Fractional Calculus and Applied Analysis, ISSN 1311-0454, 6/2014, Volume 17, Issue 2, pp. 552 - 578

In the last decades fractional calculus (FC) became an area of intensive research and development. This paper goes back and recalls important pioneers that...

pioneers | fractional calculus | Primary 26A33 | Secondary 01A55, 01A60, 34A08 | Cole | Heaviside | Lévy | Mathematics | Nigmatullin | Integral Transforms, Operational Calculus | Abstract Harmonic Analysis | Gemant | Gerasimov | Functional Analysis | Analysis | Abel | Rabotnov | Scott Blair | applications | MATHEMATICS, APPLIED | DISPERSION | RELAXATION | VISCOSITY | MATHEMATICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DIELECTRICS | Levy | ABSORPTION

pioneers | fractional calculus | Primary 26A33 | Secondary 01A55, 01A60, 34A08 | Cole | Heaviside | Lévy | Mathematics | Nigmatullin | Integral Transforms, Operational Calculus | Abstract Harmonic Analysis | Gemant | Gerasimov | Functional Analysis | Analysis | Abel | Rabotnov | Scott Blair | applications | MATHEMATICS, APPLIED | DISPERSION | RELAXATION | VISCOSITY | MATHEMATICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DIELECTRICS | Levy | ABSORPTION

Journal Article

Arabian Journal of Mathematics, ISSN 2193-5343, 3/2016, Volume 5, Issue 1, pp. 1 - 7

This paper deals with some existence and Ulam stability results for a class of partial integral equations via Hadamard’s fractional integral, by applying...

34K05 | Mathematics, general | Mathematics | 34A08

34K05 | Mathematics, general | Mathematics | 34A08

Journal Article

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, 04/2019, Volume 22, Issue 2, pp. 242 - 254

We point out a major flaw in the so-called conformable calculus. We demonstrate why it fails at defining a fractional order derivative and where exactly these...

fractional calculus | Primary 26A33 | conformable derivative | Secondary 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

fractional calculus | Primary 26A33 | conformable derivative | Secondary 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Journal Article

Calcolo, ISSN 0008-0624, 12/2016, Volume 53, Issue 4, pp. 521 - 543

In this paper, new operational matrices for shifted Legendre orthonormal polynomial are derived. This polynomial is used as a basis function for developing a...

33C45 | Fractional optimal control problem | Time delay system | Mathematics | Theory of Computation | Caputo definition | 34A08 | Riemann–Liouville definition | 65M70 | Numerical Analysis | Operational matrix | Lagrange multiplier method | Legendre polynomials | MATRIX | MATHEMATICS, APPLIED | ALGORITHM | EQUATIONS | MATHEMATICS | Riemann-Liouville definition | NUMERICAL-SOLUTIONS | TRANSIENTS | SYSTEMS | HYBRID | Performance indices | Lagrange multiplier | Basis functions | Lagrange multipliers | Integrals | Optimal control | Polynomials | Control theory | Derivatives | Delay | Approximation | Mathematical analysis | Mathematical models

33C45 | Fractional optimal control problem | Time delay system | Mathematics | Theory of Computation | Caputo definition | 34A08 | Riemann–Liouville definition | 65M70 | Numerical Analysis | Operational matrix | Lagrange multiplier method | Legendre polynomials | MATRIX | MATHEMATICS, APPLIED | ALGORITHM | EQUATIONS | MATHEMATICS | Riemann-Liouville definition | NUMERICAL-SOLUTIONS | TRANSIENTS | SYSTEMS | HYBRID | Performance indices | Lagrange multiplier | Basis functions | Lagrange multipliers | Integrals | Optimal control | Polynomials | Control theory | Derivatives | Delay | Approximation | Mathematical analysis | Mathematical models

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 04/2017, Volume 20, Issue 2, pp. 307 - 336

Since the 60s of last century Fractional Calculus exhibited a remarkable progress and presently it is recognized to be an important topic in the scientific...

