Fractional Boundary Value Problems Research Articles

Analysis of a fractional boundary value problem involving Riesz-Caputo fractional derivative

This paper concerned with study the existence and uniqueness of solutions for
 a class of fractional differential equations with boundary conditions
 involving the Riesz-Caputo type fractional derivatives. We apply the methods
 of functional analysis such that the uniqueness result is established using
 the Banach contraction principle, whereas existence results are obtained using
 Schaefer's and Krasnoslkii's fixed theorems. Some examples are given to
 illustrate our acquired results.

Qualitative Analysis for Multiterm Langevin Systems with Generalized Caputo Fractional Operators of Different Orders

In this research work, we study two types of fractional boundary value problems for multi-term Langevin systems with generalized Caputo fractional operators of different orders. The existence and uniqueness results are acquired by applying Sadovskii’s and Banach’s fixed point theorems, whereas the guarantee of the existence of solutions is shown by Ulam–Hyer’s stability. Our reported results cover many outcomes as special cases. An example is provided to illustrate and validate our results.

A study of a Langevin type fractional nonlocal nonlinear boundary value problem with finitely many nonlinearities

A study of a Langevin type fractional nonlocal nonlinear boundary value problem with finitely many nonlinearities

Analysis of a system of nonlinear Hadamard type fractional boundary value problems in a Banach space

The aim of this work is to analyze the existence of positive solutions for a coupled system of Hadamard type fractional boundary value problems. By using the five functional fixed point theorem, the conditions for the existence of positive solutions are derived. Finally, to show the applicability of the main result, an illustrative example is also involved.

Nonlocal Boundary Value Problems for Hilfer Generalized Proportional Fractional Differential Equations

In this paper, we discuss the existence and uniqueness of solutions for boundary value problems for Hilfer generalized proportional fractional differential equations with multi-point boundary conditions. Firstly, we consider the scalar case for which the uniqueness result is proved by using Banach’s fixed point theorem and the existence results are established via Krasnosel’skiĭ’s fixed point theorem and Leray–Schauder nonlinear alternative. We then establish an existence result in the Banach space case based on Mönch’s fixed point theorem and the technique of the measure of noncompactness. Examples are constructed to illustrate the application of the main results. We emphasize that, in this paper, we initiate the study of Hilfer generalized proportional fractional boundary value problems of order in (1, 2].

Explicit iteration and unbounded solutions for fractional q\u2013difference equations with boundary conditions on an infinite interval

In this work, a proposed system of fractional boundary value problems is investigated concerning its unbounded solutions’ existence for a class of nonlinear fractional q-difference equations in the context of the Riemann–Liouville fractional q-derivative on an infinite interval. The system’s solution is formulated with the help of Green’s function. A compactness criterion is established in a special space. All the obtained results of uniqueness and existence are investigated with the help of fixed-point theorems. Some essential examples are illustrated to support our main outcomes.

Existence of solutions for $n$-dimensional fractional order bvp with $\infty$–point boundary conditions via the concept of measure of noncompactness

In this paper, using the concept of measure of noncompactness we present some results on the existence of $n$-fixed points for a class of operators. Also as an application, we derive the sufficient conditions for the existence of solutions for $n$-dimensional fractional order functional boundary value problems.

Right Fractional Sobolev Space via Riemann–Liouville Derivatives on Time Scales and an Application to Fractional Boundary Value Problem on Time Scales

Using the concept of fractional derivatives of Riemann–Liouville on time scales, we first introduce right fractional Sobolev spaces and characterize them. Then, we prove the equivalence of some norms in the introduced spaces, and obtain their completeness, reflexivity, separability and some embeddings. Finally, as an application, we propose a recent method to study the existence of weak solutions of fractional boundary value problems on time scales by using variational methods and critical point theory, and, by constructing an appropriate variational setting, we obtain two existence results of the problem.

A Theoretical Analysis of a Fractional Multi-Dimensional System of Boundary Value Problems on the Methylpropane Graph via Fixed Point Technique

Few studies have investigated the existence and uniqueness of solutions for fractional differential equations on star graphs until now. The published papers on the topic are based on the assumption of existence of one junction node and some boundary nodes as the origin on a star graph. These structures are special cases and do not cover more general non-star graph structures. In this paper, we state a labeling method for graph vertices, and then we prove the existence results for solutions to a new family of fractional boundary value problems (FBVPs) on the methylpropane graph. We design the chemical compound of the methylpropane graph with vertices specified by 0 or 1, and on every edge of the graph, we consider fractional differential equations. We prove the existence of solutions for the proposed FBVPs by means of the Krasnoselskii’s and Scheafer’s fixed point theorems, and further, we study the Ulam–Hyers type stability for the given multi-dimensional system. Finally, we provide an illustrative example to examine our results.

Numerical solution of fractional boundary value problem with caputo-fabrizio and its fractional integral.

In this article, we investigate the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo-Fabrizio type. In order to study this problem we used a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo-Fabrizio, therefore, so we transformed the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and taking sufficient conditions existence and uniqueness of this solution is proven based on the results obtained. The analytical study is followed by a complete numerical study.

