Involving Riemann Liouville Fractional Derivatives Research Articles

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results.

Analysis of Neutral Stochastic Fractional Differential Equations Involving Riemann–Liouville Fractional Derivative with Retarded and Advanced Arguments

Analysis of Neutral Stochastic Fractional Differential Equations Involving Riemann–Liouville Fractional Derivative with Retarded and Advanced Arguments

Optimal control for a class of fractional order neutral evolution equations

Abstract The optimal control, for a class of nonlinear neutral evolution equations involving Riemann–Liouville fractional derivative, is investigated in this paper by using Darbo–Sadovskii fixed point theorem. An example is given in the last section to illustrate the validity of the abstract conclusions.

Regional controllability results for Riemann–Liouville fractional control systems

In this paper, we consider the problem of regional controllability for semi-linear time-fractional sub-diffusion systems involving Riemann–Liouville fractional derivative. Our goal is to steering the considered system into a desired state only in a sub-region of the spatial domain. In order to do that, we adapt the standard Hilbert Uniqueness Method (HUM), developed in Lions and Magenes (1968), Lions (1988), for our kind of system and we use two fixed point theorem depend to proposed assumptions. We also present some numerical simulations that illustrate the effectiveness of the proposed method, where we consider respectively zone actuator and pointwise one.

Global Existence for an Implicit Hybrid Differential Equation of Arbitrary Orders with a Delay

In this paper, we present a qualitative study of an implicit fractional differential equation involving Riemann–Liouville fractional derivative with delay and its corresponding integral equation. Under some sufficient conditions, we establish the global and local existence results for that problem by applying some fixed point theorems. In addition, we have investigated the continuous and integrable solutions for that problem. Moreover, we discuss the continuous dependence of the solution on the delay function and on some data. Finally, further results and particular cases are presented.

Regional Controllability of Riemann–Liouville Time-Fractional Semilinear Evolution Equations

In this paper, we discuss the exact regional controllability of fractional evolution equations involving Riemann–Liouville fractional derivative of order q ∈ 0,1 . The result is obtained with the help of the theory of fractional calculus, semigroup theory, and Banach fixed-point theorem under several assumptions on the corresponding linear system and the nonlinear term. Finally, some numerical simulations are given to illustrate the obtained result.

Analysis of fractional differential systems involving Riemann Liouville fractional derivative

This paper is devoted to studying the multiple positive solutions for a system of nonlinear fractional boundary value problems. Our analysis is based upon the Avery Peterson fixed point theorem. In addition, we include an example for the demonstration of our main result. .

MATHEMATICAL ANALYSIS OF COUPLED SYSTEMS WITH FRACTIONAL ORDER BOUNDARY CONDITIONS

In this paper, we prove the existence, uniqueness and various kinds of Ulam stability for fractional order coupled systems with fractional order boundary conditions involving Riemann–Liouville fractional derivatives. The standard fixed point theorem like Leray–Schauder alternative and Banach contraction are applied to establish the existence theory and uniqueness. Furthermore, we build sufficient conditions for the stability mentioned above by two methods. Also, an example is given to illustrate our theoretical results. The proposed problem is the generalization of third-order ordinary differential equations with classical, initial and anti-periodic boundary conditions.

A coincidence point theorem and its applications to fractional differential equations

We establish a coincidence point theorem in complete metric spaces. As a consequence, we show the existence of solutions to a system of initial value problem of fractional differential equations involving Riemann–Liouville fractional derivatives. Next, we derive several coincidence point theorems for new classes of sublinear and superlinear operators, in the context of ordered Banach spaces. Finally, we apply these results to discuss the existence of positive solutions to a class of initial value problem of fractional differential equations.

On a terminal value problem for pseudoparabolic equations involving Riemann–Liouville fractional derivatives

In this paper, we consider the terminal value problem for pseudo-parabolic equations with Riemann–Liouville fractional derivatives, from a given final value and we investigate the existence (and regularity) of mild solutions.

Characterizations of certain Hankel transform involving Riemann–Liouville fractional derivatives

In this paper, the relation between two dimensional fractional Fourier transform and fractional Hankel transform is discussed in terms of radial functions. Various operational properties of Hankel transform and fractional Hankel transform are studied involving Riemann–Liouville fractional derivatives. The application of fractional Hankel transform is given in networks with time varying parameters.

