Riemann-Liouville Fractional Differential Equations Research Articles

The Power Fractional Calculus: First Definitions and Properties with Applications to Power Fractional Differential Equations

Using the Laplace transform method and the convolution theorem, we introduce new and more general definitions for fractional operators with non-singular kernels, extending well-known concepts existing in the literature. The new operators are based on a generalization of the Mittag–Leffler function, characterized by the presence of a key parameter p. This power parameter p is important to enable researchers to choose an adequate notion of the derivative that properly represents the reality under study, to provide good mathematical models, and to predict future dynamic behaviors. The fundamental properties of the new operators are investigated and rigorously proved. As an application, we solve a Caputo and a Riemann–Liouville fractional differential equation.

On a Fractional Differential Equation with r-Laplacian Operator and Nonlocal Boundary Conditions

We study the existence and multiplicity of positive solutions of a Riemann-Liouville fractional differential equation with r-Laplacian operator and a singular nonnegative nonlinearity dependent on fractional integrals, subject to nonlocal boundary conditions containing various fractional derivatives and Riemann-Stieltjes integrals. We use the Guo–Krasnosel’skii fixed point theorem in the proof of our main results.

Positive Solutions for a Fractional Differential Equation with Sequential Derivatives and Nonlocal Boundary Conditions

We study the existence of positive solutions for a Riemann–Liouville fractional differential equation with sequential derivatives, a positive parameter and a sign-changing singular nonlinearity, subject to nonlocal boundary conditions containing varied fractional derivatives and general Riemann–Stieltjes integrals. We also present the associated Green functions and some of their properties. In the proof of the main results, we apply the Guo–Krasnosel’skii fixed point theorem. Two examples are finally given that illustrate our results.

Existence of positive solutions for boundary value problems with p-Laplacian operator

Abstract This paper is concerned with the existence of positive solutions for three point boundary value problems of Riemann-Liouville fractional differential equations with p-Laplacian operator. By means of the properties of the Green’s function and Avery-Peterson fixed point theorem, we establish a condition ensuring the existence of at least three positive solutions for the problem.

Approximate inversion for Abel integral operators of variable exponent and applications to fractional Cauchy problems

We investigate the variable-exponent Abel integral equations and corresponding fractional Cauchy problems. The main contributions of the work are enumerated as follows: (i) We develop an approximate inversion technique for variable-exponent Abel integral operators, based on which we analyze the corresponding integral and differential equations; (ii) We prove that the sensitive dependence of the well-posedness of classical Riemann-Liouville fractional differential equations on the initial value could be resolved by adjusting the initial value of the variable exponent; (iii) We prove that the singularity of the solutions to the Riemann-Liouville fractional differential equations could also be eliminated by adjusting the variable exponent and its derivatives at the initial time, which, together with (ii), demonstrates the advantages of introducing the variable exponent. The proposed approximate inversion technique provides a potential means to convert the intricate variable-exponent integral and differential problems to feasible forms, and the above findings suggest that the variable-exponent fractional problems may serve as a connection between integer-order and fractional models by adjusting the variable exponent at the initial time.

Positive solutions for fractional differential equation at resonance under integral boundary conditions

Abstract By using the theory of fixed point index and spectral theory of linear operators, we study the existence of positive solutions for Riemann-Liouville fractional differential equations at resonance. Our approach will provide some new ideas for the study of this kind of problem.

Stability of a Non-Lipschitz Stochastic Riemann-Liouville Type Fractional Differential Equation Driven by Lévy Noise

Stability of a Non-Lipschitz Stochastic Riemann-Liouville Type Fractional Differential Equation Driven by Lévy Noise

On a System of Riemann–Liouville Fractional Boundary Value Problems with ϱ-Laplacian Operators and Positive Parameters

In this paper, we study the existence and nonexistence of positive solutions of a system of Riemann–Liouville fractional differential equations with ϱ-Laplacian operators, supplemented with coupled nonlocal boundary conditions containing Riemann–Stieltjes integrals, fractional derivatives of various orders, and positive parameters. We apply the Schauder fixed point theorem in the proof of the existence result.