01A61 | 01A60 | 60G22 | fractional calculus | development | Primary 26A33 | fractional order differential equations | fractional order mathematical models | 35R11 | 01A67 | Secondary 34A08 | applications | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Analysis | Calculus | Differentiation | Mathematical analysis | Fractional calculus

01A61 | 01A60 | 60G22 | fractional calculus | development | Primary 26A33 | fractional order differential equations | fractional order mathematical models | 35R11 | 01A67 | Secondary 34A08 | applications | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Analysis | Calculus | Differentiation | Mathematical analysis | Fractional calculus

Journal Article

Boundary Value Problems, ISSN 1687-2762, 12/2016, Volume 2016, Issue 1, pp. 1 - 11

In this paper, the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order...

34B15 | mixed boundary value problem | Mathematics | fractional differential equations | 34A08 | Ordinary Differential Equations | Analysis | Difference and Functional Equations | Approximations and Expansions | Mathematics, general | impulse | Partial Differential Equations | fixed point theorem | MATHEMATICS | MATHEMATICS, APPLIED | SYSTEMS | Fixed point theory | Boundary value problems | Usage | Differential equations | Tests, problems and exercises | Theorems | Mathematical analysis | Uniqueness | Nonlinearity | Boundary conditions

34B15 | mixed boundary value problem | Mathematics | fractional differential equations | 34A08 | Ordinary Differential Equations | Analysis | Difference and Functional Equations | Approximations and Expansions | Mathematics, general | impulse | Partial Differential Equations | fixed point theorem | MATHEMATICS | MATHEMATICS, APPLIED | SYSTEMS | Fixed point theory | Boundary value problems | Usage | Differential equations | Tests, problems and exercises | Theorems | Mathematical analysis | Uniqueness | Nonlinearity | Boundary conditions

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

Numerical Algorithms, ISSN 1017-1398, 9/2016, Volume 73, Issue 1, pp. 91 - 113

This article adapts an operational matrix formulation of the collocation method for the one- and two-dimensional nonlinear fractional sub-diffusion equations...

Jacobi polynomials | 33C45 | Numeric Computing | Theory of Computation | 34A08 | Algorithms | Algebra | 65M70 | Numerical Analysis | Computer Science | Two-dimensional nonlinear fractional sub-diffusion equation | Operational matrix | Collocation method | Nonlinear fractional reaction sub-diffusion equation | ORDER | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | APPROXIMATION | GALERKIN METHOD | DIFFERENTIAL-EQUATIONS | TERM | Analysis | Methods | Discretization | Collocation | Mathematical analysis | Collocation methods | Nonlinearity | Mathematical models | Spectra | Derivatives

Jacobi polynomials | 33C45 | Numeric Computing | Theory of Computation | 34A08 | Algorithms | Algebra | 65M70 | Numerical Analysis | Computer Science | Two-dimensional nonlinear fractional sub-diffusion equation | Operational matrix | Collocation method | Nonlinear fractional reaction sub-diffusion equation | ORDER | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | APPROXIMATION | GALERKIN METHOD | DIFFERENTIAL-EQUATIONS | TERM | Analysis | Methods | Discretization | Collocation | Mathematical analysis | Collocation methods | Nonlinearity | Mathematical models | Spectra | Derivatives

Journal Article

Neural Computing and Applications, ISSN 0941-0643, 3/2018, Volume 29, Issue 5, pp. 1465 - 1479

This paper presents iterative reproducing kernel algorithm for obtaining the numerical solutions of Bagley–Torvik and Painlevé equations of fractional order....

