrals of non-integer orders | Fractional vector calculus | Fractional electrodynamics | PHYSICS, MULTIDISCIPLINARY | DIFFERENTIABILITY | ELECTROMAGNETICS | ANOMALOUS TRANSPORT | CONTINUOUS MEDIUM MODEL | CURL OPERATOR | NONLOCAL ELECTRODYNAMICS | GRADIENT | DYNAMICS | SYSTEMS | Theorems | Calculus | Theory | Physics | INTEGRALS | FRACTALS | WAVE EQUATIONS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | DIFFERENTIAL CALCULUS | ELECTRODYNAMICS | MAXWELL EQUATIONS | VECTORS

Fractal media | Derivatives and integrals of non-integer orders | Fractional vector calculus | Fractional electrodynamics | PHYSICS, MULTIDISCIPLINARY | DIFFERENTIABILITY | ELECTROMAGNETICS | ANOMALOUS TRANSPORT | CONTINUOUS MEDIUM MODEL | CURL OPERATOR | NONLOCAL ELECTRODYNAMICS | GRADIENT | DYNAMICS | SYSTEMS | Theorems | Calculus | Theory | Physics | INTEGRALS | FRACTALS | WAVE EQUATIONS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | DIFFERENTIAL CALCULUS | ELECTRODYNAMICS | MAXWELL EQUATIONS | VECTORS

Journal Article

Communications in nonlinear science & numerical simulation, ISSN 1007-5704, 2018, Volume 62, pp. 157 - 163

•Principle of nonlocality for fractional derivatives is suggested.•A criterion, which allows one to identify false fractional derivatives, is...

Caputo–Fabrizio fractional derivative | Nonlocality | Local fractional derivative | Memory | Conformable fractional derivative | Fractional derivative | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | Caputo-Fabrizio fractional derivative | Differential equations | Mathematics - Classical Analysis and ODEs

Caputo–Fabrizio fractional derivative | Nonlocality | Local fractional derivative | Memory | Conformable fractional derivative | Fractional derivative | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | Caputo-Fabrizio fractional derivative | Differential equations | Mathematics - Classical Analysis and ODEs

Journal Article

Communications in nonlinear science & numerical simulation, ISSN 1007-5704, 2016, Volume 37, pp. 31 - 61

•New type of differences of integer and non-integer orders are proposed.•Discretization of Riesz fractional differentiation and integration are...

Discretization | Non-standard differences | Finite differences | Infinite series | Fractional integral | Exact discretization | Fourier transform | Fractional difference | Fractional derivative | NUMERICAL-METHODS | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | CALCULUS | DISPERSION | EQUATIONS | PHYSICS, MATHEMATICAL | LONG-RANGE INTERACTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | SPACE-TIME | DYNAMICS | FRACTIONAL ORDER DIFFERENTIATOR | LATTICE MODEL | GEOMETRY | Integers | Operators | Algebra | Mathematical analysis | Differential equations | Mathematical models | Derivatives

Discretization | Non-standard differences | Finite differences | Infinite series | Fractional integral | Exact discretization | Fourier transform | Fractional difference | Fractional derivative | NUMERICAL-METHODS | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | CALCULUS | DISPERSION | EQUATIONS | PHYSICS, MATHEMATICAL | LONG-RANGE INTERACTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | SPACE-TIME | DYNAMICS | FRACTIONAL ORDER DIFFERENTIATOR | LATTICE MODEL | GEOMETRY | Integers | Operators | Algebra | Mathematical analysis | Differential equations | Mathematical models | Derivatives

Journal Article

Nonlinear dynamics, ISSN 1573-269X, 2016, Volume 86, Issue 3, pp. 1745 - 1759

... differential equations Vasily E. Tarasov Received: 15 July 2015 / Accepted: 27 July 2016 / Published online: 4 August 2016 © Springer Science+Business Media Dordrecht 2016...

Engineering | Vibration, Dynamical Systems, Control | Fractional dynamics | Classical Mechanics | 26A33 | Automotive Engineering | Mechanical Engineering | Nonlinear fractional equations | Fractional calculus | Fractional derivative | Nonlocal continuum | NUMERICAL-METHODS | DIMENSIONAL ISING FERROMAGNET | BOUSSINESQ EQUATION | QUANTUM HEISENBERG FERROMAGNETS | CALCULUS | ENGINEERING, MECHANICAL | NONLOCAL DISPERSIVE INTERACTIONS | ANHARMONIC CHAINS | WATER-WAVES | MECHANICS | LONG-RANGE INTERACTIONS | BURGERS-EQUATION | Nuclear physics | Differential equations | Integers | Economic models | Nonlinear equations | Euclidean geometry | Boussinesq equations | Mathematical analysis | Euclidean space | Regression analysis | Derivatives | Continuums | Standards