USER-ORIENTED INTELLIGENCE MINING UNDER THE EXISTENCE OF SOLUTIONS TO INTEGRAL BOUNDARY VALUE PROBLEMS FOR FUZZY PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

In order to accurately perceive user intention and improve the reliability of user intention information mining, the existence of solution of integral boundary value problem of fuzzy partial fractional differential equation is used to mine the information of user intention. Firstly, the periodic boundary value problem of fractional order fuzzy linear differential equation, periodic boundary value problem of fractional order fuzzy nonlinear differential equation and periodic boundary value problem of fractional order fuzzy coupled differential system are described in this research. The existence of the solution of the integral boundary value problem of the fuzzy partial fractional order differential equation is proved. Then a user-oriented information mining framework and an evaluation model based on the existence of solutions for the integral boundary value problem of fuzzy partial fractional differential equations are constructed. Finally, this research makes a case study and a comprehensive evaluation of user-oriented information mining based on the existence of solutions of integral boundary value problems for fuzzy partial fractional differential equations. The results show that the method based on the existence of the solution of the integral boundary value problem of fuzzy partial fractional order differential equation is feasible and scientific for the information mining of user intention. It is also concluded that this method is suitable for the modelling and solving of intelligence mining with complicated and unclear user intention. The research provides a good guidance for information mining oriented to user intention.

Fractional boundary value problems

We study some functionals associated with a process driven by a fractional boundary value problem (FBVP for short). By FBVP we mean a Cauchy problem with boundary condition written in terms of a fractional equation, that is an equation involving time-fractional derivative in the sense of Caputo. We focus on lifetimes and additive functionals characterizing the boundary conditions. We show that the corresponding additive functionals are related to the fractional telegraph equations. Moreover, the fractional order of the derivative gives a unified condition including the elastic and the sticky cases among the others.

EXISTENCE AND UNIQUENESS SOLUTION FOR THREE-POINT HADAMARD-TYPE FRACTIONAL VOLTERRA BVP

In this paper we study the existence and uniqueness solution for a first kind fractional Volterra boundary value problem involving Hadamard type and three-point boundary conditions. Our analysis is based on Krasnoselskii-Zabreiko’s fixed point theorem and Banach contraction principle. As an application we discuss a Hadamard type boundary value problem with fractional integral boundary conditions. We emphasize that our results are new in the context of Hadamard fractional calculus and are well illustrated with the aid of examples.

Existence of solutions for a three-point Hadamard fractional resonant boundary value problem

Abstract This article focuses on the creation of an existence theorem for a fully nonlinear Hadamard fractional boundary value problem subject to special three-point boundary conditions. By making use of the coincidence degree theory, it is proved that our governing problem makes resonance, that is, the linear part of the differential operator is non-invertible (equally, the corresponding linear problem has at least one nontrivial solution). Constructing some hypotheses on the linear part of the differential operator, nonlinearities and boundary conditions, we give an existence criterion for at least one solution of the fractional-order resonant boundary value problem under study. At the end, a numerical example is presented to illustrate the obtained theoretical results.

Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

By the concept of fractional derivative of Riemann-Liouville on time scales, we first introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones on time scales. Then, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and some imbeddings. Finally, as an application, by constructing an appropriate variational setting, using fibering mapping and Nehari manifolds, the existence of weak solutions for a class of fractional boundary value problems on time scales is studied, and a result of the existence of weak solutions for this problem is obtained.

On the novel existence results of solutions for a class of fractional boundary value problems on the cyclohexane graph

A branch of mathematical science known as chemical graph theory investigates the implications of connectedness in chemical networks. A few researchers have looked at the solutions of fractional differential equations using the concept of star graphs. Their decision to use star graphs was based on the assumption that their method requires a common point linked to other nodes but not to each other. Our goal is to broaden the scope of the method by defining the idea of a cyclohexane graph, which is a cycloalkane with the molecular formula C_{6}H_{12} and CAS number 110-82-7. It consists of a ring of six carbon atoms, each bonded with two hydrogen atoms above and below the plane with multiple junction nodes. This article examines the existence of fractional boundary value problem’ solutions on such graphs in the sense of the Caputo fractional derivative by using the well-known fixed point theorems. In addition, an example is given to support our key findings.

Fractional boundary value problem in Orlicz spaces

The aim of this paper is devoted to investigate the existence of positive continuous solutions for boundary value problem of fractional type    dα,gx(t) dtα = λu(t, x(t))[ζ(x(t)) + η(x(t))], t ∈ [a, b], 0 <α< 1, λ ∈ R+, x(a) − px(b) = h, (1) under the monotonicity conditions imposed on η and ζ. Here h ∈ R+, p ∈ [0, 1), and u is ”possibly singular” function from an appropriate Orlicz space. By the singularity of the above problem, we mean that the possibility of η(0) being undefined is allowed. To encompass the full scope of this paper, we present some examples illustrating the main results.

Existence Results for a New Fractional Boundary Value Problem by Variational Methods

Existence Results for a New Fractional Boundary Value Problem by Variational Methods

Solution of fractional boundary value problems by $ \psi $-shifted operational matrices

<abstract><p>In this paper, a numerical method is presented to solve fractional boundary value problems. In fractional calculus, the modelling of natural phenomenons is best described by fractional differential equations. So, it is important to formulate efficient and accurate numerical techniques to solve fractional differential equations. In this article, first, we introduce $ \psi $-shifted Chebyshev polynomials then project these polynomials to formulate $ \psi $-shifted Chebyshev operational matrices. Finally, these operational matrices are used for the solution of fractional boundary value problems. The convergence is analysed. It is observed that solution of non-integer order differential equation converges to corresponding solution of integer order differential equation. Finally, the efficiency and applicability of method is tested by comparison of the method with some other existing methods.</p></abstract>

On the Ulam-Hyers-Rassias stability of two structures of discrete fractional three-point boundary value problems: Existence theory

<abstract><p>We prove existence and uniqueness of solutions to discrete fractional equations that involve Riemann-Liouville and Caputo fractional derivatives with three-point boundary conditions. The results are obtained by conducting an analysis via the Banach principle and the Brouwer fixed point criterion. Moreover, we prove stability, including Hyers-Ulam and Hyers-Ulam-Rassias type results. Finally, some numerical models are provided to illustrate and validate the theoretical results.</p></abstract>