Bagley–Torvik equation | 34B15 | 35F10 | Reproducing kernel algorithm | Fourier series expansion | Data Mining and Knowledge Discovery | 47B32 | Computational Science and Engineering | 34A08 | Computational Biology/Bioinformatics | Computer Science | Image Processing and Computer Vision | Painlevé equation | Artificial Intelligence (incl. Robotics) | Fractional-order derivative | Probability and Statistics in Computer Science | Algorithms

Bagley–Torvik equation | 34B15 | 35F10 | Reproducing kernel algorithm | Fourier series expansion | Data Mining and Knowledge Discovery | 47B32 | Computational Science and Engineering | 34A08 | Computational Biology/Bioinformatics | Computer Science | Image Processing and Computer Vision | Painlevé equation | Artificial Intelligence (incl. Robotics) | Fractional-order derivative | Probability and Statistics in Computer Science | Algorithms

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 | Thermodynamics | Probability theory

stable distributions | Green’s function | 34K37 | fractional diffusions | Primary 35R11 | 60E07 | Secondary 34A08 | Thermodynamics | Probability theory

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 02/2017, Volume 20, Issue 1, pp. 159 - 172

The importance of the concept of stability in fractional order system and control has been recognized for some time now. Recently, it has become evident that...

93D21 | Primary 26A33 | 34D10 | fractional order systems | stability criterion | 93D09 | fractional order positive definite | linear matrix inequality (LMI) | 93C73 | Secondary 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | STABILIZATION

93D21 | Primary 26A33 | 34D10 | fractional order systems | stability criterion | 93D09 | fractional order positive definite | linear matrix inequality (LMI) | 93C73 | Secondary 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | STABILIZATION

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2018, Volume 2018, Issue 1, pp. 1 - 17

By using the coincidence degree theory, we present an existence result for a coupled system of nonlinear fractional differential equations with multi-point...

Coupled system | Fractional differential equations | Analysis | 34B10 | Mathematics, general | Mathematics | Applications of Mathematics | At resonance | Coincidence degree | 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | DIFFERENTIAL-EQUATIONS | Nonlinear equations | Boundary value problems | Differential equations | Research

Coupled system | Fractional differential equations | Analysis | 34B10 | Mathematics, general | Mathematics | Applications of Mathematics | At resonance | Coincidence degree | 34A08 | MATHEMATICS | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | DIFFERENTIAL-EQUATIONS | Nonlinear equations | Boundary value problems | Differential equations | Research

Journal Article

Nonlinear Dynamics, ISSN 0924-090X, 3/2019, Volume 95, Issue 4, pp. 3063 - 3073

We prove that conformable “fractional” differentiability of a function $$f:[0,\infty [\,\longrightarrow \mathbb {R}$$ f : [ 0 , ∞ [ ⟶ R is nothing else than...

Viscoelasticity | Engineering | Vibration, Dynamical Systems, Control | Fractional analysis | 74D05 | Fractional differential equations | Classical Mechanics | 26A33 | Automotive Engineering | Mechanical Engineering | Fractional derivative | 34A08 | MECHANICS | EQUATIONS | ENGINEERING, MECHANICAL | Electrical engineering | Analysis | Differential equations

Viscoelasticity | Engineering | Vibration, Dynamical Systems, Control | Fractional analysis | 74D05 | Fractional differential equations | Classical Mechanics | 26A33 | Automotive Engineering | Mechanical Engineering | Fractional derivative | 34A08 | MECHANICS | EQUATIONS | ENGINEERING, MECHANICAL | Electrical engineering | Analysis | Differential equations

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 6/2014, Volume 17, Issue 2, pp. 348 - 360

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Hadamard type fractional differential equations and integral...

Abstract Harmonic Analysis | Hadamard fractional derivative | Primary 34A08 | Functional Analysis | fixed point theorems | Analysis | Secondary 34A12, 34B15 | Mathematics | integral boundary conditions | Integral Transforms, Operational Calculus | fractional differential systems | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | POSITIVE SOLUTIONS

Abstract Harmonic Analysis | Hadamard fractional derivative | Primary 34A08 | Functional Analysis | fixed point theorems | Analysis | Secondary 34A12, 34B15 | Mathematics | integral boundary conditions | Integral Transforms, Operational Calculus | fractional differential systems | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | POSITIVE SOLUTIONS

Journal Article

Advances in Difference Equations, ISSN 1687-1839, 12/2018, Volume 2018, Issue 1, pp. 1 - 15

In this work, we study the diabetes model and its complications with the Caputo–Fabrizio fractional derivative. A deterministic mathematical model pertaining...