Correction to: The Existence of Solutions for a Nonlinear First-Order Differential Equation Involving the Riemann–Liouville Fractional-Order and Nonlocal Condition

In this paper, we study necessary conditions for the existence and uniqueness of continuous solution for a nonlocal boundary value problem with nonlinear term involving Riemann–Liouville fractional derivative. Our results are based on Schauder fixed point theorem and the Banach contraction principle fixed point theorem. Examples illustrating the obtained results are also presented.

An Identity Involving Integration with Respect to Variable Order of Fractional Derivative

Abstract An identity is presented, in which an expression involving Riemann–Liouville fractional derivative is integrated with respect to the derivative’s order. As a by-product an integral analogue of an identity for a sum of binomial coefficients is derived.

Combined effects in some initial value problems involving Riemann–Liouville fractional derivatives in bounded domains

We consider the following semilinear fractional initial value problem $$D^{\alpha }u(x)=a_{1}(x)u^{\sigma _{1}}(x)+a_{2}(x)u^{\sigma _{2}}, \quad x\in (0,1) \quad and \quad \lim_{x\longrightarrow 0^{+}}x^{1-\alpha }u(x)=0,$$ where \({0 < \alpha < 1}\), \({\sigma _{1},\sigma _{2} \in (-1,1)}\) and \({a_{1},a_{2}}\) are positive measurable functions on \({(0,1]}\) satisfying appropriate assumptions related to Karamata regular variation theory. We establish the existence and the uniqueness of a positive solution in the space of weighted continuous functions. We also give the boundary behavior of such solution, where appear the combined effects of singular and sublinear terms in the nonlinearity.

Studies on impulsive differential models with multi-term Riemann–Liouville fractional derivatives

In this article, two impulsive models for fractional differential equations are proposed. By converting the impulsive fractional differential equations into integral equations technically, some existence results on solutions of two classes of initial value problems for nonlinear singular multi-term fractional differential models with impulses involving Riemann–Liouville fractional derivatives are established. The first one has a single base point and the second one has multiple base points. In both cases, the corresponding fractional derivative of lower order is involved in the fractional differential equations. Our results improve some known theorems see Remarks 4.1 and 5.1. Examples are presented to illustrate the main theorems. Finally we give a conclusions section in which some basic errors in a recent paper involving impulsive fractional equations with the Riemann–Liouville derivative are addressed, see Claim 7.1 and Remark 7.2.

Initial value/boundary value problem for composite fractional relaxation equation

We consider initial value/boundary value problem for composite fractional relaxation equation involving Caputo fractional derivative of order 0<β<1. We prove by means of change of variable that this problem is reduced to initial value/boundary value problem for fractional diffusion equation involving Riemann–Liouville fractional derivative of order β=1-α. Then by means of eigenfunctions expansions, we establish the existence and uniqueness of solution.

Existence and uniqueness of solutions for a class of initial value problems of fractional differential systems on half lines

In this article, we establish the existence and uniqueness results for solutions of a class of initial value problems of nonlinear fractional differential systems on half lines involving Riemann–Liouville fractional derivatives. Our analysis relies on the well-known fixed point theorem of Schauder. The novelty of this paper is that the problems discussed are defined on half lines, and the nonlinearities f and g are allowed to be singular functions. Furthermore, we allow p∈(0,β) and q∈(0,α).

Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives

In this paper, using monotone iterative method, we consider the existence and uniqueness of solution of the initial value problem for fractional differential equation involving Riemann–Liouville fractional derivative.

Opial type inequalities involving Riemann–Liouville fractional derivatives of two functions with applications

A large variety of very general but basic L p ( 1 ≤ p ≤ ∞ ) form Opial type inequalities, [Z. Opial, Sur une inégalite, Ann. Polon. Math. 8 (1960) 29–32] is established involving Riemann–Liouville fractional derivatives [G.A. Anastassiou, Opial type inequalities involving fractional derivatives of functions, Nonlinear Stud. 6 (2) (1999) 207–230; Virginia Kiryakova, Generalized Fractional Calculus and Applications, in: Pitman Research Notes in Math. Series, vol. 301, Longman Scientific and Technical, Harlow; copublished in U.S.A with John Wiley & Sons, Inc., New York, 1994; Kenneth Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc. New York, 1993; Keith Oldham, Jerome Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Publications, New York, 2006] of two functions in different orders and powers. From the developed results derive several other concrete results of special interest. The sharpness of inequalities is established there. Finally applications of some of these special inequalities are given in establishing uniqueness of solution and in giving upper bounds to solutions of initial value fractional problems involving a very general system of two fractional differential equations. Also upper bounds to various Riemann–Liouville fractional derivatives of the solutions that are involved in the above systems are presented.