Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2010, Volume 367, Issue 1, pp. 260 - 272

In this paper, we shall discuss the properties of the well-known Mittag–Leffler function, and consider the existence and uniqueness of solution of the initial...

Fractional differential equation | Riemann–Liouville sequential fractional derivatives | Initial value problem | Upper solution and lower solution | Riemann-Liouville sequential fractional derivatives | EXISTENCE | MATHEMATICS, APPLIED | BOUNDARY-VALUE-PROBLEMS | INTEGRAL-EQUATIONS | UNIQUENESS | MATHEMATICS | ORDER | MONOTONE ITERATIVE TECHNIQUE | 1ST-ORDER

Fractional differential equation | Riemann–Liouville sequential fractional derivatives | Initial value problem | Upper solution and lower solution | Riemann-Liouville sequential fractional derivatives | EXISTENCE | MATHEMATICS, APPLIED | BOUNDARY-VALUE-PROBLEMS | INTEGRAL-EQUATIONS | UNIQUENESS | MATHEMATICS | ORDER | MONOTONE ITERATIVE TECHNIQUE | 1ST-ORDER

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

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2011, Volume 384, Issue 2, pp. 211 - 231

In this paper, we investigate the existence of solutions of the periodic boundary value problem for nonlinear impulsive fractional differential equation...

Impulsive | Periodic boundary value problem | Riemann–Liouville sequential fractional derivative | Riemann-Liouville sequential fractional derivative | EXISTENCE | SPACE | MATHEMATICS | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | INITIAL-VALUE PROBLEMS | UNIQUENESS

Impulsive | Periodic boundary value problem | Riemann–Liouville sequential fractional derivative | Riemann-Liouville sequential fractional derivative | EXISTENCE | SPACE | MATHEMATICS | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | INITIAL-VALUE PROBLEMS | UNIQUENESS

Journal Article

Nonlinear Dynamics, ISSN 0924-090X, 7/2008, Volume 53, Issue 1, pp. 67 - 74

Fractional mechanics describe both conservative and nonconservative systems. The fractional variational principles gained importance in studying the fractional...

Automotive and Aerospace Engineering, Traffic | Fractional Riemann–Liouville derivative | Faà di Bruno formula | Engineering | Vibration, Dynamical Systems, Control | Fractional Euler–Lagrange equations | Mechanics | Fractional Lagrangians | Mechanical Engineering | Fractional calculus | Fractional Euler-Lagrange equations | Fractional Riemann-Liouville derivative | CLASSICAL FIELDS | fractional calculus | SEQUENTIAL MECHANICS | Faa di Bruno formula | CALCULUS | fractional Lagrangians | FORMULATION | ENGINEERING, MECHANICAL | fractional Riemann-Liouville derivative | fractional Euler-Lagrange equations | MECHANICS | LINEAR VELOCITIES | SYSTEMS | VARIATIONAL-PROBLEMS | FORMALISM | Mechanics (physics) | Mathematical analysis | Classical mechanics | Euler-Lagrange equation | Equations of motion | Variational principles

Automotive and Aerospace Engineering, Traffic | Fractional Riemann–Liouville derivative | Faà di Bruno formula | Engineering | Vibration, Dynamical Systems, Control | Fractional Euler–Lagrange equations | Mechanics | Fractional Lagrangians | Mechanical Engineering | Fractional calculus | Fractional Euler-Lagrange equations | Fractional Riemann-Liouville derivative | CLASSICAL FIELDS | fractional calculus | SEQUENTIAL MECHANICS | Faa di Bruno formula | CALCULUS | fractional Lagrangians | FORMULATION | ENGINEERING, MECHANICAL | fractional Riemann-Liouville derivative | fractional Euler-Lagrange equations | MECHANICS | LINEAR VELOCITIES | SYSTEMS | VARIATIONAL-PROBLEMS | FORMALISM | Mechanics (physics) | Mathematical analysis | Classical mechanics | Euler-Lagrange equation | Equations of motion | Variational principles

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 09/2013, Volume 66, Issue 5, pp. 883 - 891

A Cauchy-type nonlinear problem for a class of fractional differential equations involving sequential derivatives is considered. Some properties and...