Engineering | Vibration, Dynamical Systems, Control | Fractional dynamics | Classical Mechanics | 26A33 | Automotive Engineering | Mechanical Engineering | Nonlinear fractional equations | Fractional calculus | Fractional derivative | Nonlocal continuum | NUMERICAL-METHODS | DIMENSIONAL ISING FERROMAGNET | BOUSSINESQ EQUATION | QUANTUM HEISENBERG FERROMAGNETS | CALCULUS | ENGINEERING, MECHANICAL | NONLOCAL DISPERSIVE INTERACTIONS | ANHARMONIC CHAINS | WATER-WAVES | MECHANICS | LONG-RANGE INTERACTIONS | BURGERS-EQUATION | Nuclear physics | Differential equations | Integers | Economic models | Nonlinear equations | Euclidean geometry | Boussinesq equations | Mathematical analysis | Euclidean space | Regression analysis | Derivatives | Continuums | Standards

Journal Article

Wave Motion, ISSN 0165-2125, 06/2016, Volume 63, pp. 18 - 22

Acoustic waves in fractal media are considered in the framework of continuum models with non-integer dimensional spaces. Using recently suggested vector...

Fractal media | Supersonic mode | Acoustic wave | Non-integer dimensional space | Heterogeneity | Wave motion | Mathematical analysis | Fractal analysis | Media | Fractals | Calculus | Vectors (mathematics)

Fractal media | Supersonic mode | Acoustic wave | Non-integer dimensional space | Heterogeneity | Wave motion | Mathematical analysis | Fractal analysis | Media | Fractals | Calculus | Vectors (mathematics)

Journal Article

Journal of Mathematical Physics, ISSN 0022-2488, 10/2015, Volume 56, Issue 10, p. 103506

Fractional-order operators for physical lattice models based on the Grunwald-Letnikov fractional differences are suggested. We use an approach based on the...

ELASTICITY | LONG-RANGE INTERACTION | VECTOR CALCULUS | MODEL | NONINTEGER DIMENSIONAL SPACE | PHYSICS, MATHEMATICAL | Operators (mathematics) | Difference equations | Computational fluid dynamics | Particle interactions | Lattices | Maxwell's equations | Mathematical models | INTEGRALS | NAVIER-STOKES EQUATIONS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MAXWELL EQUATIONS | PARTICLE INTERACTIONS | COORDINATES | INTERACTION RANGE | KERNELS

ELASTICITY | LONG-RANGE INTERACTION | VECTOR CALCULUS | MODEL | NONINTEGER DIMENSIONAL SPACE | PHYSICS, MATHEMATICAL | Operators (mathematics) | Difference equations | Computational fluid dynamics | Particle interactions | Lattices | Maxwell's equations | Mathematical models | INTEGRALS | NAVIER-STOKES EQUATIONS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MAXWELL EQUATIONS | PARTICLE INTERACTIONS | COORDINATES | INTERACTION RANGE | KERNELS

Journal Article

Applied mathematics and computation, ISSN 0096-3003, 2015, Volume 257, pp. 12 - 33

Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed...

Lattice models | Fractional dynamics | Fractional calculus | Fractional integral | Fractional derivative | Nonlocal continuum | LONG-RANGE INTERACTION | MATHEMATICS, APPLIED | DYNAMICS | SYSTEMS | MODEL | Transformations (mathematics) | Integrals | Mathematical analysis | Lattices | Continuums | Differential equations | Mathematical models | Derivatives

Lattice models | Fractional dynamics | Fractional calculus | Fractional integral | Fractional derivative | Nonlocal continuum | LONG-RANGE INTERACTION | MATHEMATICS, APPLIED | DYNAMICS | SYSTEMS | MODEL | Transformations (mathematics) | Integrals | Mathematical analysis | Lattices | Continuums | Differential equations | Mathematical models | Derivatives

Journal Article

Annals of physics, ISSN 0003-4916, 2005, Volume 318, Issue 2, pp. 286 - 307

We use the fractional integrals in order to describe dynamical processes in the fractal medium. We consider the “fractional” continuous medium model for the...

Fractal media | Hydrodynamic equations | Fractional integral | POROUS-MEDIA | fractal media | fractional integral | PHYSICS, MULTIDISCIPLINARY | hydrodynamic equations | GEOMETRY | DENSITY | FRACTALS | NAVIER-STOKES EQUATIONS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | EQUILIBRIUM | MASS | SOUND WAVES

Fractal media | Hydrodynamic equations | Fractional integral | POROUS-MEDIA | fractal media | fractional integral | PHYSICS, MULTIDISCIPLINARY | hydrodynamic equations | GEOMETRY | DENSITY | FRACTALS | NAVIER-STOKES EQUATIONS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | EQUILIBRIUM | MASS | SOUND WAVES

Journal Article

Communications in nonlinear science & numerical simulation, ISSN 1007-5704, 2006, Volume 11, Issue 8, pp. 885 - 898

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/∣ n −...

