Fuzzy Sets and Systems, ISSN 0165-0114, 11/2019, Volume 375, pp. 70 - 99

In this work, an initial value problem of Caputo–Katugampola (CK) fractional differential equations in fuzzy setting is considered and an idea of successive...

Caputo–Katugampola fractional derivative | Fractional fuzzy differential equations | Fractional fuzzy integral equations | Fractional calculus | INTERVAL | MATHEMATICS, APPLIED | CALCULUS | INTEGRAL-EQUATIONS | STATISTICS & PROBABILITY | COMPUTER SCIENCE, THEORY & METHODS | VALUED FUNCTIONS | Caputo-Katugampola fractional derivative | Analysis | Differential equations

Caputo–Katugampola fractional derivative | Fractional fuzzy differential equations | Fractional fuzzy integral equations | Fractional calculus | INTERVAL | MATHEMATICS, APPLIED | CALCULUS | INTEGRAL-EQUATIONS | STATISTICS & PROBABILITY | COMPUTER SCIENCE, THEORY & METHODS | VALUED FUNCTIONS | Caputo-Katugampola fractional derivative | Analysis | Differential equations

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 09/2018, 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

Applied Mathematics and Computation, ISSN 0096-3003, 2007, Volume 187, Issue 2, pp. 777 - 784

In this paper, we further discuss the properties of three kinds of fractional derivatives: the Grünwald–Letnikov derivative, the Riemann–Liouville derivative...

Riemann–Liouville derivative | Caputo derivative | Grünwald–Letnikov derivative | Consistency | Sequential property | Riemann-Liouville derivative | Grünwald-Letnikov derivative | sequential property | SYSTEM | ORDER | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | CHAOS SYNCHRONIZATION | DIFFERENTIAL-EQUATIONS | Grunwald-Letnikov derivative | Derivatives (Financial instruments)

Riemann–Liouville derivative | Caputo derivative | Grünwald–Letnikov derivative | Consistency | Sequential property | Riemann-Liouville derivative | Grünwald-Letnikov derivative | sequential property | SYSTEM | ORDER | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | CHAOS SYNCHRONIZATION | DIFFERENTIAL-EQUATIONS | Grunwald-Letnikov derivative | Derivatives (Financial instruments)

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 01/2018, Volume 316, pp. 504 - 515

In the computer security and for any defensive strategy, computer viruses are very significant aspect. To understand their expansion and extension is very...

Caputo-Fabrizio derivative | Fixed point theorem | Computer viruses | Fractional differential equations | Epidemiological model | MATHEMATICS, APPLIED | CALCULUS | Models | Numerical analysis | Epidemiology | Analysis | Differential equations | Water, Underground

Caputo-Fabrizio derivative | Fixed point theorem | Computer viruses | Fractional differential equations | Epidemiological model | MATHEMATICS, APPLIED | CALCULUS | Models | Numerical analysis | Epidemiology | Analysis | Differential equations | Water, Underground

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 07/2014, Volume 19, Issue 7, pp. 2354 - 2372

•Defining type-2 fuzzy fractional derivative in the sense of Caputo and deriving the related theorem.•Defining type-2 fuzzy fractional derivative and integral...

H2-differentiability | Type-2 fuzzy sets | Riemann–Liouville type-2 fuzzy fractional derivative | Caputo type-2 fuzzy fractional derivative | Type-2 fuzzy fractional differential equations | Type-2 Hukuhara difference | Riemann-Liouville type-2 fuzzy fractional derivative | differentiability | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | PHYSICS, FLUIDS & PLASMAS | H-2-differentiability | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | Fuzzy logic | Approximation | Mathematical analysis | Nonlinearity | Mathematical models | Fuzzy set theory | Derivatives | Fuzzy

H2-differentiability | Type-2 fuzzy sets | Riemann–Liouville type-2 fuzzy fractional derivative | Caputo type-2 fuzzy fractional derivative | Type-2 fuzzy fractional differential equations | Type-2 Hukuhara difference | Riemann-Liouville type-2 fuzzy fractional derivative | differentiability | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | PHYSICS, FLUIDS & PLASMAS | H-2-differentiability | DIFFERENTIAL-EQUATIONS | PHYSICS, MATHEMATICAL | Fuzzy logic | Approximation | Mathematical analysis | Nonlinearity | Mathematical models | Fuzzy set theory | Derivatives | Fuzzy

Journal Article

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 08/2016, Volume 89, pp. 552 - 559

In 2015 Caputo and Fabrizio suggested a new operator with fractional order, this derivative is based on the exponential kernel. Earlier this year 2016 Atangana...

