Fractional Boundary Value Problems Research Articles

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

NONLOCAL BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION ON THE HALF-LINE

This paper aims to investigate a class for nonlocal fractional boundary value problem on an infinite interval due to its importance in provide a powerful tool for mathematical modeling of complex phenomena in science. New existence results are acquired for the given problem by using the Krasnosel’skii’s fixed point theorem. Moreover, sufficient conditions are obtained as well as a modified compactness criterion that guarantees the existence of at least one solution. In addition, an illustrative example is given in the final part of the paper.

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.

On Graphical Symmetric Spaces, Fixed-Point Theorems and the Existence of Positive Solution of Fractional Periodic Boundary Value Problems

The rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H-(ϑ,φ)-contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems.

Lyapunov-type inequalities for Hilfer fractional boundary value problems

This paper deals with fractional boundary value problems involving the Hilfer–Hadamard fractional differential operator of order [Formula: see text] and type [Formula: see text]. We derive the corresponding Lyapunov-type inequalities for two prominent classes of Hilfer–Hadamard fractional boundary value problems (HFBVPs) involving separated and anti-periodic boundary conditions. For this purpose, we construct the associated Green’s functions and deduce their important properties.

Mixed fractional order p-Laplacian boundary value problem with a two-dimensional Kernel at resonance on an unbounded domain

In this paper, by using the Ge and Ren extension of coincidence degree theory, we established the existence of a solution for a resonant mixed fractional order p-Laplacian boundary value problem (BVP) on the half-line. In the process, we solved the corresponding homogeneous fractional order BVP for conditions critical for resonance and showed that the operator A(x,λ)(t) constructed from the abstract equation Mx(t)=Nx(t) is relatively compact. The results are demonstrated with an example.

On the Solution of Caputo Fractional High-Order Three-Point Boundary Value Problem with Applications to Optimal Control

This research paper establishes the existence and uniqueness of solutions for a non-integer high-order boundary value problem, incorporating the Caputo fractional derivative with a non-local type boundary condition. The analytical approach involves the introduction of the fractional Green’s function. To analyze our findings effectively, we apply the Banach contraction fixed point theorem as the primary principle. Furthermore, we illustrate our results through the presentation of various examples.

Solvability of a Hadamard Fractional Boundary Value Problem at Resonance on Infinite Domain

This paper investigates the existence of solutions for Hadamard fractional differential equations with integral boundary conditions at resonance on infinite domain. By constructing two suitable Banach spaces, establishing an appropriate compactness criterion, and defining appropriate projectors, we study an existence theorem upon the coincidence degree theory of Mawhin. An example is given to illustrate our main result.

Fractional Boundary Layer Flow: Lie Symmetry Analysis and Numerical Solution

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported.

Solve Riemann–Liouville boundary value problems using collocation boundary value methods with the graded mesh

The paper explores numerical approaches to a class of fractional boundary value problems that can be converted into Volterra integral equations with multiple singular kernels. We propose a collocation boundary value method with a graded mesh for the resulting weakly singular integral equations. Moreover, we demonstrate the local convergence property of the presented method using the Gronwall’s inequality. In addition, we assess the stability of the algorithm by examining the L2-norm of the approximate solution. It is found that the proposed collocation method exhibits both fast convergence order and good stability. The numerical experiments provide rigorous validation of the theoretical results.

On some even-sequential fractional boundary-value problems

On some even-sequential fractional boundary-value problems

Fractional differential equation with movable boundary conditions

In this research paper, we discuss the complex-valued solutions for the nonlinear fractional boundary value problem (FBVP) of complex order (δ = τ + ιa; 1 < τ ≤ 2, a ∈ R+) with movable boundary conditions. The fractional operators are taken in the sense of Riemann-Liouville (R-L) with complex order. By using the concept of Green’s function, the existence and uniqueness of solutions are established in this article. Also, we prove that the FBVP of complex order with movable boundary conditions is Ulam-Hyers Stable. Using illustrative examples, the results for this nonlinear FBVP have been shown.

Existence of solutions for Hadamard fractional nonlocal boundary value problems with mean curvature operator at resonance

<p>This paper aims to study the existence of solutions for Hadamard fractional nonlocal boundary value problems with mean curvature operator at resonance. Based on the coincidence degree theory, some new results are established. Moreover, an example is given to verify our main results.</p>

Nontrivial solutions for a Hadamard fractional integral boundary value problem

<abstract><p>In this paper, we studied a Hadamard-type fractional Riemann-Stieltjes integral boundary value problem. The existence of nontrivial solutions was obtained by using the fixed-point method when the nonlinearities can be superlinear, suberlinear, and have asymptotic linear growth. Our results improved and generalized some results of the existing literature.</p></abstract>

Positive Solutions for a Fractional Boundary Value Problem with Lidstone Like Boundary Conditions

We consider a higher order fractional boundary value problem with Lidstone like boundary conditions, where the nonlinearity is an L1-Carathèodory function. We first consider the lower order problem. Then, by using a convolution to construct the Green’s function for the higher order problem, we are able to apply a recent fixed point theorem to show the existence of positive solutions of the boundary value problem.

Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $ in infinite dimensional Banach spaces

<abstract><p>In this paper, we improved recent results on the existence of solutions for nonlinear fractional boundary value problems containing the Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $. We also derived the exact relations between these fractional boundary value problems and the corresponding fractional integral equations in infinite dimensional Banach spaces. We showed that the continuity assumption on the nonlinear term of these equations is insufficient, give the derived expression for the solution, and present two results about the existence and uniqueness of the solution. We examined the case of impulsive impact and provide some sufficiency conditions for the existence and uniqueness of the solution in these cases. We also demonstrated the existence and uniqueness of anti-periodic solution for the studied problems and considered the problem when the right-hand side was a multivalued function. Examples were given to illustrate the obtained results.</p></abstract>

Unified existence results for nonlinear fractional boundary value problems

<abstract><p>In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples.</p></abstract>

Positive solutions for a Riemann-Liouville-type impulsive fractional integral boundary value problem

<abstract><p>In this work, we investigate a Riemann-Liouville-type impulsive fractional integral boundary value problem. Using the fixed point index, we obtain two existence theorems on positive solutions under some conditions concerning the spectral radius of the relevant linear operator. Our method improves and generalizes some results in the literature.</p></abstract>

Solvability for a Higher Order Implicit Fractional Multi-point Boundary Value Problems at Resonance

Solvability for a Higher Order Implicit Fractional Multi-point Boundary Value Problems at Resonance