Fractional oscillator | Synchronization | Fractional Ginzburg–Landau equation | Long-range interaction | Fractional Ginzburg-Landau equation | Physics - Chaotic Dynamics

Fractional oscillator | Synchronization | Fractional Ginzburg–Landau equation | Long-range interaction | Fractional Ginzburg-Landau equation | Physics - Chaotic Dynamics

Journal Article

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Vector calculus in non-integer dimensional space and its applications to fractal media

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 02/2015, Volume 20, Issue 2, pp. 360 - 374

•Vector calculus for non-integer dimensional space is suggested.•The gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional...

Fractal media | Vector calculus | Non-integer dimensional space | EXTREMUM | MATHEMATICS, APPLIED | FRACTIONAL SCHRODINGER-EQUATION | PHYSICS, FLUIDS & PLASMAS | ELECTROMAGNETIC-FIELD | CONTINUOUS MEDIUM MODEL | FORMULATION | PHYSICS, MATHEMATICAL | QUANTUM-FIELD-THEORY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | CONTINUUM-MECHANICS | SOLIDS | WAVE-EQUATION | Mathematical analysis | Fractal analysis | Media | Fractals | Mathematical models | Calculus | Vectors (mathematics) | Cylinders

Fractal media | Vector calculus | Non-integer dimensional space | EXTREMUM | MATHEMATICS, APPLIED | FRACTIONAL SCHRODINGER-EQUATION | PHYSICS, FLUIDS & PLASMAS | ELECTROMAGNETIC-FIELD | CONTINUOUS MEDIUM MODEL | FORMULATION | PHYSICS, MATHEMATICAL | QUANTUM-FIELD-THEORY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | CONTINUUM-MECHANICS | SOLIDS | WAVE-EQUATION | Mathematical analysis | Fractal analysis | Media | Fractals | Mathematical models | Calculus | Vectors (mathematics) | Cylinders

Journal Article

Chaos, solitons and fractals, ISSN 0960-0779, 2015, Volume 81, pp. 38 - 42

Electromagnetic waves in non-integer dimensional spaces are considered in the framework of continuous models of fractal media and fields. Using the recently...

Fractal media | Electromagnetic | Effective refractive index | Waves | Non-integer dimensional space | VECTOR CALCULUS | PHYSICS, MULTIDISCIPLINARY | FIELD | EQUATIONS | PHYSICS, MATHEMATICAL | FRACTIONAL SPACE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MEDIA | Electromagnetic radiation | Electromagnetism | Electromagnetic fields | Analysis | Electromagnetic waves | Electric waves | Nuclear physics

Fractal media | Electromagnetic | Effective refractive index | Waves | Non-integer dimensional space | VECTOR CALCULUS | PHYSICS, MULTIDISCIPLINARY | FIELD | EQUATIONS | PHYSICS, MATHEMATICAL | FRACTIONAL SPACE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MEDIA | Electromagnetic radiation | Electromagnetism | Electromagnetic fields | Analysis | Electromagnetic waves | Electric waves | Nuclear physics

Journal Article

Entropy (Basel, Switzerland), ISSN 1099-4300, 2018, Volume 20, Issue 6, p. 414

In this paper, we propose criteria for the existence of memory of power-law type (PLT) memory in economic processes. We give the criterion of existence of...

Economic dynamics | Multiplier with memory | Accelerator with memory | Power-law memory | Fractional integral | Long memory | Fractional derivative | PHYSICS, MULTIDISCIPLINARY | economic dynamics | CAUSALITY | fractional derivative | RELAXATION | power-law memory | TIME | fractional integral | accelerator with memory | MODELS | KRAMERS-KRONIG | FRACTIONAL CALCULUS | DYNAMICS | multiplier with memory | long memory | EXACT DISCRETIZATION

Economic dynamics | Multiplier with memory | Accelerator with memory | Power-law memory | Fractional integral | Long memory | Fractional derivative | PHYSICS, MULTIDISCIPLINARY | economic dynamics | CAUSALITY | fractional derivative | RELAXATION | power-law memory | TIME | fractional integral | accelerator with memory | MODELS | KRAMERS-KRONIG | FRACTIONAL CALCULUS | DYNAMICS | multiplier with memory | long memory | EXACT DISCRETIZATION

Journal Article

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Lattice with long-range interaction of power-law type for fractional non-local elasticity

International Journal of Solids and Structures, ISSN 0020-7683, 08/2014, Volume 51, Issue 15-16, pp. 2900 - 2907

Lattice models with long-range interactions of power-law type are suggested as a new type of microscopic model for fractional non-local elasticity. Using the...

Fractional dynamics | Fractional elasticity | Long-range interaction | Lattice model | Non-local elasticity | MECHANICS | CALCULUS | MODEL | WAVE-PROPAGATION | Asymptotic properties | Mathematical analysis | Lattices | Continuums | Differential equations | Elasticity | Mathematical models | Derivatives | Physics - Materials Science

Fractional dynamics | Fractional elasticity | Long-range interaction | Lattice model | Non-local elasticity | MECHANICS | CALCULUS | MODEL | WAVE-PROPAGATION | Asymptotic properties | Mathematical analysis | Lattices | Continuums | Differential equations | Elasticity | Mathematical models | Derivatives | Physics - Materials Science

Journal Article