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

Computers and Mathematics with Applications, ISSN 0898-1221, 07/2015, Volume 70, Issue 2, pp. 158 - 166

In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics, namely the time fractional...

Improved fractional sub-equation method | Modified Riemann–Liouville derivative | space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation | Time fractional KdV–Zakharov–Kuznetsov (KdV–ZK) equation | Mittag-Leffler function | space-time fractional modified KdV-Zakharov-Kuznetsov (mKdV-ZK) equation | Time fractional KdV-Zakharov-Kuznetsov (KdV-ZK) equation | Modified Riemann-Liouville derivative | MATHEMATICS, APPLIED | DIFFERENTIAL-DIFFERENCE EQUATION | Nonlinear evolution equations | Construction | Mathematical models | Derivatives | Mathematical analysis | Exact solutions

Improved fractional sub-equation method | Modified Riemann–Liouville derivative | space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation | Time fractional KdV–Zakharov–Kuznetsov (KdV–ZK) equation | Mittag-Leffler function | space-time fractional modified KdV-Zakharov-Kuznetsov (mKdV-ZK) equation | Time fractional KdV-Zakharov-Kuznetsov (KdV-ZK) equation | Modified Riemann-Liouville derivative | MATHEMATICS, APPLIED | DIFFERENTIAL-DIFFERENCE EQUATION | Nonlinear evolution equations | Construction | Mathematical models | Derivatives | Mathematical analysis | Exact solutions

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 07/2012, Volume 263, Issue 2, pp. 476 - 510

In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional...

Fractional resolvent | Riemann–Liouville fractional derivative | Well-posedness | Cauchy problem | Riemann-Liouville fractional derivative | MATHEMATICS | DIFFUSION-EQUATIONS | Differential equations

Fractional resolvent | Riemann–Liouville fractional derivative | Well-posedness | Cauchy problem | Riemann-Liouville fractional derivative | MATHEMATICS | DIFFUSION-EQUATIONS | Differential equations

Journal Article

Integral Transforms and Special Functions, ISSN 1065-2469, 11/2010, Volume 21, Issue 11, pp. 797 - 814

In this paper, we study a certain family of generalized Riemann-Liouville fractional derivative operators of order α and type β, which were introduced and...

Riemann-Liouville fractional derivative operator | fractional kinetic equations | Fox-Wright hypergeometric functions | Secondary: 47B38 | Laplace transform method | Lebesgue integrable functions | Primary: 26A33 | fractional differential equations | generalized Mittag-Leffler function | Hardy-type inequalities | Volterra differintegral equations | Fractional differential equations | Fractional kinetic equations | Riemann-liouville fractional derivative operator | Generalized mittag-leffler function | Fox-wright hypergeometric functions | MATHEMATICS | MATHEMATICS, APPLIED | EQUATIONS | Mathematical functions | Calculus | Derivatives

Riemann-Liouville fractional derivative operator | fractional kinetic equations | Fox-Wright hypergeometric functions | Secondary: 47B38 | Laplace transform method | Lebesgue integrable functions | Primary: 26A33 | fractional differential equations | generalized Mittag-Leffler function | Hardy-type inequalities | Volterra differintegral equations | Fractional differential equations | Fractional kinetic equations | Riemann-liouville fractional derivative operator | Generalized mittag-leffler function | Fox-wright hypergeometric functions | MATHEMATICS | MATHEMATICS, APPLIED | EQUATIONS | Mathematical functions | Calculus | Derivatives

Journal Article

Mediterranean Journal of Mathematics, ISSN 1660-5446, 12/2017, Volume 14, Issue 6, pp. 1 - 26

In this paper, we introduce a new type of fractional derivative, which we called truncated $${\mathcal {V}}$$ V -fractional derivative, for $$\alpha $$ α...