Boundary value problem of Riemann-Liouville fractional differential equations in the variable exponent Lebesgue spaces Lp(.)

This manuscript deals with the existence, uniqueness and stability of solutions to the boundary value problem (BVP) of Riemann-Liouville (RL) fractional differential equations (FDEs) in the variable exponent Lebesgue spaces (Lp(.)). The generalized intervals and piece-wise constant functions are utilized to extract the aims of current paper. The variable exponent Lebesgue spaces (Lp(.)) are converted to the classical Lebesgue spaces. Further, the Banach contraction principle is used, the Ulam-Hyers-stability is examined and finally, an illustrative example is given to the validity of the observed results.

Positive Solutions for a System of Fractional Boundary Value Problems with r-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters

In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change the unknown functions such that the new boundary conditions have no positive parameters, and then, by using the corresponding Green functions, we equivalently write this new problem as a system of nonlinear integral equations. By constructing an appropriate operator A, the solutions of the integral system are the fixed points of A. Following some assumptions regarding the nonlinearities of the system, we show (by applying the Schauder fixed-point theorem) that operator A has at least one fixed point, which is a positive solution of our problem, when the positive parameters belong to some intervals. Then, we present intervals for the parameters for which our problem has no positive solution.

Existence Results for Coupled Implicit \({\psi}\)-Riemann–Liouville Fractional Differential Equations with Nonlocal Conditions

In this paper, we study the existence and uniqueness of solutions for a coupled implicit system involving ψ-Riemann–Liouville fractional derivative with nonlocal conditions. We first transformed the coupled implicit problem into an integral system and then analyzed the uniqueness and existence of this integral system by means of Banach fixed-point theorem and Krasnoselskiis fixed-point theorem. Some known results in the literature are extended. Finally, an example is given to illustrate our theoretical result.

Electrothermal response optimization of nozzle structure for multi-material rapid prototyping based on fuzzy adaptive control

PurposeMost rapid prototyping (RP) relies on energy fields to handle materials, among which electricity has been much more utilized, resulting in distinctive responsiveness of non-linear, overshoot, variable inertia, etc. The purpose of this paper is to eliminate the drawbacks of array nozzle clogging, stringing, melt sagging, particularly in multi-material RP, by focusing on the electrothermal response so as to adaptively distribute thermal more accurate, rapid and balanced.Design/methodology/approachThis paper presents an electrothermal response optimization method of nozzle structure for multi-material RP based on fuzzy adaptive control (FAC). The structural, physical and control model are successively logically built. The fractional order electrothermal model is identified by Riemann Liouville fractional differential equation, using the bisection method to approximate the physical model via least square method to minimize residual sum of squares. The FAC is thereafter implemented by defining fuzzy proportion integration differentiation control rules and fuzzy membership functions for fuzzy inference and defuzzification.FindingsThe transient thermodynamic and structural statics, as well as flow field analysis, are conducted. The response time, mean temperature difference and thermal deformation can be found using thermal-solid coupling finite element analysis. In physical experimental research, temperature change, together with material extrusion loading, were measured. Both numerical and physical studies have revealed findings that the electrothermal responsiveness varies with the three-dimensional structure, materials and energy sources, which can be optimized by FAC.Originality/valueThe proposed FAC provides an optimization method for extrusion-based multi-material RP between the balance of thermal response and energy efficiency through fulfilling potential of the hardware configuration. The originality may be widely adopted alongside increasing requirements on high quality and high efficiency RP.

Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives

<abstract><p>Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riemann-Liouville fractional derivative of orders from $ (1, 2) $. We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We establish integral presentations of the solutions in both cases and we note that these presentations are useful for furure studies of existence, stability and other qualitative properties of the solutions.</p></abstract>

Fixed point results for a new contraction mapping with integral and fractional applications

<abstract><p>The purpose of this manuscript is to present some fixed point results for a $ \Lambda $-Ćirić mapping in the setting of non-triangular metric spaces. Also, two numerical examples are given to support the theoretical study. Finally, under suitable conditions, the existence and uniqueness of a solution to a general Fredholm integral equation, a Riemann-Liouville fractional differential equation and a Caputo non-linear fractional differential equation are discussed as applications.</p></abstract>

Attractivity of solutions of Riemann–Liouville fractional differential equations

Some new weakly singular integral inequalities are established by a new method, which generalize some results of this type in some previous papers. By these new integral inequalities, we present the attractivity of solutions for Riemann–Liouville fractional differential equations. Finally, several examples are given to illustrate our main results.

Positive Solutions of a Singular Fractional Boundary Value Problem with r-Laplacian Operators

We investigate the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators and nonnegative singular nonlinearities depending on fractional integrals, supplemented with nonlocal uncoupled boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. In the proof of our main results we apply the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type.

Controllability and Ulam–Hyers stability of fractional order linear systems with variable coefficients

In this paper we study the Controllability and the generalized Ulam-Hyers stability of Riemann-Liouville fractional order linear differential equations with variable coefficients in Banach spaces. We obtained the desired results by means of fixed point principle and fractional calculus. Furthermore, we used the Gronwall’s inequality to investigate above type of stability for Riemann-Liouville fractional order linear differential equation with time dependent coefficients under some specific assumptions. An example is also provided at the end to illustrate the theoretical work.

Linear first order Riemann-Liouville fractional differential and perturbed Abel's integral equations

Linear first order Riemann-Liouville fractional differential equations are studied. These new equations unify and generalize the Riemann-Liouville, modified Caputo and Caputo fractional differential equations. The equivalences between the fractional differential equations and the corresponding perturbed Abel's integral equations are obtained. These results are useful not only to study the initial or boundary value problems for nonlinear first order Riemann-Liouville fractional differential equations but also to study the solutions of the perturbed Abel's integral equations arising in a problem of mechanics and many other physical problems. The well-known Tonelli's result on solvability of the Abel's integral equation is generalized. We exhibit that there are nonconstant equilibria for some first order Caputo fractional equations. This is different from nonlinear first order ordinary differential equations which have only constant equilibria. The equivalence results are applied to generalize the classical Mean Value Theorem to the first order Riemann-Liouville fractional derivatives.

Hyers–Ulam stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives

• In this paper, we investigate the boundary value problem with Riemann-Liouville fractional differential equations for a class of impulsive implicit nonlinear fractional order integro-differential equations. • We are using some sufficient conditions to achieve the existence and uniqueness results of the projected model with the help of Banach contraction principle, Schauder’s fixed point theorem and Kranoselskii’s fixed point theorem. • Moreover, we present different types of stability such as Hyers-Ulam stability, generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability and generalized Hyers-Ulam-Rassias stability using the classical technique of functional analysis. • To illustrate our theoretical results, at the end we give an example. In this article, we investigate the existence, uniqueness, and stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives. We analyze the existence and uniqueness of the projected model with the help of Banach contraction principle, Schauder’s fixed point theorem, and Krasnoselskii’s fixed point theorem. Moreover, we present different types of stability using the classical technique of functional analysis. To illustrate our theoretical results, at the end we give an example.

Multi-delayed perturbation of Mittag-Leffler type matrix functions

In this paper, we introduce a multivariate determining function and propose a multi-delayed perturbation of Mittag-Leffler type matrix function. It is an extension of the classical Mittag-Leffler type matrix function and delayed Mittag-Leffler type matrix function. With the help of the multi-delayed perturbation of Mittag-Leffler type matrix function, we give an explicit analytical solution to linear nonhomogeneous Riemann-Liouville fractional multi-delay differential equations of order l−1<α≤l. Our results are new even for the case 1<α<2 and for the classical (α=2) second order single delay and multi-delay linear (fractional) differential equations.