35C05 | Mathematics | Picard–Lindelof approach | 34A08 | Ordinary Differential Equations | Fixed point theorem | Functional Analysis | Analysis | Homotopy analysis method | Difference and Functional Equations | 35A22 | Fractional diabetes model | Mathematics, general | 35A20 | Laplace transform | Partial Differential Equations | SYSTEM | MATHEMATICS | MATHEMATICS, APPLIED | DOMAIN | LEFFLER TYPE KERNEL | Picard-Lindelof approach | MATHEMATICAL-MODELS | Fixed points (mathematics) | Mathematical models | Diabetes | Laplace transforms | Computer simulation | Diabetes mellitus

35C05 | Mathematics | Picard–Lindelof approach | 34A08 | Ordinary Differential Equations | Fixed point theorem | Functional Analysis | Analysis | Homotopy analysis method | Difference and Functional Equations | 35A22 | Fractional diabetes model | Mathematics, general | 35A20 | Laplace transform | Partial Differential Equations | SYSTEM | MATHEMATICS | MATHEMATICS, APPLIED | DOMAIN | LEFFLER TYPE KERNEL | Picard-Lindelof approach | MATHEMATICAL-MODELS | Fixed points (mathematics) | Mathematical models | Diabetes | Laplace transforms | Computer simulation | Diabetes mellitus

Journal Article

Mathematica Slovaca, ISSN 0139-9918, 06/2019, Volume 69, Issue 3, pp. 599 - 610

In this paper we investigate the asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < < 2 and in particular existence...

fractional evolution equations | 34C25 | asymptotically periodic solutions | sectorial operators | 34A08 | EXISTENCE | MATHEMATICS

fractional evolution equations | 34C25 | asymptotically periodic solutions | sectorial operators | 34A08 | EXISTENCE | MATHEMATICS

Journal Article

Advances in Nonlinear Analysis, ISSN 2191-9496, 06/2017, Volume 8, Issue 1, pp. 482 - 496

This paper deals with the existence of periodic solutions of fractional differential equations with periodic impulses. The first part of the paper is devoted...

34C25 | 34A37 | Fractional differential equations | periodic solutions | impulses | existence result | 34A08 | MATHEMATICS | MATHEMATICS, APPLIED

34C25 | 34A37 | Fractional differential equations | periodic solutions | impulses | existence result | 34A08 | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

International Journal of Control, ISSN 0020-7179, 06/2017, Volume 90, Issue 6, pp. 1230 - 1244

In this manuscript, we report a new operational technique for approximating the numerical solution of fractional optimal control (FOC) problems. The...

Gauss quadrature | 33C45 | fractional optimal control problem | 65M70 | Lagrange multiplier method | Orthonormal polynomials | operational matrix | 34A08 | POLYNOMIALS | CABLE | MODELS | CALCULUS | QUADRATURE | NUMERICAL TECHNIQUE | AUTOMATION & CONTROL SYSTEMS | Chebyshev approximation | Legendre functions | Lagrange multipliers | Optimal control

Gauss quadrature | 33C45 | fractional optimal control problem | 65M70 | Lagrange multiplier method | Orthonormal polynomials | operational matrix | 34A08 | POLYNOMIALS | CABLE | MODELS | CALCULUS | QUADRATURE | NUMERICAL TECHNIQUE | AUTOMATION & CONTROL SYSTEMS | Chebyshev approximation | Legendre functions | Lagrange multipliers | Optimal control

Journal Article

20.
Full Text
Iterative learning control with pulse compensation for fractional differential systems

Mathematica Slovaca, ISSN 0139-9918, 06/2018, Volume 68, Issue 3, pp. 563 - 574

In this paper, we explore PD-type ILC schemes of fractional version with pulse compensation for single-input-single-output fractional differential systems....

Secondary 34K35, 93B05 | pulse compensation | sign function | type iterative learning control | Primary 26A33, 34A08 | & | fractional differential systems | P&D

Secondary 34K35, 93B05 | pulse compensation | sign function | type iterative learning control | Primary 26A33, 34A08 | & | fractional differential systems | P&D

Journal Article

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