Riemann–Liouville integral | 26A42 | Riemann–Liouville derivative | {\mathcal {V}}$$ V -fractional integral | 26A24 | {\mathcal {V}}$$ V -fractional | 26A33 | Mathematics, general | 26A06 | 26A39 | Mathematics | Mittag–Leffler functions | V-fractional integral | V-fractional | MATHEMATICS | MATHEMATICS, APPLIED | Mittag-Leffler functions | Riemann-Liouville derivative | CALCULUS | V-fractional derivative | Riemann-Liouville integral

Riemann–Liouville integral | 26A42 | Riemann–Liouville derivative | {\mathcal {V}}$$ V -fractional integral | 26A24 | {\mathcal {V}}$$ V -fractional | 26A33 | Mathematics, general | 26A06 | 26A39 | Mathematics | Mittag–Leffler functions | V-fractional integral | V-fractional | MATHEMATICS | MATHEMATICS, APPLIED | Mittag-Leffler functions | Riemann-Liouville derivative | CALCULUS | V-fractional derivative | Riemann-Liouville integral

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

Applied Mathematics and Computation, ISSN 0096-3003, 2011, Volume 218, Issue 3, pp. 860 - 865

The paper presents a new fractional integration, which generalizes the Riemann–Liouville and Hadamard fractional integrals into a single form. Conditions are...

Hadamard fractional derivative | Generalized fractional integral | Riemann–Liouville fractional derivative | Generalized fractional derivative | Semigroup property | Riemann-Liouville fractional derivative | MATHEMATICS, APPLIED | Operators | Mathematical models | Derivatives | Computation | Integrals | Mathematical analysis

Hadamard fractional derivative | Generalized fractional integral | Riemann–Liouville fractional derivative | Generalized fractional derivative | Semigroup property | Riemann-Liouville fractional derivative | MATHEMATICS, APPLIED | Operators | Mathematical models | Derivatives | Computation | Integrals | Mathematical analysis

Journal Article

Mathematical and Computer Modelling, ISSN 0895-7177, 2010, Volume 52, Issue 5, pp. 862 - 874

In this paper we focus on establishing stability theorems for fractional differential system with Riemann–Liouville derivative, in particular our analysis...

Fractional differential system | Riemann–Liouville derivative | Stability | Mittag-Leffler function | Riemann-Liouville derivative | MATHEMATICS, APPLIED

Fractional differential system | Riemann–Liouville derivative | Stability | Mittag-Leffler function | Riemann-Liouville derivative | MATHEMATICS, APPLIED

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

力学学报：英文版, ISSN 0567-7718, 2015, Volume 31, Issue 2, pp. 153 - 161

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem...

广义二阶流体 | Levenberg-Marquardt算法 | 分数阶导数 | 加热 | 估计 | 黎曼 | 逆问题 | gamma函数 | Engineering | Computational Intelligence | Implicit numerical method | Riemann–Liouville fractional derivative | Engineering Fluid Dynamics | Generalized second grade fluid | Fractional sensitivity equation | Theoretical and Applied Mechanics | Classical Continuum Physics | Inverse problem | UNSTEADY-FLOW | MECHANICS | Riemann-Liouville fractional derivative | VISCOELASTIC FLUID | ENGINEERING, MECHANICAL | Models | Algorithms

广义二阶流体 | Levenberg-Marquardt算法 | 分数阶导数 | 加热 | 估计 | 黎曼 | 逆问题 | gamma函数 | Engineering | Computational Intelligence | Implicit numerical method | Riemann–Liouville fractional derivative | Engineering Fluid Dynamics | Generalized second grade fluid | Fractional sensitivity equation | Theoretical and Applied Mechanics | Classical Continuum Physics | Inverse problem | UNSTEADY-FLOW | MECHANICS | Riemann-Liouville fractional derivative | VISCOELASTIC FLUID | ENGINEERING, MECHANICAL | Models | Algorithms

Journal Article

Applied Numerical Mathematics, ISSN 0168-9274, 04/2015, Volume 90, pp. 22 - 37

A one dimensional fractional diffusion model with the Riemann–Liouville fractional derivative is studied. First, a second order discretization for this...