Caputo–Fabrizio derivatives | Allen–Cahn model | Atangana–Baleanu derivatives | Caputo-Fabrizio derivatives | Allen-Cahn model | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Atangana-Baleanu derivatives | PHYSICS, MULTIDISCIPLINARY | PHYSICS, MATHEMATICAL | EQUATION | Analysis | Models | Numerical analysis

Caputo–Fabrizio derivatives | Allen–Cahn model | Atangana–Baleanu derivatives | Caputo-Fabrizio derivatives | Allen-Cahn model | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Atangana-Baleanu derivatives | PHYSICS, MULTIDISCIPLINARY | PHYSICS, MATHEMATICAL | EQUATION | Analysis | Models | Numerical analysis

Journal Article

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 12/2017, Volume 105, pp. 111 - 119

•Numerical approximation of the Caputo–Fabrizio fractional derivative.•New three-step fractional Adams–Bashforth method.•Stability analysis of the...

Numerical simulations | Caputo–Fabrizio derivative | Error analysis | Nonlinear chaotic systems | Fractional Adams–Bashforth method | Caputo-Fabrizio derivative | Fractional Adams-Bashforth method | CHAOTIC FLOWS | PHYSICS, MULTIDISCIPLINARY | DIFFERENTIAL-EQUATIONS | DIFFUSION EQUATION | SIMULATION | PHYSICS, MATHEMATICAL | SPACE | SPECTRAL METHOD | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Water, Underground | Numerical analysis | Differential equations

Numerical simulations | Caputo–Fabrizio derivative | Error analysis | Nonlinear chaotic systems | Fractional Adams–Bashforth method | Caputo-Fabrizio derivative | Fractional Adams-Bashforth method | CHAOTIC FLOWS | PHYSICS, MULTIDISCIPLINARY | DIFFERENTIAL-EQUATIONS | DIFFUSION EQUATION | SIMULATION | PHYSICS, MATHEMATICAL | SPACE | SPECTRAL METHOD | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Water, Underground | Numerical analysis | Differential equations

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 11/2018, Volume 509, pp. 703 - 716

In this paper, we obtain analytical solutions for the fractional cubic isothermal auto-catalytic chemical system with Caputo–Fabrizio and Atangana–Baleanu...

q-HATM | Atangana–Baleanu | [formula omitted]-curves | Fractional isothermal auto-catalytic chemical systems | Caputo–Fabrizio | h-curves | Atangana-Baleanu | LONG-WAVE EQUATION | NUMERICAL-SOLUTION | TRANSPORT | PHYSICS, MULTIDISCIPLINARY | Caputo-Fabrizio | TIME | MODEL | HOMOTOPY ANALYSIS METHOD

q-HATM | Atangana–Baleanu | [formula omitted]-curves | Fractional isothermal auto-catalytic chemical systems | Caputo–Fabrizio | h-curves | Atangana-Baleanu | LONG-WAVE EQUATION | NUMERICAL-SOLUTION | TRANSPORT | PHYSICS, MULTIDISCIPLINARY | Caputo-Fabrizio | TIME | MODEL | HOMOTOPY ANALYSIS METHOD

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 02/2014, Volume 259, pp. 33 - 50

In the present work, first, a new fractional numerical differentiation formula (called the L1-2 formula) to approximate the Caputo fractional derivative of...

L1 formula | Sub-diffusion | Fractional numerical differentiation formula | Caputo fractional derivative | L1-2 formula | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | SUB-DIFFUSION EQUATIONS | PHYSICS, MATHEMATICAL | FINITE-DIFFERENCE SCHEME | Interpolation | Accuracy | Approximation | Mathematical analysis | Differential equations | Mathematical models | Derivatives | Finite difference method | Formulas (mathematics)

L1 formula | Sub-diffusion | Fractional numerical differentiation formula | Caputo fractional derivative | L1-2 formula | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | SUB-DIFFUSION EQUATIONS | PHYSICS, MATHEMATICAL | FINITE-DIFFERENCE SCHEME | Interpolation | Accuracy | Approximation | Mathematical analysis | Differential equations | Mathematical models | Derivatives | Finite difference method | Formulas (mathematics)

Journal Article

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 06/2017, Volume 99, pp. 171 - 179

•Caputo–Fabrizio fractional derivative approximation in Riemann–Liouville sense.•Approximation of space fractional derivative with exponential decay...