Fractional differential equation | Riemann–Liouville fractional derivative | Sequential fractional derivative | Fractional derivatives | Riemann-Liouville fractional derivative | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | CALCULUS | ABSTRACT DIFFERENTIAL-EQUATION

Fractional differential equation | Riemann–Liouville fractional derivative | Sequential fractional derivative | Fractional derivatives | Riemann-Liouville fractional derivative | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | CALCULUS | ABSTRACT DIFFERENTIAL-EQUATION

Journal Article

中国物理B：英文版, ISSN 1674-1056, 2012, Volume 21, Issue 8, pp. 302 - 306

In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A...

衍生 | 运动微分方程 | 分数阶微分方程 | 工具 | 哈密顿正则方程 | 黎曼 | 哈密顿原理 | 拉格朗日方程 | fractional Hamilton canonical equation | fractional Hamilton principle | combined Riemann-Liouville fractional derivative | fractional Lagrange equation | SEQUENTIAL MECHANICS | PHYSICS, MULTIDISCIPLINARY | SYSTEMS | VARIATIONAL-PROBLEMS | FORMULATION | HAMILTON FORMALISM | EULER-LAGRANGE | Operators (mathematics) | Mathematical analysis | Hamilton's principle | Differential equations | Paper | Derivatives | Variational principles | Mechanical systems

衍生 | 运动微分方程 | 分数阶微分方程 | 工具 | 哈密顿正则方程 | 黎曼 | 哈密顿原理 | 拉格朗日方程 | fractional Hamilton canonical equation | fractional Hamilton principle | combined Riemann-Liouville fractional derivative | fractional Lagrange equation | SEQUENTIAL MECHANICS | PHYSICS, MULTIDISCIPLINARY | SYSTEMS | VARIATIONAL-PROBLEMS | FORMULATION | HAMILTON FORMALISM | EULER-LAGRANGE | Operators (mathematics) | Mathematical analysis | Hamilton's principle | Differential equations | Paper | Derivatives | Variational principles | Mechanical systems

Journal Article

International Journal of Non-Linear Mechanics, ISSN 0020-7462, 04/2017, Volume 90, pp. 32 - 38

Based on Riemann-Liouville fractional derivatives, conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems are...

Conserved quantity | Riemann-Liouville fractional derivative | Adiabatic invariant | Generalized Birkhoffian system | SEQUENTIAL MECHANICS | CALCULUS | PERTURBATION | LAGRANGIANS | MEI SYMMETRY | FORMULATION | MECHANICS | DYNAMICAL-SYSTEMS | NOETHERS THEOREM | Generalized Birlchoffian system | VARIATIONAL-PROBLEMS | DERIVATIVES

Conserved quantity | Riemann-Liouville fractional derivative | Adiabatic invariant | Generalized Birkhoffian system | SEQUENTIAL MECHANICS | CALCULUS | PERTURBATION | LAGRANGIANS | MEI SYMMETRY | FORMULATION | MECHANICS | DYNAMICAL-SYSTEMS | NOETHERS THEOREM | Generalized Birlchoffian system | VARIATIONAL-PROBLEMS | DERIVATIVES

Journal Article

Advances in Difference Equations, ISSN 1687-1839, 12/2017, Volume 2017, Issue 1, pp. 1 - 10

We discuss the existence and uniqueness of solutions for a nonlocal three-point boundary value problem of sequential fractional differential equations on an...