Weighted average methods | Von Neumann stability analysis | Riemann–Liouville derivative | Fractional diffusion equations | Riemann-Liouville derivative | DYNAMICS APPROACH | ORDER | MATHEMATICS, APPLIED | ADVECTION-DISPERSION EQUATION | MOTION | APPROXIMATIONS | ALGORITHMS | SOLUTE TRANSPORT | Discretization | Mathematical analysis | Mathematical models | Derivatives | Diffusion | Physical properties | Convergence | Finite difference method

Weighted average methods | Von Neumann stability analysis | Riemann–Liouville derivative | Fractional diffusion equations | Riemann-Liouville derivative | DYNAMICS APPROACH | ORDER | MATHEMATICS, APPLIED | ADVECTION-DISPERSION EQUATION | MOTION | APPROXIMATIONS | ALGORITHMS | SOLUTE TRANSPORT | Discretization | Mathematical analysis | Mathematical models | Derivatives | Diffusion | Physical properties | Convergence | Finite difference method

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 10/2015, Volume 27, Issue 1-3, pp. 1 - 11

•We do apply fractional calculus to reaction–diffusion equations.•Give solutions of these fractional derivative type equations suitable for numerical analysis...

Riesz derivative | H-function | Riemann–Liouville fractional derivative | Caputo derivative | Fourier transform | Riesz–Feller fractional derivative | Mittag–Leffler function | Laplace transform | Riesz-Feller fractional derivative | Mittag-Leffler function | Riemann-Liouville fractional derivative | MATHEMATICS, APPLIED | KINETIC-EQUATION | PHYSICS, FLUIDS & PLASMAS | TERMS | PHYSICS, MATHEMATICAL | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | Fourier transforms | Computation | Mathematical analysis | Nonlinearity | Mathematical models | Derivatives | Reaction-diffusion equations | Diffusion

Riesz derivative | H-function | Riemann–Liouville fractional derivative | Caputo derivative | Fourier transform | Riesz–Feller fractional derivative | Mittag–Leffler function | Laplace transform | Riesz-Feller fractional derivative | Mittag-Leffler function | Riemann-Liouville fractional derivative | MATHEMATICS, APPLIED | KINETIC-EQUATION | PHYSICS, FLUIDS & PLASMAS | TERMS | PHYSICS, MATHEMATICAL | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | Fourier transforms | Computation | Mathematical analysis | Nonlinearity | Mathematical models | Derivatives | Reaction-diffusion equations | Diffusion

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 04/2015, Volume 257, pp. 566 - 580

We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann–Liouville and the Hadamard fractional...

Stirling numbers of the 2nd kind | Riemann–Liouville derivative | Hadamard derivative | Generalized fractional derivative | Recurrence relations | Mellin transform | Riemann-Liouville derivative | Erdélyi-Kober operators | Hidden Pascal triangles | MATHEMATICS, APPLIED | INEQUALITIES | CALCULUS | Erdelyi-Kober operators | Operators | Mellin transforms | Computation | Integrals | Mathematical models | Polynomials | Derivatives | Combinatorial analysis

Stirling numbers of the 2nd kind | Riemann–Liouville derivative | Hadamard derivative | Generalized fractional derivative | Recurrence relations | Mellin transform | Riemann-Liouville derivative | Erdélyi-Kober operators | Hidden Pascal triangles | MATHEMATICS, APPLIED | INEQUALITIES | CALCULUS | Erdelyi-Kober operators | Operators | Mellin transforms | Computation | Integrals | Mathematical models | Polynomials | Derivatives | Combinatorial analysis

Journal Article

Advances in Difference Equations, ISSN 1687-1839, 12/2018, Volume 2018, Issue 1, pp. 1 - 18

We aim to investigate the following nonlinear boundary value problems of fractional differential equations:...

