Fractional Calculus and Applied Analysis, ISSN 1311-0454, 08/2016, Volume 19, Issue 4, pp. 806 - 831

Recently, in series of papers we have proposed different concepts of solutions of impulsive fractional differential equations (IFDE). This paper is a survey of...

fractional calculus | Primary 26A33 | noninstantaneous impulsive fractional differential equations | 35R11 | Mittag-Leffler type functions | Secondary 33E12 | 34K37 | fractional ordinary and partial differential equations | 34A08 | nonlocal impulsive fractional switched systems | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Boundary values | Solutions | Mathematical analysis | Differential equations

fractional calculus | Primary 26A33 | noninstantaneous impulsive fractional differential equations | 35R11 | Mittag-Leffler type functions | Secondary 33E12 | 34K37 | fractional ordinary and partial differential equations | 34A08 | nonlocal impulsive fractional switched systems | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Boundary values | Solutions | Mathematical analysis | Differential equations

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 10/2016, Volume 19, Issue 5, pp. 1222 - 1249

Over the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than...

fractional-order derivative | image processing | fractional calculus | Primary 26A33 | 35R11 | 34K37 | Secondary 34A08 | Fractional-Order Derivative | Fractional calculus | MATHEMATICS, APPLIED | ENCRYPTION | RECOGNITION | MULTILEVEL ALGORITHM | MODEL | COUPLED NEURAL-NETWORK | MATHEMATICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | EDGE-DETECTION | FOURIER-TRANSFORM | DIFFUSION | DIFFERENTIATION | REGULARIZATION | Usage | Image processing | Mathematical analysis | Methods

fractional-order derivative | image processing | fractional calculus | Primary 26A33 | 35R11 | 34K37 | Secondary 34A08 | Fractional-Order Derivative | Fractional calculus | MATHEMATICS, APPLIED | ENCRYPTION | RECOGNITION | MULTILEVEL ALGORITHM | MODEL | COUPLED NEURAL-NETWORK | MATHEMATICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | EDGE-DETECTION | FOURIER-TRANSFORM | DIFFUSION | DIFFERENTIATION | REGULARIZATION | Usage | Image processing | Mathematical analysis | Methods

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 06/2018, Volume 21, Issue 3, pp. 786 - 800

In this paper, we initiate the question of the attractivity of solutions for fractional evolution equations with almost sectorial operators. We establish...

fractional evolution equations | Secondary 34K37, 37L05, 47J35 | Primary 26A33 | attractivity | Caputo derivative | Riemann-Liuoville derivative | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DIFFERENTIAL-EQUATIONS | Operators | Mathematical analysis | Chaos theory | Evolution | Euclidean space | Cases (containers) | Quantum theory

fractional evolution equations | Secondary 34K37, 37L05, 47J35 | Primary 26A33 | attractivity | Caputo derivative | Riemann-Liuoville derivative | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DIFFERENTIAL-EQUATIONS | Operators | Mathematical analysis | Chaos theory | Evolution | Euclidean space | Cases (containers) | Quantum theory

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Nonlinear dynamics, ISSN 1573-269X, 2016, Volume 87, Issue 2, pp. 815 - 826

This paper describes a robust, accurate and efficient scheme based on a cubic spline interpolation. The proposed scheme is applied to approximate...

65O15 | Classical Mechanics | 26A33 | 34K37 | 33F05 | Fractional calculus | Spline approximation | Engineering | Vibration, Dynamical Systems, Control | Oscillatory dynamic systems | Variable-order derivative | Automotive Engineering | Mechanical Engineering | Functional differential equations | 41A15 | STABILITY | CHAOTIC BEHAVIOR | OSCILLATOR | ENGINEERING, MECHANICAL | SYNCHRONIZATION | MECHANICS | NEURAL-NETWORKS | OPERATOR | MODELS | DYNAMICS | SYSTEMS | FINITE-DIFFERENCE | Analysis | Differential equations | Interpolation | Nonlinear equations | Robustness (mathematics) | Mathematical analysis | Time lag | Accuracy | Approximation | Efficiency | Splines | Mathematical models | Delay | Convergence

65O15 | Classical Mechanics | 26A33 | 34K37 | 33F05 | Fractional calculus | Spline approximation | Engineering | Vibration, Dynamical Systems, Control | Oscillatory dynamic systems | Variable-order derivative | Automotive Engineering | Mechanical Engineering | Functional differential equations | 41A15 | STABILITY | CHAOTIC BEHAVIOR | OSCILLATOR | ENGINEERING, MECHANICAL | SYNCHRONIZATION | MECHANICS | NEURAL-NETWORKS | OPERATOR | MODELS | DYNAMICS | SYSTEMS | FINITE-DIFFERENCE | Analysis | Differential equations | Interpolation | Nonlinear equations | Robustness (mathematics) | Mathematical analysis | Time lag | Accuracy | Approximation | Efficiency | Splines | Mathematical models | Delay | Convergence

