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

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 33425

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Prof. Dr. Francisco Rodríguez

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Department of Applied Mathematics and Multidisciplinary Institute for Environmental Studies (IMEM), University of Alicante, Apdo. 99, 03080 Alicante, Spain
Interests: delay differential equations; diffusion and heat conduction models with delay; mathematical biology
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Prof. Dr. Juan Carlos Cortés López

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Department of Applied Mathematics and Institute for Multidisciplinary Mathematics (im2), Universitat Politècnica de València, 46022 Valencia, Spain
Interests: differential equations with randomness; mathematical modelling
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Prof. Dr. María Ángeles Castro

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Guest Editor

Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain
Interests: delay differential equations; numerical methods; non-Fourier heat conduction models
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Special Issue Information

Dear Colleagues,

Models of differential equations with delay have pervaded many scientific and technical fields in the last decades. The use of delay differential equations (DDE) and partial delay differential equations (PDDE) to model problems with the presence of lags or hereditary effects have demonstrated a valuable balance between realism and tractability. Of special interest in recent years is the development and analysis of models with interactions between delay and random effects, through the use of stochastic and random delay differential equation (SDDE and RDDE). In this Special Issue, we are inviting submissions of original papers dealing with the theory and applications of differential equations with delay (DDE, PDDE, SDDE, and RDDE), including, but not limited to, construction of exact solutions, numerical methods, dynamical properties, and applications to mathematical modeling of phenomena and processes in biology, medicine, economics, engineering, and the social sciences.

Prof. Dr. Francisco Rodríguez
Prof. Dr. Juan Carlos Cortés López
Prof. Dr. María Ángeles Castro
Guest Editors

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Keywords

Delay differential equations
Partial delay differential equations
Random and stochastic delay differential equations
Numerical methods
Exact solutions and dynamical properties
Diffusion and heat conduction models with delay
Uncertainty quantification with delay differential equations and simulation
Models with delay in biology, economics, and engineering

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Published Papers (13 papers)

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Research

21 pages, 628 KiB

Open AccessArticle

Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay

by Abraham J. Arenas, Gilberto González-Parra, Jhon J. Naranjo, Myladis Cogollo and Nicolás De La Espriella

Mathematics 2021, 9(3), 257; https://doi.org/10.3390/math9030257 - 28 Jan 2021

Cited by 13 | Viewed by 4144

Abstract

We propose a mathematical model based on a set of delay differential equations that describe intracellular HIV infection. The model includes three different subpopulations of cells and the HIV virus. The mathematical model is formulated in such a way that takes into account the time between viral entry into a target cell and the production of new virions. We study the local stability of the infection-free and endemic equilibrium states. Moreover, by using a suitable Lyapunov functional and the LaSalle invariant principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. In addition, we designed a non-standard difference scheme that preserves some relevant properties of the continuous mathematical model. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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19 pages, 4917 KiB

Open AccessArticle

Dependence of Dynamics of a System of Two Coupled Generators with Delayed Feedback on the Sign of Coupling

by Alexandra Kashchenko

Mathematics 2020, 8(10), 1790; https://doi.org/10.3390/math8101790 - 15 Oct 2020

Cited by 4 | Viewed by 1466

Abstract

In this paper, we study the nonlocal dynamics of a system of delay differential equations with large parameters. This system simulates coupled generators with delayed feedback. Using the method of steps, we construct asymptotics of solutions. By these asymptotics, we construct a special finite-dimensional map. This map helps us to determine the structure of solutions. We study the dependence of solutions on the coupling parameter and show that the dynamics of the system is significantly different in the case of positive coupling and in the case of negative coupling. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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8 pages, 274 KiB

Open AccessArticle

A Class of Fractional Degenerate Evolution Equations with Delay

by Amar Debbouche and Vladimir E. Fedorov

Mathematics 2020, 8(10), 1700; https://doi.org/10.3390/math8101700 - 3 Oct 2020

Cited by 3 | Viewed by 1725

Abstract

We establish a class of degenerate fractional differential equations involving delay arguments in Banach spaces. The system endowed by a given background and the generalized Showalter–Sidorov conditions which are natural for degenerate type equations. We prove the results of local unique solvability by using, mainly, the method of contraction mappings. The obtained theory via its abstract results is applied to the research of initial-boundary value problems for both Scott–Blair and modified Sobolev systems of equations with delays. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

