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Fractal Fract
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Advances in Boundary Value Problems for Fractional Differential Equations, 3rd...
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Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (20 April 2025) | Viewed by 2840

Special Issue Editor

Prof. Dr. Rodica Luca

E-Mail Website
Guest Editor
Department of Mathematics, "Gheorghe Asachi" Technical University of Iasi, Blvd. Carol I, nr. 11, 700506 Iasi, Romania
Interests: fractional differential equations; ordinary differential equations; partial differential equations; finite difference equations; boundary value problems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional differential equations have extensive applications in the mathematical modelling of real-world phenomena which occur in scientific and engineering disciplines. This Special Issue will cover new aspects of the recent developments in the theory and applications of fractional differential equations, inclusions, inequalities, and systems of fractional differential equations with Riemann-Liouville, Caputo, and Hadamard derivatives or other generalized fractional derivatives, subject to various initial and boundary conditions. Problems such as the existence, uniqueness, multiplicity, and nonexistence of solutions or positive solutions, the stability of solutions, and numerical computations for these models are of great interest for readers who work in this field.

Also, please feel free to read and download all the published articles in our first volume:

https://www.mdpi.com/journal/fractalfract/special_issues/BVP_FDE

Our second volume is also available：

https://www.mdpi.com/journal/fractalfract/special_issues/BVP_FDE2

Prof. Dr. Rodica Luca
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional differential equations
  • fractional differential inclusions
  • fractional differential inequalities
  • initial value problems
  • boundary value problems
  • existence and nonexistence
  • uniqueness and multiplicity
  • stability
  • numerical computations

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Further information on MDPI's Special Issue policies can be found here.

Related Special Issue

Published Papers (6 papers)

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Research

17 pages, 22156 KiB  
Article
Variable-Fractional-Order Nosé–Hoover System: Chaotic Dynamics and Numerical Simulations
by D. K. Almutairi, Dalal M. AlMutairi, Nidal E. Taha, Mohammed E. Dafaalla and Mohamed A. Abdoon
Fractal Fract. 2025, 9(5), 277; https://doi.org/10.3390/fractalfract9050277 - 25 Apr 2025
Abstract
This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing [...] Read more.
This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing how changes in the order function influence a system’s behavior. The variable-order behavior is shown by phase space orbits and time series for various variable orders α. We look at how the system acts by using numerical solutions and numerical simulations. The phase space orbits and time series for different α show variable-order effects. The findings emphasize the role of variable-order derivatives in enhancing chaotic behavior, offering novel insights into their impact on dynamical systems. Full article
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)
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15 pages, 293 KiB  
Article
Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications
by Yongqing Wang
Fractal Fract. 2025, 9(4), 261; https://doi.org/10.3390/fractalfract9040261 - 19 Apr 2025
Viewed by 132
Abstract
In this article, we study a fractional lower-order differential equation, [...] Read more.
In this article, we study a fractional lower-order differential equation, −D0+αΥ(ξ)+a(ξ)Υ(ξ)=y(ξ),ξ∈(0,1),α∈(1,2), with a Dirichlet-type boundary condition, where a(ξ)∈L1[0,1] permits singularity. When the coefficient of perturbation term a(ξ) is continuous on [0,1], Graef et al. derived the associated Green’s function under certain conditions on a, but failed to obtain its positivity. We obtain some positive properties of the Green’s function of this problem under certain conditions. The results will fill the gap in the above literature. As for application of the theoretical results, a singular fractional differential equation with a perturbation term is considered. The unique positive solution is obtained by using the fixed point theorem, and an iterative sequence is established to approximate it. In addition, a semipositone fractional differential equation, −D0+αΥ(ξ)=μF(ξ,Υ(ξ)),ξ∈(0,1),α∈(1,2), is also considered. The existence of positive solutions is determined under a more general condition, F(ξ,x)≥−b(ξ)x−e(ξ), where b(ξ),e(ξ)∈L1[0,1] are non-negative functions. Relevant examples are listed to manifest the theoretical results. Full article
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)
14 pages, 292 KiB  
Article
Positive Normalized Solutions to a Kind of Fractional Kirchhoff Equation with Critical Growth
by Shiyong Zhang and Qiongfen Zhang
Fractal Fract. 2025, 9(3), 193; https://doi.org/10.3390/fractalfract9030193 - 20 Mar 2025
Viewed by 131
Abstract
In this paper, we have investigated the existence of normalized solutions for a class of fractional Kirchhoff equations involving nonlinearity and critical nonlinearity. The nonlinearity satisfies L2-supercritical conditions. We transform the problem into an extremal problem within the framework of Lagrange [...] Read more.
In this paper, we have investigated the existence of normalized solutions for a class of fractional Kirchhoff equations involving nonlinearity and critical nonlinearity. The nonlinearity satisfies L2-supercritical conditions. We transform the problem into an extremal problem within the framework of Lagrange multipliers by utilizing the energy functional of the equation in the fractional Sobolev space and applying the mass constraint condition (i.e., for given m>0,∫RN|u|2dx=m2). We introduced a new set and proved that it is a natural constraint. The proof is based on a constrained minimization method and some characterizations of the mountain pass levels are given in order to prove the existence of ground state normalized solutions. Full article
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)
21 pages, 374 KiB  
Article
A Class of ψ-Hilfer Fractional Pantograph Equations with Functional Boundary Data at Resonance
by Bingzhi Sun, Shuqin Zhang, Tianhu Yu and Shanshan Li
Fractal Fract. 2025, 9(3), 186; https://doi.org/10.3390/fractalfract9030186 - 17 Mar 2025
Viewed by 181
Abstract
In this paper, we explore the outcomes related to the existence of nonlocal functional boundary value problems associated with pantograph equations utilizing ψ-Hilfer fractional derivatives. The nonlinear term relies on unknown functions which contain a proportional delay term and their fractional derivatives [...] Read more.
In this paper, we explore the outcomes related to the existence of nonlocal functional boundary value problems associated with pantograph equations utilizing ψ-Hilfer fractional derivatives. The nonlinear term relies on unknown functions which contain a proportional delay term and their fractional derivatives in a higher order. We discuss various existence results for the different “smoothness” requirements of the unknown function by means of Mawhin’s coincidence theory at resonance. We wrap up by providing a detailed explanation accompanied by an illustration of one of the outcomes. Full article
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)
13 pages, 289 KiB  
Article
Existence, Uniqueness, and Stability of Solutions for Nabla Fractional Difference Equations
by Nikolay D. Dimitrov and Jagan Mohan Jonnalagadda
Fractal Fract. 2024, 8(10), 591; https://doi.org/10.3390/fractalfract8100591 - 8 Oct 2024
Cited by 1 | Viewed by 1039
Abstract
In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure [...] Read more.
In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure existence and uniqueness results. Also, we establish some conditions under which the solution to the considered problem is generalized Ulam–Hyers–Rassias stable. In the end, some examples are included in order to illustrate our main results. Full article
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)
27 pages, 406 KiB  
Article
Positive Solutions to a System of Coupled Hadamard Fractional Boundary Value Problems
by Alexandru Tudorache and Rodica Luca
Fractal Fract. 2024, 8(9), 543; https://doi.org/10.3390/fractalfract8090543 - 19 Sep 2024
Cited by 2 | Viewed by 685
Abstract
We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard [...] Read more.
We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel’skii fixed-point theorem. Full article
(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)
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