Riemann-Liouville fractional integral | 34B10 | existence | Mathematics | Caputo fractional derivative | 34A08 | Ordinary Differential Equations | Functional Analysis | fixed point | Analysis | Difference and Functional Equations | Mathematics, general | sequential fractional derivative | Partial Differential Equations | MATHEMATICS | MATHEMATICS, APPLIED | Theorems (Mathematics) | Boundary value problems | Usage | Differential equations | Boundary conditions | Fixed points (mathematics) | Mathematical analysis | Uniqueness

Riemann-Liouville fractional integral | 34B10 | existence | Mathematics | Caputo fractional derivative | 34A08 | Ordinary Differential Equations | Functional Analysis | fixed point | Analysis | Difference and Functional Equations | Mathematics, general | sequential fractional derivative | Partial Differential Equations | MATHEMATICS | MATHEMATICS, APPLIED | Theorems (Mathematics) | Boundary value problems | Usage | Differential equations | Boundary conditions | Fixed points (mathematics) | Mathematical analysis | Uniqueness

Journal Article

Differential Equations and Dynamical Systems, ISSN 0971-3514, 7/2017, Volume 25, Issue 3, pp. 373 - 383

This paper adopts the inverse fractional differential operator method for obtaining the explicit particular solution to a linear sequential fractional...

Riemann–Liouville derivative | Inverse fractional differential operators | Fractional differential equations | Mathematics, general | Mathematics | Inverse differential operators | Engineering, general | Computer Science, general | Jumarie’s fractional derivation | Differential equations

Riemann–Liouville derivative | Inverse fractional differential operators | Fractional differential equations | Mathematics, general | Mathematics | Inverse differential operators | Engineering, general | Computer Science, general | Jumarie’s fractional derivation | Differential equations

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 3/2013, Volume 16, Issue 1, pp. 171 - 188

In this paper we establish some bounds for the solution of a Cauchytype problem for a class of fractional differential equations with a weighted sequential...

34A34 | 34A12 | 26A33 | fractional differential equation | Mathematics | Erdélyi-Kober fractional derivative | Integral Transforms, Operational Calculus | 34A08 | Abstract Harmonic Analysis | Functional Analysis | Analysis | Riemann-Liouville fractional derivative | 45J05 | sequential fractional derivatives | generalized fractional calculus | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DIFFERENTIAL-EQUATION | DIFFUSION | Erdelyi-Kober fractional derivative

34A34 | 34A12 | 26A33 | fractional differential equation | Mathematics | Erdélyi-Kober fractional derivative | Integral Transforms, Operational Calculus | 34A08 | Abstract Harmonic Analysis | Functional Analysis | Analysis | Riemann-Liouville fractional derivative | 45J05 | sequential fractional derivatives | generalized fractional calculus | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DIFFERENTIAL-EQUATION | DIFFUSION | Erdelyi-Kober fractional derivative

Journal Article

Journal of Applied Mathematics and Computing, ISSN 1598-5865, 2/2018, Volume 56, Issue 1, pp. 367 - 389

This paper studies the existence of solutions for a six-point boundary value problem of coupled system of nonlinear Caputo (Liouville–Caputo) type sequential...

Computational Mathematics and Numerical Analysis | 34B15 | Coupled system | 34A12 | Mathematics | Theory of Computation | Integral boundary conditions | 34A08 | Riemann–Liouville | Mathematics of Computing | Caputo derivative | Mathematical and Computational Engineering | Six-point | Existence | SYSTEM | MATHEMATICS, APPLIED | NONEXISTENCE | POSITIVE SOLUTIONS | DIFFERENTIAL-EQUATIONS | CAUCHY-PROBLEM | MATHEMATICS | Riemann-Liouville | Analysis | Differential equations | Boundary conditions | Boundary value problems | Mathematical analysis | Integrals

Computational Mathematics and Numerical Analysis | 34B15 | Coupled system | 34A12 | Mathematics | Theory of Computation | Integral boundary conditions | 34A08 | Riemann–Liouville | Mathematics of Computing | Caputo derivative | Mathematical and Computational Engineering | Six-point | Existence | SYSTEM | MATHEMATICS, APPLIED | NONEXISTENCE | POSITIVE SOLUTIONS | DIFFERENTIAL-EQUATIONS | CAUCHY-PROBLEM | MATHEMATICS | Riemann-Liouville | Analysis | Differential equations | Boundary conditions | Boundary value problems | Mathematical analysis | Integrals

Journal Article

Boundary Value Problems, ISSN 1687-2762, 12/2012, Volume 2012, Issue 1, pp. 1 - 14