Riemann–Liouville definition | Exponential time differencing | Numerical simulations | Caputo–Fabrizio derivative | Fractional nonlinear PDEs | Exponential decay-law | Finite difference method | Caputo-Fabrizio derivative | SPECTRAL METHOD | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Riemann-Liouville definition | PHYSICS, MULTIDISCIPLINARY | MODEL | PHYSICS, MATHEMATICAL | Water, Underground | Numerical analysis | Differential equations | Analysis

Riemann–Liouville definition | Exponential time differencing | Numerical simulations | Caputo–Fabrizio derivative | Fractional nonlinear PDEs | Exponential decay-law | Finite difference method | Caputo-Fabrizio derivative | SPECTRAL METHOD | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Riemann-Liouville definition | PHYSICS, MULTIDISCIPLINARY | MODEL | PHYSICS, MATHEMATICAL | Water, Underground | Numerical analysis | Differential equations | Analysis

Journal Article

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 08/2016, Volume 89, pp. 539 - 546

In order to control the movement of waves on the area of shallow water, the newly derivative with fractional order proposed by Caputo and Fabrizio was used. To...

Caputo–Fabrizio fractional derivative | Stability and uniqueness | Shallow water model | Fixed-point theorem | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | PHYSICS, MATHEMATICAL | EQUATION | Caputo-Fabrizio fractional derivative | Water, Underground | Numerical analysis | Analysis | Differential equations

Caputo–Fabrizio fractional derivative | Stability and uniqueness | Shallow water model | Fixed-point theorem | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | PHYSICS, MATHEMATICAL | EQUATION | Caputo-Fabrizio fractional derivative | Water, Underground | Numerical analysis | Analysis | Differential equations

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 01/2019, Volume 346, pp. 247 - 260

In this paper, we present a new definition of fractional-order derivative with a smooth kernel based on the Caputo–Fabrizio fractional-order operator which...

Modified-Caputo–Fabrizio derivative | Multi step homotopy analysis method | Fractional calculus | Homotopy analysis method | SPACE | MATHEMATICS, APPLIED | PARTIAL-DIFFERENTIAL-EQUATIONS | Modified-Caputo-Fabrizio derivative | TRANSFORM METHOD | SYSTEMS | Methods | Differential equations

Modified-Caputo–Fabrizio derivative | Multi step homotopy analysis method | Fractional calculus | Homotopy analysis method | SPACE | MATHEMATICS, APPLIED | PARTIAL-DIFFERENTIAL-EQUATIONS | Modified-Caputo-Fabrizio derivative | TRANSFORM METHOD | SYSTEMS | Methods | Differential equations

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 02/2018, Volume 491, pp. 406 - 424

A reaction–diffusion system can be represented by the Gray–Scott model. The reaction–diffusion dynamic is described by a pair of time and space dependent...

Adams method | Atangana–Baleanu–Caputo derivative of variable-order | Reaction–diffusion | Gray–Scott model | Liouville–Caputo derivative of variable-order | Fractional derivative | Gray-Scott model | PATTERN-FORMATION | MORPHOGENESIS | PHYSICS, MULTIDISCIPLINARY | Reaction-diffusion | Atangana-Baleanu-Caputo derivative of variable-order | STIRRED TANK REACTOR | Liouville-Caputo derivative of variable-order | AUTOCATALYTIC REACTIONS | Models | Differential equations

Adams method | Atangana–Baleanu–Caputo derivative of variable-order | Reaction–diffusion | Gray–Scott model | Liouville–Caputo derivative of variable-order | Fractional derivative | Gray-Scott model | PATTERN-FORMATION | MORPHOGENESIS | PHYSICS, MULTIDISCIPLINARY | Reaction-diffusion | Atangana-Baleanu-Caputo derivative of variable-order | STIRRED TANK REACTOR | Liouville-Caputo derivative of variable-order | AUTOCATALYTIC REACTIONS | Models | Differential equations

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 12/2017, Volume 350, pp. 1 - 15

In this paper, a finite difference scheme is proposed to solve time–space fractional diffusion equation which has second-order accuracy in both time and space...

