Fractional Boundary Value Problems Research Articles

Nonlinear Contractions Employing Digraphs and Comparison Functions with an Application to Singular Fractional Differential Equations

After the initiation of Jachymski’s contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem.

Analysis of fractional Euler-Bernoulli bending beams using Green’s function method

This paper deals with the effect of fractional order on the fractional Euler-Bernoulli beams model. This model is based on the elastic curve that can be approximated using tangent lines and the osculating circle. We introduce a general form of the fractional Green’s function for the highest-order term 1<α≤2, which gives a unique solution for the geometrical fractional bending beams with an inhomogeneous fractional differential equation. The analytical closed-form responses were obtained by inverse problem technique and integration over Green’s function kernel. The theory and properties of fractional Green’s function method are described by generalizing classical theory to the initial and two-point fractional boundary value problems. Static analysis of the numerical examples, such as determinate and indeterminate beams with typical loads and beam property boundary conditions, are presented and discussed. The results were verified using the direct integration method. The fractional conventional ratio indicates the beam's deformation deviation from its classical state. The analysis results reveal that changing the fractional parameter significantly impacts the deflection behavior of flexural beams due to the beam’s characteristics and the fractional order. The behavior of the fractional beam is identical to the classical state when the fractional parameter is an integer number.

Legendre-Galerkin spectral algorithm for fractional-order BVPs: Application to the Bagley-Torvik equation

&lt;p&gt;Herein, we provide an efficient spectral Galerkin algorithm, according to a special type of shifted Legendre basis for finding a semi-analytic solution to the Liouville-Caputo fractional boundary value problem. The algorithm’s main goal is to transform the fractional differential problem into a linear system with efficiently invertible, well-structured matrices. The convergence rates of the algorithm are carefully obtained as well as the error bound.&lt;/p&gt;

EXISTENCE AND STABILITY RESULTS OF FRACTIONAL DIFFERENTIAL EQUATIONS MITTAG-LEFFLER KERNEL

This paper presents the following AB-Caputo fractional boundary value problem [Formula: see text] with integral-type boundary conditions [Formula: see text] of order [Formula: see text]. Schauder and Krasnoselskii’s fixed point theorems are used to find existence results. Uniqueness is obtained via the Banach contraction principle. To investigate the stability of a given problem, Hyers–Ulam stability is discussed. An example is provided to validate our results.

Stability analysis and solutions of fractional boundary value problem on the cyclopentasilane graph

The study is being applied to a model involving silane and on cyclopentasilane graph. We consider a graph with labeled vertices by 0 or 1 inspired by the molecular structure of cyclopentasilane. In this paper, we first study the existence of solutions to fractional conformable boundary value problem on the cyclopentasilane graph by applying Scheafer and Krasnoselskii fixed point theorems. Furthermore, we investigate different kinds of Ulam stability such as Ulam-Hyers stable, generalized Ulam-Hyers stable, Ulam-Hyers-Rassias stable and generalized Ulam-Hyers-Rassias stable for the given problem. Finally, we give an example to support our important results.

Existence of solutions to multi-point boundary value problem of fractional order on the half-line

The purpose of this paper is to establish the existence of solutions to multi-point fractional boundary value problem on an infinite interval. Using the fixed point theory, sufficient conditions are obtained that guarantee the existence of at least one solution. At the end, an example is presented to illustrate the main results

Existence and uniqueness of well-posed fractional boundary value problem.

In this paper, the existence and uniqueness of solution for a fractional differential model involving well-posed boundary conditions and implicit fractional differential equation is considered. The desired goals are achieved by using Banach contraction principle and Scheafer's fixed point theorem. To show the results applicability some examples are presented. The basic mathematical concept of well-posed fractional boundary value issues is investigated in this study. It dives into the existence and uniqueness of these difficulties, offering light on the conditions that allow for both practical and singular solutions. This study contributes to a better knowledge of fractional calculus and its applications in a variety of scientific and technical areas, giving significant insights for both scholars and practitioners.

Efficient semianalytical investigation of a fractional model describing human cornea shape

Purpose: This study presents a novel application of the semianalytical residual power series method to investigate a one-dimensional fractional anisotropic curvature equation describing the human cornea, the outermost layer of the eye. The fractional boundary value problem, involving the fractional derivative of curvature, poses challenges that conventional methods struggle to address. Methods: The analytical results are obtained by utilizing the simple and efficient residual power series method. The proposed method is accessible to researchers in all medical fields and is extendable to various models in disease spread and control. Results: The derived solution is a crucial outcome of this study. Through the application of the proposed method to the corneal shape model, an explicit formula for the curvature profile is obtained. To validate the solution, direct comparisons are made with numerical solutions for the integer case and other analytical solutions available in the literature for the fractional case. Conclusion: Our findings highlight the potential of the proposed method to significantly contribute to the diagnosis and treatment of various ophthalmological conditions.