A Cauchy-type nonlinear problem for a class of fractional differential equations with sequential derivatives is considered in the space of weighted continuous...

fractional derivatives | Ordinary Differential Equations | Analysis | Riemann-Liouville fractional derivative | Difference and Functional Equations | Approximations and Expansions | fractional differential equation | Mathematics, general | Mathematics | sequential fractional derivative | Partial Differential Equations | Fractional differential equation | Sequential fractional derivative | Fractional derivatives | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | ABSTRACT DIFFERENTIAL-EQUATION | Functions, Continuous | Usage | Research | Integral equations | Mathematical research | Cauchy problem | Boundary value problems | Mathematical analysis | Uniqueness | Classification | Differential equations | Nonlinearity | Derivatives

fractional derivatives | Ordinary Differential Equations | Analysis | Riemann-Liouville fractional derivative | Difference and Functional Equations | Approximations and Expansions | fractional differential equation | Mathematics, general | Mathematics | sequential fractional derivative | Partial Differential Equations | Fractional differential equation | Sequential fractional derivative | Fractional derivatives | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | ABSTRACT DIFFERENTIAL-EQUATION | Functions, Continuous | Usage | Research | Integral equations | Mathematical research | Cauchy problem | Boundary value problems | Mathematical analysis | Uniqueness | Classification | Differential equations | Nonlinearity | Derivatives

Journal Article

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, ISSN 2081-545X, 12/2018, Volume 17, Issue 1, pp. 103 - 125

In this paper we study existence and uniqueness of solutions for a coupled system consisting of fractional differential equations of Caputo type, subject to...

coupled system | uniqueness | existence | Caputo fractional derivative | Hyers–Ulam stability | Riemann–Liouville fractional integral | fixed point theorem

coupled system | uniqueness | existence | Caputo fractional derivative | Hyers–Ulam stability | Riemann–Liouville fractional integral | fixed point theorem

Journal Article

Electronic Journal of Qualitative Theory of Differential Equations, ISSN 1417-3875, 2011, Volume 2011, Issue 87, pp. 1 - 13

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence of solution of the periodic boundary value...

Fractional differential equation | Periodic boundary value problem | Upper solution and lower solution | Riemann-Liouville sequential fractional derivatives | MATHEMATICS | MATHEMATICS, APPLIED | periodic boundary value problem | fractional differential equation | upper solution and lower solution | riemann-liouville sequential fractional derivatives

Fractional differential equation | Periodic boundary value problem | Upper solution and lower solution | Riemann-Liouville sequential fractional derivatives | MATHEMATICS | MATHEMATICS, APPLIED | periodic boundary value problem | fractional differential equation | upper solution and lower solution | riemann-liouville sequential fractional derivatives

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2010, Volume 371, Issue 1, pp. 57 - 68

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: D α u ( t ) + f ( t , u ( t ) , D μ u ( t...

Fractional differential equation | Singular Dirichlet problem | Riemann–Liouville fractional derivative | Positive solution | Riemann-Liouville fractional derivative | MATHEMATICS | ORDER | MATHEMATICS, APPLIED | BOUNDARY-VALUE PROBLEM

Fractional differential equation | Singular Dirichlet problem | Riemann–Liouville fractional derivative | Positive solution | Riemann-Liouville fractional derivative | MATHEMATICS | ORDER | MATHEMATICS, APPLIED | BOUNDARY-VALUE PROBLEM

Journal Article

Journal of Applied Mathematics and Computing, ISSN 1598-5865, 2/2012, Volume 38, Issue 1, pp. 641 - 652

In this paper, we consider the following sequential fractional differential equation with initial value problem: where 0<α≤1 and f:[0,1]×ℝ×ℝ→ℝ is continuous....