Trapezoidal rule | Riesz derivative | Fractional diffusion equation | Caputo derivative | NUMERICAL-METHODS | FINITE-DIFFERENCE SCHEMES | SUBDIFFUSION EQUATION | ALGORITHMS | HIGH-ORDER APPROXIMATION | PHYSICS, MATHEMATICAL | CAPUTO DERIVATIVES | SPECTRAL METHOD | DIMENSIONS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MODELS | COMPACT ADI SCHEME | Analysis | Numerical analysis | Differential equations

Trapezoidal rule | Riesz derivative | Fractional diffusion equation | Caputo derivative | NUMERICAL-METHODS | FINITE-DIFFERENCE SCHEMES | SUBDIFFUSION EQUATION | ALGORITHMS | HIGH-ORDER APPROXIMATION | PHYSICS, MATHEMATICAL | CAPUTO DERIVATIVES | SPECTRAL METHOD | DIMENSIONS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MODELS | COMPACT ADI SCHEME | Analysis | Numerical analysis | Differential equations

Journal Article

15.
Full Text
Analysis of the Keller-Segel model with a fractional derivative without singular kernel

Entropy, ISSN 1099-4300, 2015, Volume 17, Issue 6, pp. 4439 - 4453

Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the...

Special solution | Keller-Segel model | Fixed-point theorem | Caputo-Fabrizio fractional derivative | fixed-point theorem | PHYSICS, MULTIDISCIPLINARY | special solution | EQUATION | Kernels | Computer simulation | Uniqueness | Joining | Mathematical models | Entropy | Derivatives | Iterative methods | Caputo–Fabrizio fractional derivative | Keller–Segel model

Special solution | Keller-Segel model | Fixed-point theorem | Caputo-Fabrizio fractional derivative | fixed-point theorem | PHYSICS, MULTIDISCIPLINARY | special solution | EQUATION | Kernels | Computer simulation | Uniqueness | Joining | Mathematical models | Entropy | Derivatives | Iterative methods | Caputo–Fabrizio fractional derivative | Keller–Segel model

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 05/2013, Volume 37, Issue 9, pp. 6183 - 6190

The flow through porous media can be better described by fractional models than the classical ones since they include inherently memory effects caused by...

Fractional differential equation | Variational iteration method | Caputo derivative | APPROXIMATE SOLUTIONS | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | FLUID | PARTIAL-DIFFERENTIAL-EQUATIONS | PREDICTOR-CORRECTOR APPROACH | HOMOTOPY PERTURBATION METHOD | Analysis | Aquatic resources | Methods

Fractional differential equation | Variational iteration method | Caputo derivative | APPROXIMATE SOLUTIONS | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | FLUID | PARTIAL-DIFFERENTIAL-EQUATIONS | PREDICTOR-CORRECTOR APPROACH | HOMOTOPY PERTURBATION METHOD | Analysis | Aquatic resources | Methods

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 04/2016, Volume 447, pp. 467 - 481

In this paper we present an alternative representation of the diffusion equation and the diffusion–advection equation using the fractional calculus approach,...

Caputo–Fabrizio fractional derivative | Non-local transport processes | Non-Fickian diffusion | Dissipative dynamics | Fractional calculus | Anomalous diffusion | Caputo-Fabrizio fractional derivative | PHYSICS, MULTIDISCIPLINARY | CALCULUS | TERMS | EQUATION | Local transit | Analysis | Approximation | Mathematical analysis | Fractal analysis | Mathematical models | Calculus | Derivatives | Representations | Diffusion

Caputo–Fabrizio fractional derivative | Non-local transport processes | Non-Fickian diffusion | Dissipative dynamics | Fractional calculus | Anomalous diffusion | Caputo-Fabrizio fractional derivative | PHYSICS, MULTIDISCIPLINARY | CALCULUS | TERMS | EQUATION | Local transit | Analysis | Approximation | Mathematical analysis | Fractal analysis | Mathematical models | Calculus | Derivatives | Representations | Diffusion

Journal Article