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Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods...
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Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application

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 (31 March 2025) | Viewed by 12694

Special Issue Editor

Prof. Dr. Wenying Feng

E-Mail Website
Guest Editor
1. Department of Mathematics, Trent University Durham Greater Toronto Area, Oshawa, ON L1J5Y1, Canada
2. Department of Computer Science, Trent University Durham Greater Toronto Area, Oshawa, ON L1J5Y1, Canada
Interests: fixed point theory and operator equations; fractional differential equations, boundary value problems, dynamical systems; nonlinear spectral theory and applications; computational approaches for data analytics; neural networks

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to the broad research areas involving Boundary Value Problems (BVPs) of Nonlinear Fractional Differential Equations. The study of nonlinear BVPs for Ordinary Differential Equations (ODEs), Partial Differential Equations (PEDs), Fractional Differential Equations (FDEs), and their discrete counterparts in the form of Difference Equations has a long history and various applications in sciences, engineering, social activities, and natural phenomenon. In particular, BVPs for fractional-order differential equations have attracted more and more interest and have achieved significant improvements recently, partly due to their new applications in physics, control theory, quantitative finance, econometrics, and signal processing.

It is known that fractional-order equations have different behavior from the corresponding integer order equations. Although the traditional topological and numerical methods in dealing with differential equations are applicable to some fractional problems, new methods and techniques have been developed particularly for FDEs. For example, it has been shown that neural networks are efficient in solving and analyzing certain types of FDEs. Fractional techniques have also been applied to train deep learning neural networks to achieve better learning effect for artificial intelligence.

We are interested in the most recent advances in the theory, methods, and applications of FDEs. Topics include, but are not limited to:

Existence and positivity of solutions;

Uniqueness and multiplicity of solutions;

Stability and equilibrium;

Fixed point methods and applications;

Modeling with FDEs;

Numerical solutions;

Neural networks and FDEs;

Eigenvalue problems;

Fractional q-differential equations.

Prof. Dr. Wenying Feng
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

  • existence and positivity of solutions
  • uniqueness and multiplicity of solutions
  • stability and equilibrium
  • fixed point methods and applications
  • modeling with FDEs
  • numerical solutions
  • neural networks and FDEs
  • eigenvalue problems
  • fractional q-differential equations

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Published Papers (12 papers)