Computational Mathematics and Numerical Analysis | Riemann-Liouville fractional derivatives | Fixed point theorem | Mathematics of Computing | Initial value problem | 34A12 | Appl.Mathematics/Computational Methods of Engineering | 26A33 | Sequential fractional differential equation | Mathematics | Theory of Computation | 34A40 | Differential equations | Studies | Mathematical analysis | Initial value problems | Mathematical models | Computation | Uniqueness

Computational Mathematics and Numerical Analysis | Riemann-Liouville fractional derivatives | Fixed point theorem | Mathematics of Computing | Initial value problem | 34A12 | Appl.Mathematics/Computational Methods of Engineering | 26A33 | Sequential fractional differential equation | Mathematics | Theory of Computation | 34A40 | Differential equations | Studies | Mathematical analysis | Initial value problems | Mathematical models | Computation | Uniqueness

Journal Article

Journal of Automation and Information Sciences, ISSN 1064-2315, 2008, Volume 40, Issue 6, pp. 1 - 11

Consideration is given to inhomogeneous linear systems of differential equations with classical Riemann-Liouville fractional derivatives as well as the...

Riemann-Liouville fractional derivatives | Sequential derivatives | Fractional differentiation | Derivation of fractional order | fractional differentiation | sequential derivatives | AUTOMATION & CONTROL SYSTEMS | derivation of fractional order

Riemann-Liouville fractional derivatives | Sequential derivatives | Fractional differentiation | Derivation of fractional order | fractional differentiation | sequential derivatives | AUTOMATION & CONTROL SYSTEMS | derivation of fractional order

Journal Article

18.
On solvability of some boundary value problems for a fractional analogue of the Helmholtz equation

New York Journal of Mathematics, ISSN 1076-9803, 2014, Volume 20, pp. 1237 - 1251

In this paper we study some boundary value problems for fractional analogue of Helmholtz equation in a rectangular and in a half-band. Theorems about existence...

Fractional differential equation | Laplace operator | Boundary value problem | Mittag-Leffler function | Helmholtz equation | Riemann–Liouville operator | Sequential derivative | Caputo operator | MATHEMATICS | Riemann-Liouville operator | fractional differential equation | sequential derivative | boundary value problem

Fractional differential equation | Laplace operator | Boundary value problem | Mittag-Leffler function | Helmholtz equation | Riemann–Liouville operator | Sequential derivative | Caputo operator | MATHEMATICS | Riemann-Liouville operator | fractional differential equation | sequential derivative | boundary value problem

Journal Article

Computational and Applied Mathematics, ISSN 0101-8205, 5/2018, Volume 37, Issue 2, pp. 1012 - 1026

Eigenfunctions associated with Riemann–Liouville and Caputo fractional differential operators are obtained by imposing a restriction on the fractional...

Computational Mathematics and Numerical Analysis | Mittag-Leffler functions | Linear fractional differential equations | Mathematical Applications in Computer Science | Riemann–Liouville derivatives | 26A33 Fractional derivatives and integrals | Mathematics | Applications of Mathematics | Mathematical Applications in the Physical Sciences | Caputo derivatives | MATHEMATICS, APPLIED | CALCULUS | MODEL | Riemann-Liouville derivatives

Computational Mathematics and Numerical Analysis | Mittag-Leffler functions | Linear fractional differential equations | Mathematical Applications in Computer Science | Riemann–Liouville derivatives | 26A33 Fractional derivatives and integrals | Mathematics | Applications of Mathematics | Mathematical Applications in the Physical Sciences | Caputo derivatives | MATHEMATICS, APPLIED | CALCULUS | MODEL | Riemann-Liouville derivatives

Journal Article

Fractional Calculus and Applied Analysis, ISSN 1311-0454, 2015, Volume 18, Issue 1, pp. 192 - 207

In this paper, we define fractional derivative of arbitrary complex order of the distributions concentrated on R+, based on convolutions of generalized...

Dirac delta function | Caputo derivative and Riemann-Liouville derivative | Convolution | Gamma function | Abel's equation | Distribution | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DISTRIBUTIONAL PRODUCTS | convolution | distribution

Dirac delta function | Caputo derivative and Riemann-Liouville derivative | Convolution | Gamma function | Abel's equation | Distribution | MATHEMATICS | MATHEMATICS, APPLIED | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DISTRIBUTIONAL PRODUCTS | convolution | distribution

Journal Article

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