Journal Article

International journal of nonlinear sciences and numerical simulation, ISSN 2191-0294, 2019, Volume 20, Issue 1, pp. 1 - 16

In this paper, we investigate the existence of mild solutions of impulsive fractional integrodifferential evolution equations with nonlocal conditions via the...

mild solutions | fractional cosine family | 34K30 | 34K37 | 47H08 | nonlocal conditions | 34A08 | MATHEMATICS, APPLIED | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | DIFFERENTIAL-EQUATIONS | SYSTEMS | PHYSICS, MATHEMATICAL | Evolution | Theorems | Fixed points (mathematics) | Mathematical analysis

mild solutions | fractional cosine family | 34K30 | 34K37 | 47H08 | nonlocal conditions | 34A08 | MATHEMATICS, APPLIED | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | DIFFERENTIAL-EQUATIONS | SYSTEMS | PHYSICS, MATHEMATICAL | Evolution | Theorems | Fixed points (mathematics) | Mathematical analysis

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 02/2017, Volume 20, Issue 1, pp. 52 - 70

Many phenomena in inter-disciplinary fields can be explained naturally by coordinated behavior of agents with fractional-order dynamics. Under the assumption...

consensus | leader-following consensus | input time delay | Secondary 68T42 | fractional-order multi-agent systems | Primary 34K37 | 93C85 | TOPOLOGY | MATHEMATICS, APPLIED | SUFFICIENT CONDITIONS | STABILITY | EQUATIONS | ALGORITHMS | DISTRIBUTED COORDINATION | MATHEMATICS | COMMUNICATION DELAYS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

consensus | leader-following consensus | input time delay | Secondary 68T42 | fractional-order multi-agent systems | Primary 34K37 | 93C85 | TOPOLOGY | MATHEMATICS, APPLIED | SUFFICIENT CONDITIONS | STABILITY | EQUATIONS | ALGORITHMS | DISTRIBUTED COORDINATION | MATHEMATICS | COMMUNICATION DELAYS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Journal Article

Nonlinear dynamics, ISSN 1573-269X, 2017, Volume 90, Issue 3, pp. 2137 - 2143

The calculus of arbitrary order, known as the fractional calculus, allows for real- and complex-valued orders. Interest in time-varying order is growing in...

Engineering | Vibration, Dynamical Systems, Control | Variable-order fractional operator | Classical Mechanics | 26A33 | Automotive Engineering | 34K37 | Fractional-order dynamics | Mechanical Engineering | Memory measure | Fractional calculus | MECHANICS | CALCULUS | ENGINEERING, MECHANICAL | Energy conservation | Analysis | Memory | Operators (mathematics)

Engineering | Vibration, Dynamical Systems, Control | Variable-order fractional operator | Classical Mechanics | 26A33 | Automotive Engineering | 34K37 | Fractional-order dynamics | Mechanical Engineering | Memory measure | Fractional calculus | MECHANICS | CALCULUS | ENGINEERING, MECHANICAL | Energy conservation | Analysis | Memory | Operators (mathematics)

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 02/2019, Volume 22, Issue 1, pp. 128 - 138

Stable distributions are a class of distributions that have important uses in probability theory. They also have a applications in the theory of fractional...

stable distributions | Green’s function | 34K37 | fractional diffusions | Primary 35R11 | 60E07 | Secondary 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Green's function | Thermodynamics | Probability theory

stable distributions | Green’s function | 34K37 | fractional diffusions | Primary 35R11 | 60E07 | Secondary 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Green's function | Thermodynamics | Probability theory

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Distributed coordination of fractional order multi-agent systems with communication delays

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 3/2014, Volume 17, Issue 1, pp. 23 - 37

Because of the complexity of the practical environments, many distributed multi-agent systems can not be illustrated with the integer-order dynamics and can...

Abstract Harmonic Analysis | fractional calculus | multi-agent systems | Functional Analysis | Analysis | Primary 34K37 | Secondary 68T42, 93C85 | Mathematics | communication delays | distributed coordination | Integral Transforms, Operational Calculus | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | STABILITY | CONSENSUS | AGENTS

Abstract Harmonic Analysis | fractional calculus | multi-agent systems | Functional Analysis | Analysis | Primary 34K37 | Secondary 68T42, 93C85 | Mathematics | communication delays | distributed coordination | Integral Transforms, Operational Calculus | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | STABILITY | CONSENSUS | AGENTS

Journal Article

Mathematica Slovaca, ISSN 0139-9918, 06/2019, Volume 69, Issue 3, pp. 583 - 598

Caputo fractional delay differential equations with non-instantaneous impulses are studied. Initially a brief overview of the basic two approaches in the...