Critical point theory | Riemann–Liouville and Caputo fractional derivative | 34B10 | Mathematics | Nonlinear fractional differential equations | Existence of solutions | 34A08 | 47H10 | Ordinary Differential Equations | Functional Analysis | Analysis | Difference and Functional Equations | Mathematics, general | Method of Nehari manifold | Partial Differential Equations | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | CALCULUS | Riemann-Liouville and Caputo fractional derivative | Manifolds | Nonlinear equations | Boundary value problems | Mathematical analysis | Differential equations

Critical point theory | Riemann–Liouville and Caputo fractional derivative | 34B10 | Mathematics | Nonlinear fractional differential equations | Existence of solutions | 34A08 | 47H10 | Ordinary Differential Equations | Functional Analysis | Analysis | Difference and Functional Equations | Mathematics, general | Method of Nehari manifold | Partial Differential Equations | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | CALCULUS | Riemann-Liouville and Caputo fractional derivative | Manifolds | Nonlinear equations | Boundary value problems | Mathematical analysis | Differential equations

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

Journal of Computational Physics, ISSN 0021-9991, 01/2016, Volume 305, pp. 1 - 28

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann–Liouville (RL)...

Well-conditioned collocation methods | (Modified) Riemann–Liouville fractional derivative | Fractional differential equations | Interpolation basis polynomials | Caputo fractional derivative | Fractional Birkhoff interpolation | (Modified) Riemann-Liouville fractional derivative | SPECTRAL COLLOCATION | APPROXIMATIONS | PHYSICS, MATHEMATICAL | POLYNOMIALS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | INTEGRATION | DIFFUSION | Usage | Algorithms | Analysis | Methods | Linear systems | Interpolation | Collocation | Computation | Mathematical analysis | Collocation methods | Polynomials | Formulas (mathematics)

Well-conditioned collocation methods | (Modified) Riemann–Liouville fractional derivative | Fractional differential equations | Interpolation basis polynomials | Caputo fractional derivative | Fractional Birkhoff interpolation | (Modified) Riemann-Liouville fractional derivative | SPECTRAL COLLOCATION | APPROXIMATIONS | PHYSICS, MATHEMATICAL | POLYNOMIALS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | INTEGRATION | DIFFUSION | Usage | Algorithms | Analysis | Methods | Linear systems | Interpolation | Collocation | Computation | Mathematical analysis | Collocation methods | Polynomials | Formulas (mathematics)

Journal Article

Advances in Difference Equations, ISSN 1687-1839, 12/2018, Volume 2018, Issue 1, pp. 1 - 26

In this manuscript, we investigate a sort of fractional neutral integro-differential equations with impulsive outcomes and extend the formula of general...

34A37 | Impulsive | Riemann–Liouville fractional derivatives | Mathematics | 34A08 | Ordinary Differential Equations | Functional Analysis | Fractional differential equations | Analysis | Difference and Functional Equations | Mathematics, general | 35R12 | 45J05 | Partial Differential Equations | MATHEMATICS | MATHEMATICS, APPLIED | Riemann-Liouville fractional derivatives | DIFFERENTIAL-EQUATIONS | APPROXIMATE CONTROLLABILITY | EVOLUTION-EQUATIONS | Operators (mathematics) | Banach spaces | Banach space | Differential equations

34A37 | Impulsive | Riemann–Liouville fractional derivatives | Mathematics | 34A08 | Ordinary Differential Equations | Functional Analysis | Fractional differential equations | Analysis | Difference and Functional Equations | Mathematics, general | 35R12 | 45J05 | Partial Differential Equations | MATHEMATICS | MATHEMATICS, APPLIED | Riemann-Liouville fractional derivatives | DIFFERENTIAL-EQUATIONS | APPROXIMATE CONTROLLABILITY | EVOLUTION-EQUATIONS | Operators (mathematics) | Banach spaces | Banach space | Differential equations

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