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Fractal Fract
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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, Second 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: 30 September 2026 | Viewed by 1699

Special Issue Editors

Prof. Dr. Wenying Feng

E-Mail Website
Guest Editor
1. Department of Mathematics, Trent University, Peterborough, ON K9L 0G2, Canada
2. Department of Computer Science, Trent University, Peterborough, ON K9L 0G2, 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 Issues, Collections and Topics in MDPI journals
Prof. Dr. Feng Gao

E-Mail Website
Guest Editor
School of Science, Qingdao University of Technology, Qingdao 266520, China
Interests: numerical analysis; ODE; PDE; fractional calculus; machine learning; mathematical modelling

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 an extensive history and various applications in sciences, engineering, social activities, and natural phenomena. In particular, BVPs for fractional-order differential equations have attracted increasing interest and have achieved significant improvements, 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 behaviors compared to corresponding integer-order equations. Although the traditional topological and numerical methods for 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 effects for artificial intelligence.

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

Boundary Value Problems;
Eigenvalue problems;
Existence and positivity of solutions;
Uniqueness and multiplicity of solutions;
Stability and equilibrium;
Numerical solutions;
Fixed point methods and applications;
Nonlinear differential and Difference Equations;
Modeling with FDEs;
Neural networks and FDEs;
Fractional q-differential equations.

Prof. Dr. Wenying Feng
Prof. Dr. Feng Gao
Guest Editors

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 250 words) can be sent to the Editorial Office for assessment.

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

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

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Related Special Issue

Published Papers (2 papers)

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Research

23 pages, 480 KB  
Article
Impulsive Tempered Ψ-Fractional Differential Equations with Boundary and Integral Conditions
by Chayapat Sudprasert, Suphawat Asawasamrit, Sotiris K. Ntouyas and Jessada Tariboon
Fractal Fract. 2026, 10(2), 113; https://doi.org/10.3390/fractalfract10020113 - 5 Feb 2026
Viewed by 601
Abstract
This paper studies mixed impulsive boundary value problems involving tempered Ψ-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The [...] Read more.
This paper studies mixed impulsive boundary value problems involving tempered Ψ-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The generalized tempered Ψ-operator unifies several existing fractional derivatives, offering enhanced flexibility for modeling complex dynamical phenomena. Impulsive effects and integral boundary conditions are incorporated to describe processes subject to sudden changes and historical dependence. The problem is reformulated as a Volterra integral equation, and fixed-point theory is employed to establish analytical results. Existence and uniqueness of solutions are proven using the Banach Contraction Mapping Principle, while the Leray–Schauder nonlinear alternative ensures existence in non-contractive cases. The proposed framework provides a rigorous analytical basis for modeling phenomena characterized by both fading memory and sudden perturbations, with potential applications in physics, control theory, population dynamics, and epidemiology. A numerical example is presented to illustrate the validity and applicability of the main theoretical results. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application, Second Edition)
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22 pages, 385 KB  
Article
Analysis of a Coupled System of Implicit Fractional Differential Equations of Order α ∈ (1, 2] with Anti-Periodic Boundary Conditions
by Areen Al-Khateeb, Muath Awadalla, Murugesan Manigandan and Salma Trabelsi
Fractal Fract. 2025, 9(12), 768; https://doi.org/10.3390/fractalfract9120768 - 25 Nov 2025
Cited by 1 | Viewed by 755
Abstract
This paper investigates a coupled system of nonlinear implicit fractional differential equations of order α∈(1,2] subject to anti-periodic boundary conditions. The analysis is conducted using the ψ-Caputo fractional derivative, a generalized operator that incorporates several well-known [...] Read more.
This paper investigates a coupled system of nonlinear implicit fractional differential equations of order α∈(1,2] subject to anti-periodic boundary conditions. The analysis is conducted using the ψ-Caputo fractional derivative, a generalized operator that incorporates several well-known fractional derivatives. The system features implicit coupling, where each equation depends on both unknown functions and their first derivatives, as well as an implicit dependence on the fractional derivatives themselves. The boundary value problem is transformed into an equivalent system of integral equations. Sufficient conditions for the existence and uniqueness of solutions are established using Banach’s and Krasnoselskii’s fixed-point theorems in an appropriately chosen Banach space. Furthermore, the Ulam–Hyers stability of the system is analyzed. The applicability of the theoretical results is demonstrated through a detailed example of a coupled system where all hypotheses are verified. Full article
(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application, Second Edition)
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