Riemann-Liouville Fractional Derivative Of Order Research Articles

High order numerical method for a subdiffusion problem

We consider a subdiffusive fractional differential problem characterized by an equation that incorporates a time Riemann-Liouville fractional derivative of order 1−α, α∈(0,1), on its right-hand side, while the diffusive coefficient is allowed to vary with both space and time. An high order numerical method for the subdiffusion problem is derived based on the fractional splines of degree β∈(1,2]. The main purpose of this work is to apply fractional splines for approximating the fractional integral in the definition of the Riemann-Liouville fractional derivative, and hence explain how to solve the subdiffusion problem using this approach. It is discussed how the rate of convergence of the numerical method depends on the solution, the degree of the spline and the order of the fractional derivative.

On Impulsive Boundary Value Problem with Riemann-Liouville Fractional Order Derivative

Our manuscript is devoted to investigating a class of impulsive boundary value problems under the concept of the Riemann-Liouville fractional order derivative. The subject problem is of implicit type. We develop some adequate conditions for the existence and puniness of a solution to the proposed problem. For our required results, we utilize the classical fixed point theorems from Banach and Scheafer. It is to be noted that the impulsive boundary value problem under the fractional order derivative of the Riemann-Liouville type has been very rarely considered in literature. Finally, to demonstrate the obtained results, we provide some pertinent examples.

Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

The article presents a method for solving the inverse problem of a two-dimensional anomalous diffusion equation with a Riemann–Liouville fractional-order derivative. In the first part of the present study, the authors present a numerical solution of the direct problem. For this purpose, a differential scheme was developed based on the alternating direction implicit method. The presented method was accompanied by examples illustrating its accuracy. The second part of the study concerned the inverse problem of recreating the model parameters, including the orders of the fractional derivative, in the anomalous diffusion equation. Equations of this type can be used to describe, inter alia, the heat conductivity in porous materials. The ant colony optimization algorithm was used to solve this problem. The authors investigated the impact of the distribution of measurement points, the use of different mesh sizes, and the input data errors on the obtained results.

Existence and uniqueness results for a nonlinear singular fractional differential equation of order $ \sigma\in(1, 2) $

<abstract><p>The first objective of this article is to discuss the local existence of the solution to an initial value problem involving a non-linear differential equation in the sense of Riemann-Liouville fractional derivative of order $ \sigma\in(1, 2), $ when the nonlinear term has a singularity at zero of its independent argument. Hereafter, by using some tools of Lebesgue spaces such as Hölder inequality, we obtain Nagumo-type, Krasnoselskii-Krein-type and Osgood-type uniqueness theorems for the problem.</p></abstract>

Stability analysis of Riemann-Liouville fractional-order neural networks with reaction-diffusion terms and mixed time-varying delays

The stability analysis of time-delay neural networks with reaction-diffusion terms in the sense of Riemann–Liouville derivative is still an open problem, which will be considered in this paper. We first extend a new inequality on Riemann–Liouville fractional-order derivative, which plays an important role in the subsequent proof. Using the Lyapunov direct method, Jensen’s integral inequality, and the linear matrix inequality (LMI) method, several easy-to-test criteria expressed by system parameters and given parameters are given to ensure the stability of the system under consideration. The advantage of our method is that we can directly calculate the integral-order derivative of the Lyapunov function, which can be very convenient to test the stability of practical system. Finally, the validity and conciseness of the results are verified by numerical simulation.

Mathematical description of the bulk fluid flow and that of the contained impurity dispersion which uses Caputo or Riemann-Liouville fractional order partial derivatives is nonobjective

In this paper it is shown that the mathematical description of a Newtonian, incompressible, viscous bulk fluid flow and that of the contained impurity dispersion which uses Caputo or Riemann-Liouville fractional order derivative, having integral representation on finite interval, is nonobjective. This means that, two different observers describing the flow or the contained impurity dispersion with these tools obtain two different results which cannot be reconciled i.e. transformed into each other using only formulas that link the coordinates of a point in two fixed orthogonal reference frames and formulas that link the numbers representing a moment of time in two different choices of the origin of time measuring. This is not an academic curiosity! It is rather a problem: which of the obtained results is correct?

