Note on the fractional Mittag-Leffler functions by applying the modified Riemann-Liouville derivatives

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

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A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application

IMA Journal of Applied Mathematics ◽

10.1093/imamat/hxx019 ◽

2017 ◽

Vol 82 (5) ◽

pp. 909-944 ◽

Cited By ~ 3

Author(s):

Hengfei Ding ◽

Changpin Li ◽

Qian Yi

Keyword(s):

Fractional Derivatives ◽

Second Order ◽

Stability And Convergence ◽

Approximation Formula ◽

Finite Difference Schemes ◽

First Order ◽

Liouville Derivative ◽

Two Space Dimensions ◽

Fractional Differential ◽

Nicolson Scheme

Abstract Compared to the classical first-order Grünwald–Letnikov formula at time $t_{k+1}\; (\text{or}\; t_{k})$, we firstly propose a second-order numerical approximate formula for discretizing the Riemann–Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank–Nicolson scheme for the fractional differential equations with time fractional derivatives. The established formula has the following form RLD0,tαu(t)| t=tk+12=τ−α∑ℓ=0kϖℓ(α)u(tk−ℓτ)+O(τ2),k=0,1,…,α∈(0,1), where the coefficients $\varpi_{\ell}^{(\alpha)}$$(\ell=0,1,\ldots,k)$ can be determined via the following generating function G(z)=(3α+12α−2α+1αz+α+12αz2)α,|z|<1. Next, applying the formula to the time fractional Cable equations with Riemann–Liouville derivative in one and two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.

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Analytical Approach for the Space-time Nonlinear Partial Differential Fractional Equation

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2013-0145 ◽

2014 ◽

Vol 15 (7-8) ◽

Cited By ~ 16

Author(s):

Ahmet Bekir ◽

Özkan Güner

Keyword(s):

Function Method ◽

Analytical Approach ◽

Fractional Derivatives ◽

Space Time ◽

Variable Method ◽

Partial Differential ◽

Functional Variable ◽

Liouville Derivative ◽

Functional Variable Method ◽

Fractional Equation

AbstractIn this paper, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the functional variable method, exp-function method and

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On Riesz derivative

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0019 ◽

2019 ◽

Vol 22 (2) ◽

pp. 287-301 ◽

Cited By ~ 7

Author(s):

Min Cai ◽

Changpin Li

Keyword(s):

Caputo Derivative ◽

Fractional Derivatives ◽

Fractional Laplacian ◽

One Dimension ◽

Riesz Derivative ◽

Liouville Derivative

Abstract This paper focuses on studying Riesz derivative. An interesting investigation on properties of Riesz derivative in one dimension indicates that it is distinct from other fractional derivatives such as Riemann-Liouville derivative and Caputo derivative. In the existing literatures, Riesz derivative is commonly considered as a proxy for fractional Laplacian on ℝ. We show the equivalence between Riesz derivative and fractional Laplacian on ℝn with n ≥ 1 in details.

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Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4025770 ◽

2014 ◽

Vol 9 (2) ◽

Cited By ~ 28

Author(s):

H. Jafari ◽

H. Tajadodi ◽

D. Baleanu

Keyword(s):

Exact Solutions ◽

Fractional Derivatives ◽

Evolution Equations ◽

Telegraph Equation ◽

Fisher Equation ◽

Fractional Evolution Equations ◽

Liouville Derivative ◽

Nonlinear Telegraph Equation ◽

Homogeneous Balance Method ◽

Homogeneous Balance

The fractional Fan subequation method of the fractional Riccati equation is applied to construct the exact solutions of some nonlinear fractional evolution equations. In this paper, a powerful algorithm is developed for the exact solutions of the modified equal width equation, the Fisher equation, the nonlinear Telegraph equation, and the Cahn–Allen equation of fractional order. Fractional derivatives are described in the sense of the modified Riemann–Liouville derivative. Some relevant examples are investigated.

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Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation

Mathematical Problems in Engineering ◽

10.1155/2013/543848 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 10

Author(s):

Asma Ali Elbeleze ◽

Adem Kılıçman ◽

Bachok M. Taib

Keyword(s):

Iteration Method ◽

Fractional Derivatives ◽

Financial Models ◽

Gordon Equation ◽

Burgers Equation ◽

Variational Iteration Method ◽

Variational Iteration ◽

Liouville Derivative ◽

Black Scholes ◽

Klein Gordon

We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics and in financial models. The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated.