Caputo fractional Dini derivative | Lyapunov functions | non-instantaneous impulses | Mittag-Leffler stability | Primary 34K37 | Secondary 34K45 | Caputo fractional derivative | 34K20 | 34A08 | MATHEMATICS | VARIATIONAL APPROACH

Caputo fractional Dini derivative | Lyapunov functions | non-instantaneous impulses | Mittag-Leffler stability | Primary 34K37 | Secondary 34K45 | Caputo fractional derivative | 34K20 | 34A08 | MATHEMATICS | VARIATIONAL APPROACH

Journal Article

Journal of inequalities and applications, ISSN 1029-242X, 2019, Volume 2019, Issue 1, pp. 1 - 23

Sobolev-type nonlocal fractional differential systems with Clarke’s subdifferential are studied. Sufficient conditions for controllability and constrained...

34B10 | 34K37 | Nonlocal condition | Mathematics | 34A08 | 93C10 | Generalized Clarke subdifferential | Analysis | 93B05 | Contraction mapping principle | Mathematics, general | Controllability | Applications of Mathematics | Sobolev-type Hilfer fractional differential equation | Constrained controllability | MATHEMATICS, APPLIED | INCLUSIONS | EQUATIONS | MATHEMATICS | EVOLUTION | HADAMARD | APPROXIMATE CONTROLLABILITY

34B10 | 34K37 | Nonlocal condition | Mathematics | 34A08 | 93C10 | Generalized Clarke subdifferential | Analysis | 93B05 | Contraction mapping principle | Mathematics, general | Controllability | Applications of Mathematics | Sobolev-type Hilfer fractional differential equation | Constrained controllability | MATHEMATICS, APPLIED | INCLUSIONS | EQUATIONS | MATHEMATICS | EVOLUTION | HADAMARD | APPROXIMATE CONTROLLABILITY

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 12/2014, Volume 17, Issue 4, pp. 1016 - 1038

We study the multiplicity of solutions for fractional differential equations subject to boundary value conditions and impulses. After introducing the notions...

Abstract Harmonic Analysis | fractional differential equations t]impulsive conditions | weak solution | Functional Analysis | Analysis | classical solution | Secondary 34K37, 34K45, 49J40 | three critical points theorem | Mathematics | Integral Transforms, Operational Calculus | Primary 34B37 | Classical solution | Weak solution | Fractional differential equations | Impulsive conditions | Three critical points theorem | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | impulsive conditions | fractional differential equations

Abstract Harmonic Analysis | fractional differential equations t]impulsive conditions | weak solution | Functional Analysis | Analysis | classical solution | Secondary 34K37, 34K45, 49J40 | three critical points theorem | Mathematics | Integral Transforms, Operational Calculus | Primary 34B37 | Classical solution | Weak solution | Fractional differential equations | Impulsive conditions | Three critical points theorem | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | impulsive conditions | fractional differential equations

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 7/2017, Volume 174, Issue 1, pp. 47 - 64

In this paper, an evolution system with a Riemann–Liouville fractional derivative is proposed and analyzed. With the help of a resolvent technique, a suitable...

Riemann–Liouville derivative | 34K37 | Mathematics | Theory of Computation | 47A10 | Optimization | Resolvent | Calculus of Variations and Optimal Control; Optimization | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Optimal controls | 49J15 | DIFFERENTIAL-INCLUSIONS | EXISTENCE | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Riemann-Liouville derivative | RESOLVENTS | SOLVABILITY | EQUATIONS | APPROXIMATE CONTROLLABILITY | DERIVATIVES | Control systems | Evolution | Continuity | Optimal control

Riemann–Liouville derivative | 34K37 | Mathematics | Theory of Computation | 47A10 | Optimization | Resolvent | Calculus of Variations and Optimal Control; Optimization | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Optimal controls | 49J15 | DIFFERENTIAL-INCLUSIONS | EXISTENCE | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Riemann-Liouville derivative | RESOLVENTS | SOLVABILITY | EQUATIONS | APPROXIMATE CONTROLLABILITY | DERIVATIVES | Control systems | Evolution | Continuity | Optimal control

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 08/2017, Volume 20, Issue 4, pp. 963 - 987

This paper treats the approximate controllability of fractional differential systems of Sobolev type in Banach spaces. We first characterize the properties on...

