The Delay Differential Equations and Their Applications

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

Deadline for manuscript submissions: 31 August 2024 | Viewed by 2320

Special Issue Editors


E-Mail Website
Guest Editor
Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al-Ain 15551, United Arab Emirates
Interests: applied mathematics; stochastic delay differential equations; biomathematics

E-Mail Website
Guest Editor
Department of Applied Mathematics and Modeling, University of Plovdiv, 4000 Plovdiv, Bulgaria
Interests: applied mathematics; differential equations; delay systems; impulsive differential equations; fractional differential equations; modeling with differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Delay differential equations (DDEs) are a type of differential equation in which the derivative of a function depends not only on its current value, but also on its past values. In other words, the rate of change of the function at a given time depends on its values at previous times. This introduces a time delay into the system, hence the name "delay" differential equations.

DDEs find applications in various fields, including biology, physics, chemistry, economics, and engineering. They are particularly useful for modeling systems with memory effects or systems in which time delays play a crucial role. Some examples include population dynamics, with time delays in birth or death rates, chemical reactions with delayed effects, and control systems with communication delays.

Solving DDEs analytically can be challenging due to the presence of time delays. Numerical methods, such as the method of steps, collocation methods, and numerical continuation, are commonly employed to approximate solutions to DDEs. These methods discretize the time domain and approximate the delayed terms using appropriate interpolation techniques. 

The primary objective of this Special Issue is twofold: first, to provide a comprehensive review of the current state-of-the-art knowledge in Delay Differential Equations (DDEs) and their applications in engineering and biological systems. The second aim is to delve into the emerging challenges and open problems in mathematics that have arisen as a result of these novel and intricate models. These DDE models, spanning various classes, have been developed with the aim of unraveling the complexities of diverse phenomena, and their analysis has led to intriguing mathematical questions that require further investigation.

Keywords: bifurcation; fractional-order; epidemic models; Lyapunov functionals; predator–prey model; sensitivity; stability; stationary distribution; stochastic perturbations; time delays

In this Special Issue, original research articles and reviews are welcome. Research areas may include (but are not limited to) the following:

  • Qualitative behaviors of DDEs;
  • Fractional-order and Stochastic DDEs and applications;
  • Optimal control in biological systems, medicine/spread of disease;
  • Analysis and numerical implementation of models described by ODEs, DDEs, and PDEs systems;
  • Numerical schemes for DDEs and stochastic DDEs;
  • Dynamics, including stability, bifurcation, and chaos of DDEs.

I look forward to receiving your contributions.

Dr. Hebatallah J. Alsakaji
Prof. Dr. Snezhana Hristova
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 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

  • bifurcation
  • fractional-order
  • epidemic models
  • Lyapunov functionals
  • predator–prey model
  • sensitivity
  • stability
  • stationary distribution
  • stochastic perturbations
  • time delays

Published Papers (2 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

14 pages, 598 KiB  
Article
Further Results on the Input-to-State Stability of a Linear Disturbed System with Control Delay
by Daniela Enciu, Adrian Toader and Ioan Ursu
Mathematics 2024, 12(5), 634; https://doi.org/10.3390/math12050634 - 21 Feb 2024
Viewed by 460
Abstract
In this paper, a theorem is obtained that gives sufficient input-to-state stability conditions for linear systems with control delay and additive disturbances. Stabilizing feedback is considered available in the absence of delay and disturbances. The mathematical tools are the Lyapunov–Krasovskii functional, the Jensen [...] Read more.
In this paper, a theorem is obtained that gives sufficient input-to-state stability conditions for linear systems with control delay and additive disturbances. Stabilizing feedback is considered available in the absence of delay and disturbances. The mathematical tools are the Lyapunov–Krasovskii functional, the Jensen inequality and the double Hadamard inequality. The critical delay is highlighted. Full article
(This article belongs to the Special Issue The Delay Differential Equations and Their Applications)
Show Figures

Figure 1

34 pages, 863 KiB  
Article
A Systematic Approach to Delay Functions
by Christopher N. Angstmann, Stuart-James M. Burney, Bruce I. Henry, Byron A. Jacobs and Zhuang Xu
Mathematics 2023, 11(21), 4526; https://doi.org/10.3390/math11214526 - 2 Nov 2023
Viewed by 1013
Abstract
We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into [...] Read more.
We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions. Full article
(This article belongs to the Special Issue The Delay Differential Equations and Their Applications)
Show Figures

Figure 1

Back to TopTop