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Research

14 pages, 1208 KiB  
Article
Neural Network Method for Solving Time Fractional Diffusion Equations
by Feng Gao and Chunmei Chi
Fractal Fract. 2025, 9(6), 338; https://doi.org/10.3390/fractalfract9060338 - 23 May 2025
Abstract
In this paper, we propose a neural network method to solve time-fractional diffusion equations with Dirichlet boundary conditions by using a combination of machine learning techniques and Method of Lines. We first used the Method of Lines to discretize the equation in the [...] Read more.
In this paper, we propose a neural network method to solve time-fractional diffusion equations with Dirichlet boundary conditions by using a combination of machine learning techniques and Method of Lines. We first used the Method of Lines to discretize the equation in the space domain while keeping the time domain continuous, and represent the solution of the diffusion equation using a neural network. Then we used Gauss–Jacobi quadrature to approximate the fractional derivative in the time domain, thereby obtaining the loss function for the neural network. We used TensorFlow to carry out the gradient descent process to train this neural network. We conducted numerical tests in 1D and 2D cases and compared the results with the exact solutions. The numerical tests showed that this method is effective and easy to manipulate for many time-fractional diffusion problems. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
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16 pages, 650 KiB  
Article
Qualitative Analysis of a Three-Dimensional Dynamical System of Fractal-Fractional-Order Evolution Differential Equations with Terminal Boundary Conditions
by F. Gassem, Arshad Ali, Khaled Aldwoah, Ria H. Egami, Osman Osman, Bakri Younis and Amel Touati
Fractal Fract. 2025, 9(4), 259; https://doi.org/10.3390/fractalfract9040259 - 18 Apr 2025
Viewed by 231
Abstract
In this research article, we investigate a three-dimensional dynamical system governed by fractal-fractional-order evolution differential equations subject to terminal boundary conditions. We derive existence and uniqueness results using Schaefer’s and Banach’s fixed-point theorems, respectively. Additionally, the Hyers–Ulam stability approach is employed to analyze [...] Read more.
In this research article, we investigate a three-dimensional dynamical system governed by fractal-fractional-order evolution differential equations subject to terminal boundary conditions. We derive existence and uniqueness results using Schaefer’s and Banach’s fixed-point theorems, respectively. Additionally, the Hyers–Ulam stability approach is employed to analyze the system’s stability. We employ vector terminology for the proposed problem to make the analysis simple. To illustrate the practical relevance of our findings, we apply the derived results to a numerical example and graphically illustrate the solution for different fractal-fractional orders, emphasizing the effect of the derivative’s order on system behavior. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
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25 pages, 1605 KiB  
Article
Analysis of an Acute Diarrhea Piecewise Modified ABC Fractional Model: Optimal Control, Stability and Simulation
by Yasir A. Madani, Mohammed A. Almalahi, Osman Osman, Blgys Muflh, Khaled Aldwoah, Khidir Shaib Mohamed and Nidal Eljaneid
Fractal Fract. 2025, 9(2), 68; https://doi.org/10.3390/fractalfract9020068 - 23 Jan 2025
Cited by 1 | Viewed by 676
Abstract
Acute diarrhea poses a significant global health challenge, especially in settings with poor sanitation. This study develops a mathematical model of diarrhea, employing a piecewise modified ABC (pmABC) fractional derivative to capture the disease’s transmission dynamics, including crossover effects between classical and fractional [...] Read more.
Acute diarrhea poses a significant global health challenge, especially in settings with poor sanitation. This study develops a mathematical model of diarrhea, employing a piecewise modified ABC (pmABC) fractional derivative to capture the disease’s transmission dynamics, including crossover effects between classical and fractional behaviors. We analyze the local and global stability of the disease-free equilibrium and calculate the basic reproduction number R0 using the next-generation matrix method. Furthermore, we formulate an optimal control model that incorporates both strategies to reduce contact between susceptible and infected individuals, and to treat infected patients. Numerical simulations demonstrate the model’s behavior, illustrating that enhanced hygiene compliance reduces R0 by decreasing contact rates, while increased effective contact rates elevate R0. Additionally, the simulations reveal a positive correlation between higher concentrations of acute diarrhea bacteria and increased rates of subsequent infections. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
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32 pages, 409 KiB  
Article
Regularity Results for Hybrid Proportional Operators on Hölder Spaces
by Mieczysław Cichoń, Hussein A. H. Salem and Wafa Shammakh
Fractal Fract. 2025, 9(2), 58; https://doi.org/10.3390/fractalfract9020058 - 21 Jan 2025
Viewed by 635
Abstract
Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on [...] Read more.
Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equivalence of differential and integral problems for proportional calculus problems. The operators are always studied in the appropriate function spaces. Furthermore, the investigation extends these results to encompass the more general notion of Hilfer hybrid derivatives. The primary aim of this study is to preserve the maximal regularity of solutions for this class of problems. To this end, we consider such operators not only in spaces of absolutely continuous functions, but also in particular in little Hölder spaces. It is widely acknowledged that these spaces offer a natural framework for the study of classical Riemann–Liouville integral operators as inverse operators with derivatives of fractional order. This paper presents a comprehensive study of this problem for proportional derivatives and demonstrates the application of the obtained results to Langevin-type boundary problems. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
22 pages, 393 KiB  
Article
Certain Novel Fixed-Point Theorems Applied to Fractional Differential Equations
by Muhammad Din, Umar Ishtiaq, Khalid A. Alnowibet, Tania A. Lazăr, Vasile L. Lazăr and Liliana Guran
Fractal Fract. 2024, 8(12), 701; https://doi.org/10.3390/fractalfract8120701 - 27 Nov 2024
Cited by 1 | Viewed by 1011
Abstract
In this paper, we introduce a new class of contractions in normed spaces, referred to as generalized enriched Kannan contractions. These contractions expand the familiar enriched Kannan contractions to three-point versions, broadening the scope of Kannan contractions. These mappings are typically discontinuous, except [...] Read more.
In this paper, we introduce a new class of contractions in normed spaces, referred to as generalized enriched Kannan contractions. These contractions expand the familiar enriched Kannan contractions to three-point versions, broadening the scope of Kannan contractions. These mappings are typically discontinuous, except at the fixed points, where they exhibit continuity, similar to enriched Kannan mappings. However, through suitable examples, we demonstrate that these two classes of mappings are distinct from one another. We present new results for generalized enriched Kannan contractions. Additionally, by incorporating conditions of continuity and asymptotic regularity, we extend the class of operators to which fixed-point methods can be applied. Additionally, we derive two more results for generalized enriched Kannan contractions in normed spaces, without the requirement that they be Banach spaces. Finally, we use our main result to demonstrate the existence of solutions for a boundary value problem involving a fractional differential equation. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
13 pages, 799 KiB  
Article
Integral Operators in b-Metric and Generalized b-Metric Spaces and Boundary Value Problems
by Christopher Middlebrook and Wenying Feng
Fractal Fract. 2024, 8(11), 674; https://doi.org/10.3390/fractalfract8110674 - 19 Nov 2024
Viewed by 991
Abstract
We study fixed-point theorems of contractive mappings in b-metric space, cone b-metric space, and the newly introduced extended b-metric space. To generalize an existence and uniqueness result for the so-called Φs functions in the b-metric space to the extended b-metric space and [...] Read more.
We study fixed-point theorems of contractive mappings in b-metric space, cone b-metric space, and the newly introduced extended b-metric space. To generalize an existence and uniqueness result for the so-called Φs functions in the b-metric space to the extended b-metric space and the cone b-metric space, we introduce the class of ΦM functions and apply the Hölder continuous condition in the extended b-metric space. The obtained results are applied to prove the existence and uniqueness of solutions and positive solutions for nonlinear integral equations and fractional boundary value problems. Examples and numerical simulation are given to illustrate the applications. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
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13 pages, 309 KiB  
Article
On One Point Singular Nonlinear Initial Boundary Value Problem for a Fractional Integro-Differential Equation via Fixed Point Theory
by Said Mesloub, Eman Alhazzani and Hassan Eltayeb Gadain
Fractal Fract. 2024, 8(9), 526; https://doi.org/10.3390/fractalfract8090526 - 10 Sep 2024
Viewed by 1030
Abstract
In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular [...] Read more.
In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order θ∈[0,1]. The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
37 pages, 485 KiB  
Article
Existence and Stability of Solutions for p-Proportional ω-Weighted κ-Hilfer Fractional Differential Inclusions in the Presence of Non-Instantaneous Impulses in Banach Spaces
by Feryal Aladsani and Ahmed Gamal Ibrahim
Fractal Fract. 2024, 8(8), 475; https://doi.org/10.3390/fractalfract8080475 - 14 Aug 2024
Cited by 1 | Viewed by 885
Abstract
In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the p-proportional ω-weighted κ-Hilfer derivative includes an exponential function, [...] Read more.
In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the p-proportional ω-weighted κ-Hilfer derivative includes an exponential function, Da,λσ,ρ,p,κ,ω, and then we consider a non-instantaneous impulse differential inclusion containing Da,λσ,ρ,p,κ,ω with order σ∈(1,2) and of kind ρ∈[0,1] in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the (p,ω,κ)-generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the (p,ω,κ)-generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator Da,λσ,ρ,p,κ,ω, our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