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On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2021-0065 ◽

2021 ◽

Vol 24 (5) ◽

pp. 1559-1570

Author(s):

Riccardo Droghei

Keyword(s):

Differential Equation ◽

Fractional Derivatives ◽

Special Function ◽

Special Functions ◽

Wright Function ◽

Fractional Partial Differential Equation ◽

Caputo Fractional Derivatives ◽

Multi Index ◽

Multiple Parameters ◽

Bessel Differential Equation

Abstract In this paper we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the α-Mittag-Leffler function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffler functions. As an application, we mention that this new generalization Wright function nis an isochronous solution of a nonlinear fractional partial differential equation.

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The Four-Parameters Wright Function of the Second kind and its Applications in FC

Mathematics ◽

10.3390/math8060970 ◽

2020 ◽

Vol 8 (6) ◽

pp. 970 ◽

Cited By ~ 1

Author(s):

Yuri Luchko

Keyword(s):

Fractional Calculus ◽

Fractional Derivatives ◽

Dimensional Space ◽

Wide Spectrum ◽

Operational Calculus ◽

Fractional Diffusion ◽

Wright Function ◽

Scale Invariant ◽

First Case

In this survey paper, we present both some basic properties of the four-parameters Wright function and its applications in Fractional Calculus. For applications in Fractional Calculus, the four-parameters Wright function of the second kind is especially important. In the paper, three case studies illustrating a wide spectrum of its applications are presented. The first case study deals with the scale-invariant solutions to a one-dimensional time-fractional diffusion-wave equation that can be represented in terms of the Wright function of the second kind and the four-parameters Wright function of the second kind. In the second case study, we consider a subordination formula for the solutions to a multi-dimensional space-time-fractional diffusion equation with different orders of the fractional derivatives. The kernel of the subordination integral is a special case of the four-parameters Wright function of the second kind. Finally, in the third case study, we shortly present an application of an operational calculus for a composed Erdélyi-Kober fractional operator for solving some initial-value problems for the fractional differential equations with the left- and right-hand sided Erdélyi-Kober fractional derivatives. In particular, we present an example with an explicit solution in terms of the four-parameters Wright function of the second kind.

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Applications of the Novel (G′/G)-Expansion Method for a Time Fractional Simplified Modified Camassa-Holm (MCH) Equation

Abstract and Applied Analysis ◽

10.1155/2014/601961 ◽

2014 ◽

Vol 2014 ◽

pp. 1-16 ◽

Cited By ~ 4

Author(s):

Muhammad Shakeel ◽

Qazi Mahmood Ul-Hassan ◽

Jamshad Ahmad

Keyword(s):

Exact Solutions ◽

Fractional Derivatives ◽

Traveling Wave Solutions ◽

Rational Functions ◽

Expansion Method ◽

The Novel ◽

Trigonometric Functions ◽

Hyperbolic Functions ◽

Liouville Derivative ◽

Free Parameters

We use the fractional derivatives in modified Riemann-Liouville derivative sense to construct exact solutions of time fractional simplified modified Camassa-Holm (MCH) equation. A generalized fractional complex transform is properly used to convert this equation to ordinary differential equation and, as a result, many exact analytical solutions are obtained with more free parameters. When these free parameters are taken as particular values, the traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. Moreover, the numerical presentations of some of the solutions have been demonstrated with the aid of commercial software Maple. The recital of the method is trustworthy and useful and gives more new general exact solutions.

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Why fractional derivatives with nonsingular kernels should not be used

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0032 ◽

2020 ◽

Vol 23 (3) ◽

pp. 610-634 ◽

Cited By ~ 8

Author(s):

Kai Diethelm ◽

Roberto Garrappa ◽

Andrea Giusti ◽

Martin Stynes

Keyword(s):

Fractional Calculus ◽

Fundamental Theorem ◽

Fractional Derivatives ◽

Mathematical Reasoning ◽

Singular Kernel ◽

Initial Time ◽

Left Inverse ◽

Convolution Integral ◽

Caputo Fractional Derivatives ◽

Liouville Derivative

AbstractIn recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.

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