On the existence and uniqueness results for fuzzy fractional boundary value problem involving Caputo iractional derivative

In this paper, we investigate the existence and uniqueness of solutions for fuzzy boundary value problems involving fuzzy Caputo fractional derivatives of order q 2 (2, 3). As a preliminary step, we construct a generic structure of the solution associated with our proposed model by utilizing the Green’s function. We establish the existence of a unique solution of the proposed model paired with the given initial conditions by using Banach fixed point theorem. At last, as application, an illustrative example is given to show the applicability of our theoritical results.

A novel approach on the sequential type ψ-Hilfer pantograph fractional differential equation with boundary conditions

This article investigates sufficient conditions for the existence and uniqueness of solutions to the ψ-Hilfer sequential type pantograph fractional boundary value problem. Considering the system depends on a lower-order fractional derivative of an unknown function, the study is carried out in a special working space. Standard fixed point theorems such as the Banach contraction principle and Krasnosel’skii’s fixed point theorem are applied to prove the uniqueness and the existence of a solution, respectively. Finally, an example demonstrating our results with numerical simulations is presented.

Positive solutions for a Hadamard-type fractional p-Laplacian integral boundary value problem

In this paper we study the existence of positive solutions to a Hadamard-type fractional integral boundary value problem using fixed point index. We construct a new linear operator and obtain our main results under some conditions concerning the spectral radius of this linear operator. Our method improves and generalizes some results in the literature.

Mixed Hilfer and Caputo Fractional Riemann–Stieltjes Integro-Differential Equations with Non-Separated Boundary Conditions

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach’s fixed point theorem, while by applying Leray–Schauder’s nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained.

Existence of Solutions to a System of Fractional q-Difference Boundary Value Problems

We are investigating the existence of solutions to a system of two fractional q-difference equations containing fractional q-integral terms, subject to multi-point boundary conditions that encompass q-derivatives and fractional q-derivatives of different orders. In our main results, we rely on various fixed point theorems, such as the Leray–Schauder nonlinear alternative, the Schaefer fixed point theorem, the Krasnosel’skii fixed point theorem for the sum of two operators, and the Banach contraction mapping principle. Finally, several examples are provided to illustrate our findings.

A collocation method for an RLC fractional derivative two-point boundary value problem with a singular solution

A collocation method for an RLC fractional derivative two-point boundary value problem with a singular solution

Fractional order iterative boundary value problem

In the present paper, we establish the existence and uniqueness solution for an iterative differential equation involving caputo fractional derivative of order $1\alpha2$. The existence results is proved by using Schauder fixed point. To prove the extremal solution, we demonstrate some fractional inequalities. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained result.

Fractional periodic boundary value and Cauchy problems with Hilfer–Prabhakar operator

We introduce a successive approximations method to study one fractional periodic boundary value problem of the Hilfer-Prabhakar type. The problem is associated to the corresponding Cauchy problem, whose solution depends on an unknown initial value. To find this value we numerically solve the so-called ’determining system’ of algebraic or transcendental equations. As a result, we determine an approximate solution of the studied problem, written in a closed form. Finally, we evaluate efficiency of our method on a nonlinear numerical example.

Existence and stability results for fractional boundary value problems in Banach spaces

Existence and stability results for fractional boundary value problems in Banach spaces

Applications of new smart algorithm based on kernel method for variable fractional functional boundary value problems

Applications of new smart algorithm based on kernel method for variable fractional functional boundary value problems

Some Properties of a Falling Function and Related Inequalities on Green’s Functions

Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and presents some properties of this function in a delta fractional model with Green’s function kernel. In the deterministic case, Green’s function will be non-negative, and this shows that the function has an upper bound for its maximum point. More precisely, in this paper, based on the properties of the Taylor monomial falling function, we investigate Lyapunov-type inequalities for a delta fractional boundary value problem of Riemann–Liouville type.

On Generalized Sehgal–Guseman-Like Contractions and Their Fixed-Point Results with Applications to Nonlinear Fractional Differential Equations and Boundary Value Problems for Homogeneous Transverse Bars

The goal of this study is to describe the class of modified Sehgal–Guseman-like contraction mappings and set up some fixed-point results in S-metric spaces. The class of generalized Sehgal–Guseman-like contraction mappings contains enhancements of Banach contractions, Kannan contractions, Chatterjee contractions, Chatterjee-type contractions, quasi-contractions, Ćirić–Reich–Rus-type contractions, Hardy–Rogers-type contractions, Reich-type contractions, interpolative Kannan contractions, interpolative Chatterjee contractions, among others, with their generalizations in S-metric spaces. We offer significant examples to substantiate the reliability of our results. This study also establishes consequential fixed-point results and applies them to nonlinear fractional differential equations and the boundary value problem for homogeneous transverse bars. At the end of the manuscript, we present an important open problem.