20 pages, 394 KiB  
Article
A Unified Approach to Solvability and Stability of Multipoint BVPs for Langevin and Sturm–Liouville Equations with CH–Fractional Derivatives and Impulses via Coincidence Theory
by Kaihong Zhao, Juqing Liu and Xiaojun Lv
Fractal Fract. 2024, 8(2), 111; https://doi.org/10.3390/fractalfract8020111 - 13 Feb 2024
Cited by 26 | Viewed by 1955
Abstract
The Langevin equation is a model for describing Brownian motion, while the Sturm–Liouville equation is an important mechanical model. This paper focuses on the solvability and stability of nonlinear impulsive Langevin and Sturm–Liouville equations with Caputo–Hadamard (CH) fractional derivatives and multipoint boundary value [...] Read more.
The Langevin equation is a model for describing Brownian motion, while the Sturm–Liouville equation is an important mechanical model. This paper focuses on the solvability and stability of nonlinear impulsive Langevin and Sturm–Liouville equations with Caputo–Hadamard (CH) fractional derivatives and multipoint boundary value conditions. To unify the two types of equations, we investigate a general nonlinear impulsive coupled implicit system. By cleverly constructing relevant operators involving impulsive terms, we establish the coincidence degree theory and obtain the solvability. We explore the stability of solutions using nonlinear analysis and inequality techniques. As the most direct application, we naturally obtained the solvability and stability of the Langevin and Sturm–Liouville equations mentioned above. Finally, an example is provided to demonstrate the validity and availability of our major findings. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
40 pages, 451 KiB  
Article
The Impulsive Coupled Langevin ψ-Caputo Fractional Problem with Slit-Strip-Generalized-Type Boundary Conditions
by Haroon Niaz Ali Khan, Akbar Zada, Ioan-Lucian Popa and Sana Ben Moussa
Fractal Fract. 2023, 7(12), 837; https://doi.org/10.3390/fractalfract7120837 - 25 Nov 2023
Cited by 3 | Viewed by 1191
Abstract
In this paper, the existence of a unique solution is established for a coupled system of Langevin fractional problems of ψ-Caputo fractional derivatives with generalized slit-strip-type integral boundary conditions and impulses using the Banach contraction principle. We also find at least one [...] Read more.
In this paper, the existence of a unique solution is established for a coupled system of Langevin fractional problems of ψ-Caputo fractional derivatives with generalized slit-strip-type integral boundary conditions and impulses using the Banach contraction principle. We also find at least one solution to the aforementioned system using some assumptions and Schaefer’s fixed point theorem. After that, Ulam–Hyers stability is discussed. Finally, to provide additional support for the main results, pertinent examples are presented. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
19 pages, 342 KiB  
Article
Fractional Simpson-like Inequalities with Parameter for Differential s-tgs-Convex Functions
by Meriem Merad, Badreddine Meftah, Hamid Boulares, Abdelkader Moumen and Mohamed Bouye
Fractal Fract. 2023, 7(11), 772; https://doi.org/10.3390/fractalfract7110772 - 24 Oct 2023
Cited by 1 | Viewed by 1167
Abstract
In this paper, we first prove a new parameterized identity. Based on this identity we establish some parametrized Simpson-like type symmetric inequalities, for functions whose first derivatives are s-tgs-convex via Reimann–Liouville frational operators. Some special cases are discussed. [...] Read more.
In this paper, we first prove a new parameterized identity. Based on this identity we establish some parametrized Simpson-like type symmetric inequalities, for functions whose first derivatives are s-tgs-convex via Reimann–Liouville frational operators. Some special cases are discussed. Applications to numerical quadrature are provided. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
19 pages, 365 KiB  
Article
An Outlook on Hybrid Fractional Modeling of a Heat Controller with Multi-Valued Feedback Control
by Shorouk M. Al-Issa, Ahmed M. A. El-Sayed and Hind H. G. Hashem
Fractal Fract. 2023, 7(10), 759; https://doi.org/10.3390/fractalfract7100759 - 15 Oct 2023
Cited by 12 | Viewed by 1650
Abstract
In this study, we extend the investigations of fractional-order models of thermostats and guarantee the solvability of hybrid Caputo fractional models for heat controllers, satisfying some nonlocal hybrid multi-valued conditions with multi-valued feedback control, which involves the Chandrasekhar kernel, by using hybrid Dhage’s [...] Read more.
In this study, we extend the investigations of fractional-order models of thermostats and guarantee the solvability of hybrid Caputo fractional models for heat controllers, satisfying some nonlocal hybrid multi-valued conditions with multi-valued feedback control, which involves the Chandrasekhar kernel, by using hybrid Dhage’s fixed point theorem. A part of this study is dedicated to transforming this problem into an equivalent integral representation and then proving some existence results to achieve our aims. Furthermore, the continuous dependence of the unique solution on the control variable and on the set of selections will be discussed. Moreover, we provide an illustration to support our results. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)
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