Secondary 34K37 | approximate controllability | Primary 45N05 | compact operators | Sobolev type differential equations | 93B05 | 26A33 | fractional derivative | 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | INCLUSIONS | SYSTEMS | Research | Vector spaces | Mathematical research | Operator theory | Differential equations | Operators (mathematics) | Controllability | Properties (attributes) | Fixed points (mathematics)

Secondary 34K37 | approximate controllability | Primary 45N05 | compact operators | Sobolev type differential equations | 93B05 | 26A33 | fractional derivative | 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | INCLUSIONS | SYSTEMS | Research | Vector spaces | Mathematical research | Operator theory | Differential equations | Operators (mathematics) | Controllability | Properties (attributes) | Fixed points (mathematics)

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 12/2018, Volume 21, Issue 6, pp. 1524 - 1541

Time optimal control problems governed by Riemann-Liouville fractional differential system are considered in this paper. Firstly, the existence results are...

Primary 93C23 | mild solution | Secondary 49J15 | 34K37 | Riemann-Liouville fractional derivatives | time optimal controls | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | RESOLVENTS | APPROXIMATE CONTROLLABILITY | EVOLUTION-EQUATIONS | Time optimal control | Optimal control | Continuity (mathematics)

Primary 93C23 | mild solution | Secondary 49J15 | 34K37 | Riemann-Liouville fractional derivatives | time optimal controls | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | RESOLVENTS | APPROXIMATE CONTROLLABILITY | EVOLUTION-EQUATIONS | Time optimal control | Optimal control | Continuity (mathematics)

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 2015, Volume 18, Issue 1, pp. 146 - 162

We show, using invariant subspace method, how to derive exact solutions to the time fractional Korteweg-de Vries (KdV) equation, potential KdV equation with...

Invariant subspace method | Kilbas-Saigo function | Regular α-singular point | α-analytic function | Mittag-Leffler function | α-ordinary point | MATHEMATICS | ORDER | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | alpha-ordinary point | regular alpha-singular point | alpha-analytic function | invariant subspace method | Invariant subspaces | Differential equations, Nonlinear | Difference equations | Differential equations, Partial | Analysis

Invariant subspace method | Kilbas-Saigo function | Regular α-singular point | α-analytic function | Mittag-Leffler function | α-ordinary point | MATHEMATICS | ORDER | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | alpha-ordinary point | regular alpha-singular point | alpha-analytic function | invariant subspace method | Invariant subspaces | Differential equations, Nonlinear | Difference equations | Differential equations, Partial | Analysis

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On fractional order derivatives and Darboux problem for implicit differential equations

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 6/2012, Volume 15, Issue 2, pp. 168 - 182

In this paper we prove some relations between the Riemann-Liouville and the Caputo fractional order derivatives, and we investigate the existence and...

partial hyperbolic differential equation | Primary 26A33 | mixed regularized derivative | Mathematics | Integral Transforms, Operational Calculus | fractional order | Abstract Harmonic Analysis | left-sided mixed Riemann-Liouville integral | Functional Analysis | solution | fixed point | Analysis | Secondary 35R11, 34K37 | Fractional order | Partial hyperbolic differential equation | Mixed regularized derivative | Left-sided mixed Riemann-Liouville integral | Solution | Fixed point | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

partial hyperbolic differential equation | Primary 26A33 | mixed regularized derivative | Mathematics | Integral Transforms, Operational Calculus | fractional order | Abstract Harmonic Analysis | left-sided mixed Riemann-Liouville integral | Functional Analysis | solution | fixed point | Analysis | Secondary 35R11, 34K37 | Fractional order | Partial hyperbolic differential equation | Mixed regularized derivative | Left-sided mixed Riemann-Liouville integral | Solution | Fixed point | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 12/2018, Volume 21, Issue 6, pp. 1506 - 1523

This paper aims at obtaining a high precision numerical approximation for fractional partial differential equations. This is achieved through appropriate...

extrapolation process | Caputo’s fractional derivative | 34K37 | Secondary 33D45 | fractional partial differential equations | Primary 35R11 | 94A11 | 34A08 | orthogonal polynomials | Caputo's fractional derivative | MATHEMATICS | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Chebyshev approximation | Polynomials | Algorithms | Partial differential equations | Discretization

extrapolation process | Caputo’s fractional derivative | 34K37 | Secondary 33D45 | fractional partial differential equations | Primary 35R11 | 94A11 | 34A08 | orthogonal polynomials | Caputo's fractional derivative | MATHEMATICS | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Chebyshev approximation | Polynomials | Algorithms | Partial differential equations | Discretization

Journal Article