Noether symmetry for fractional Hamiltonian system

Fractional Hamiltonian systems within combined Riemann-Liouville fractional order derivative and combined Caputo fractional order derivative are established. Then Noether quasi-symmetry and conserved quantity for the fractional Hamiltonian systems are presented. Thirdly, perturbation to Noether quasi-symmetry and adiabatic invariant are studied. Several special cases are discussed in each section. And finally, two applications, i.e., the fractional Lotka biochemical oscillator model and the fractional isotropic harmonic oscillator model, are discussed to illustrate the results and methods.

Approximate Controllability and Existence of Mild Solutions for Riemann-Liouville Fractional Stochastic Evolution Equations with Nonlocal Conditions of Order 1 < α < 2

Abstract In this paper, we consider the existence of mild solutions and approximate controllability for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2. As far as we know, there are few articles investigating on this issue. Firstly, the mild solutions to the equations are proved using Laplace transform of the Riemann-Liouville derivative. Moreover, the estimations of resolve operators involving the Riemann-Liouville fractional derivative of order 1 < α < 2 are given. Then, the existence results are obtained via the noncompact measurement strategy and the Mönch fixed point theorem. The approximate controllability of this nonlinear Riemann-Liouville fractional nonlocal stochastic systems of order 1 < α < 2 is concerned under the assumption that the associated linear system is approximately controllable. Finally, the approximate controllability results are obtained by using Lebesgue dominated convergence theorem.

Inverse problem for $2b$-order differential equation with a time-fractional derivative

We study the inverse problem for a differential equation of order $2b$ with the Riemann-Liouville fractional derivative of order $\beta\in (0,1)$ in time and given Schwartz type distributions in the right-hand sides of the equation and the initial condition. The problem is to find the pair of functions $(u, g)$: a generalized solution $u$ to the Cauchy problem for such equation and the time dependent multiplier $g$ in the right-hand side of the equation. As an additional condition, we use an analog of the integral condition $$(u(\cdot,t),\varphi_0(\cdot))=F(t), \;\;\; t\in [0,T],$$ where the symbol $(u(\cdot,t),\varphi_0(\cdot))$ stands for the value of an unknown distribution $u$ on the given test function $\varphi_0$ for every $t\in [0,T]$, $F$ is a given continuous function.
 We prove a theorem for the existence and uniqueness of a generalized solution of the Cauchy problem, obtain its representation using the Green's vector-function. The proof of the theorem is based on the properties of conjugate Green's operators of the Cauchy problem on spaces of the Schwartz type test functions and on the structure of the Schwartz type distributions.
 We establish sufficient conditions for a unique solvability of the inverse problem and find a representation of anunknown function $g$ by means of a solution of a certain Volterra integral equation of the second kind with an integrable kernel.

Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945–964], error estimates in L2 (Ω)- and H1 (Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H01(Ω) and v ∈ H01(Ω) are established. For nonsmooth data, that is, v ∈ L2 (Ω), the optimal L2 (Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞ (Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

THREE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM FOR DIFFERENTIAL EQUATION WITH RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE

In this paper, by using the Avery-Peterson fixed point theorem, we establish the existence result of at least three positive solutions of boundary value problem of nonlinear differential equation with Riemann-Liouville's fractional order derivative. An example illustrating our main result is given. Our results complements and extends previous work in the area of boundary value problems of nonlinear fractional differential equations.

Solvability of triple-point integral boundary value problems for a class of impulsive fractional differential equations

This paper is concerned with a class of triple-point integral boundary value problems for impulsive fractional differential equations involving the Riemann-Liouville fractional derivative of order α (2<alphaleq3). Some sufficient criteria for the existence of solutions are obtained by applying the contraction mapping principle and the fixed point theorem. As an application, one example is given to demonstrate the validity of our main results.

Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem

A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Riemann-Liouville fractional derivative of order 2 ź ź with 0 <ź < 1. It is shown that any solution of such a problem can be expressed in terms of solutions to two associated weakly singular Volterra integral equations of the second kind. As a consequence, existence and uniqueness of a solution to the boundary value problem are proved, the structure of this solution is elucidated, and sharp bounds on its derivatives (in terms of the parameter ź) are derived. These results show that in general the first-order derivative of the solution will blow up at x = 0, so accurate numerical solution of this class of problems is not straightforward. The reformulation of the boundary problem in terms of Volterra integral equations enables the construction of two distinct numerical methods for its solution, both based on piecewise-polynomial collocation. Convergence rates for these methods are proved and numerical results are presented to demonstrate their performance.

Lyapunov’s type inequalities for hybrid fractional differential equations

We investigate new results about Lyapunov-type inequalities by considering hybrid fractional boundary value problems. We give necessary conditions for the existence of nontrivial solutions for a class of hybrid boundary value problems involving Riemann-Liouville fractional derivative of order $2<\alpha\le3$ . The investigation is based on a construction of Green’s functions and on finding its corresponding maximum value. In order to illustrate the results, we provide numerical examples.

GENERALIZATION OF THE WIMAN-VALIRON METHOD FOR FRACTIONAL DERIVATIVES

We generalize the Wiman-Valiron method for fractional derivatives proving that |z|Df(z) ∼ (ν(r, f))f(z) holds in a neighborhood of a maximum modulus point outside an exceptional set of values of |z| as |z| → ∞, where D is the Riemann-Liouville fractional derivative of order q > 0, ν(r, f) is the central index of the Taylor representation of f . We use this result to find the precise value for the order of growth of solutions of a fractional differential equation. AMS Subject Classification: 30E15, 26A33, 34A08

FORCED OSCILLATION FOR A CLASS OF FRACTIONAL PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

We investigate the oscillation of class of time fractional partial dierential equationof the formfor (x; t) 2 R+ = G; R+ = [0;1); where is a bounded domain in RN with a piecewisesmooth boundary @ ; 2 (0; 1) is a constant, D +;t is the Riemann-Liouville fractional derivativeof order of u with respect to t and is the Laplacian operator in the Euclidean N- space RNsubject to the Neumann boundary conditionWe will obtain sucient conditions for the oscillation of class of fractional partial dierentialequations by utilizing generalized Riccatti transformation technique and the integral averagingmethod. We illustrate the main results through examples.

Finite element method for space-time fractional diffusion equation

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order s (1 <s ≤ 2), and the first order time derivative by a Caputo fractional derivative of order ? (0 <? ≤ 1). For the 0 <? < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(?2?? + h2) and O(?2 + h2), respectively, in which ? is the time step size and h is the space step size. And for the case ? = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(?2 + h2) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.

Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment

The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann–Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann–Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises.

Relaxation Equations: Fractional Models

The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by introducing a fractional differential operator written in terms of the Riemann-Liouville fractional derivative of order γ, 0<γ≤1. In order to solve the generalized equations, the Laplace transform methodology is introduced and the corresponding solutions are then presented, in terms of Mittag-Leffler functions. In the case in which the derivative’s order is γ = 1, the traditional relaxation functions are recovered. Finally, we presented some 2D graphs of these function.

Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators

We consider the following state dependent boundary-value problemD0+αy(t)-pD0+βg(t,y(σ(t)))+f(t,y(τ(t)))=0,0&lt;t&lt;1;y(0)=0,ηy(σ(1))=y(1),whereDαis the standard Riemann-Liouville fractional derivative of order1&lt;α&lt;2,0&lt;η&lt;1,p≤0,0&lt;β&lt;1,β+1-α≥0the functiongis defined asg(t,u):[0,1]×[0,∞)→[0,∞), andg(0,0)=0the functionfis defined asf(t,u):[0,1]×[0,∞)→[0,∞)σ(t),τ(t)are continuous ontand0≤σ(t),τ(t)≤t. Using Banach contraction mapping principle and Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references.