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Mathematics
Special Issues
Research on Delay Differential Equations and Their Applications
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Research on Delay Differential Equations and Their Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 30 September 2025 | Viewed by 816

Special Issue Editor

Dr. Qifeng Zhang

E-Mail Website
Guest Editor
Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China
Interests: high accurate and fast algorithms for nonlinear evolution equations; stability and numerical simulation of delayed differential equations; efficient numerical methods for fractional differential equations

Special Issue Information

Dear Colleagues,

Delay differential equations (DDEs) are a type of differential equation where the derivative of the unknown function at a certain time depends on the values of the function at previous times. This is in contrast to ordinary differential equations (ODEs), where the derivative depends only on the current value of the function. Their applications span a wide range of disciplines, from biology and engineering to economics and physics. Despite the challenges in solving and analyzing DDEs, they provide a more accurate representation of many real-world processes compared to ODEs. This Special Issue aims to gather original contributions including, but not limited to, the following:

  • Analytical/numerical methods for solving DDEs.
  • Constant delay DDEs.
  • Time-dependent delay DDEs.
  • State-dependent delay DDEs.
  • Neutral DDEs.
  • Fractional DDEs.
  • Stochastic DDEs.

Dr. Qifeng Zhang
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • numerical solution
  • analytical solution
  • modeling
  • stability analysis
  • convergence
  • delay diffusion equations
  • delay wave-type equations
  • partial differential equations with delay
  • partial functional differential equations
  • fractional delay partial differential equations

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Published Papers (2 papers)

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Research

27 pages, 1024 KiB  
Article
Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative
by Liang Hong and Lipu Zhang
Mathematics 2025, 13(13), 2044; https://doi.org/10.3390/math13132044 - 20 Jun 2025
Viewed by 159
Abstract
Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay [...] Read more.
Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay in affective states, while the nonlinear component regulates connection strength through emotional difference thresholds. Mathematical analysis establishes the existence and uniqueness of solutions with continuous dependence on initial conditions and proves the local asymptotic stability of network equilibria (Wij*=1δsech2(∥Ei−Ej∥), e.g., W*≈1.40 under typical parameters η=0.5, δ=0.3). We further derive closed-form expressions for the steady-state variance under stochastic perturbations (Var(Wij)=σζ22ηδ) and demonstrate a less than 6% deviation between simulated and theoretical values when σζ=0.1. Numerical experiments using the Euler–Maruyama method validate the convergence of connection weights toward the predicted equilibrium, reveal Gaussian features in the stationary distributions, and confirm power-law scaling between noise intensity and variance. The numerical accuracy of the fractional system is further verified through L1 discretization, with observed error convergence consistent with theoretical expectations for μ=0.5. This framework advances the mechanistic understanding of co-evolutionary dynamics in emotion-modulated social networks, supporting applications in clinical intervention design, collective sentiment modeling, and psychophysiological coupling research. Full article
(This article belongs to the Special Issue Research on Delay Differential Equations and Their Applications)
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17 pages, 497 KiB  
Article
Analyzing Coupled Delayed Fractional Systems: Theoretical Insights and Numerical Approaches
by Meraa Arab, Mohammed S. Abdo, Najla Alghamdi and Muath Awadalla
Mathematics 2025, 13(7), 1113; https://doi.org/10.3390/math13071113 - 28 Mar 2025
Viewed by 476
Abstract
In this work, we investigate the theoretical properties of a generalized coupled system of finite-delay fractional differential equations involving Caputo derivatives. We establish rigorous criteria to ensure the existence and uniqueness of solutions under appropriate assumptions on the problem parameters and constituent functions, [...] Read more.
In this work, we investigate the theoretical properties of a generalized coupled system of finite-delay fractional differential equations involving Caputo derivatives. We establish rigorous criteria to ensure the existence and uniqueness of solutions under appropriate assumptions on the problem parameters and constituent functions, employing contraction mapping principles and Schauder’s fixed-point theorem. Then, we examine the Ulam–Hyers stability of the proposed system. To illustrate the main findings, three examples are provided. Moreover, we provide numerical solutions using the Adams–Bashforth–Moulton method. The practical significance of our results is demonstrated through illustrative examples, highlighting applications in predator–prey dynamics and control systems. Full article
(This article belongs to the Special Issue Research on Delay Differential Equations and Their Applications)
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