20 pages, 329 KiB

Open AccessArticle

Theoretical Analysis (Convergence and Stability) of a Difference Approximation for Multiterm Time Fractional Convection Diffusion-Wave Equations with Delay

by A. S. Hendy and R. H. De Staelen

Mathematics 2020, 8(10), 1696; https://doi.org/10.3390/math8101696 - 3 Oct 2020

Cited by 1 | Viewed by 1684

Abstract

In this paper, we introduce a high order numerical approximation method for convection diffusion wave equations armed with a multiterm time fractional Caputo operator and a nonlinear fixed time delay. A temporal second-order scheme which is behaving linearly is derived and analyzed for [...] Read more.

L_{2} - 1_{σ}

and the order reduction technique. By means of the discrete energy method, convergence and stability of the proposed compact difference scheme are estimated unconditionally. A numerical example is provided to illustrate the theoretical results. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

19 pages, 1299 KiB

Open AccessArticle

Delay Stability of n-Firm Cournot Oligopolies

by Akio Matsumoto and Ferenc Szidarovszky

Mathematics 2020, 8(9), 1615; https://doi.org/10.3390/math8091615 - 18 Sep 2020

Cited by 1 | Viewed by 2109

Abstract

The dynamic behavior of n-firm oligopolies is examined without product differentiation and with linear price and cost functions. Continuous time scales are assumed with best response dynamics, in which case the equilibrium is asymptotically stable without delays. The firms are assumed to face both implementation and information delays. If the delays are equal, then the model is a single delay case, and the equilibrium is oscillatory stable if the delay is small, at the threshold stability is lost by Hopf bifurcation with cyclic behavior, and for larger delays, the trajectories show expanding cycles. In the case of the non-equal delays, the stability switching curves are constructed and the directions of stability switches are determined. In the case of growth rate dynamics, the local behavior of the trajectories is similar to that of the best response dynamics. Simulation studies verify and illustrate the theoretical findings. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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18 pages, 462 KiB

Open AccessArticle

The Convergence Analysis of a Numerical Method for a Structured Consumer-Resource Model with Delay in the Resource Evolution Rate

by Luis M. Abia, Óscar Angulo, Juan C. López-Marcos and Miguel A. López-Marcos

Mathematics 2020, 8(9), 1440; https://doi.org/10.3390/math8091440 - 27 Aug 2020

Viewed by 2096

Abstract

In this paper, we go through the development of a new numerical method to obtain the solution to a size-structured population model that describes the evolution of a consumer feeding on a dynamical resource that reacts to the environment with a lag-time response. The problem involves the coupling of the partial differential equation that represents the population evolution and an ordinary differential equation with a constant delay that describes the evolution of the resource. The numerical treatment of this problem has not been considered before when a delay is included in the resource evolution rate. We analyzed the numerical scheme and proved a second-order rate of convergence by assuming enough regularity of the solution. We numerically confirmed the theoretical results with an academic test problem. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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17 pages, 706 KiB

Open AccessFeature PaperArticle

Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay

by Julia Calatayud, Juan Carlos Cortés, Marc Jornet and Francisco Rodríguez

Mathematics 2020, 8(9), 1417; https://doi.org/10.3390/math8091417 - 24 Aug 2020

Cited by 3 | Viewed by 2277

Abstract

In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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16 pages, 446 KiB

Open AccessArticle

L^p-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term

by Juan Carlos Cortés and Marc Jornet

Mathematics 2020, 8(6), 1013; https://doi.org/10.3390/math8061013 - 20 Jun 2020

Cited by 9 | Viewed by 2232

Abstract

This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay

τ > 0

, by adding a random forcing term

f (t)

that varies with time: [...] Read more.

This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay

τ > 0

, by adding a random forcing term

f (t)

that varies with time:

x^{'} (t) = a x (t) + b x (t - τ) + f (t)

t \geq 0

, with initial condition

x (t) = g (t)

- τ \leq t \leq 0

. The coefficients a and b are assumed to be random variables, while the forcing term

f (t)

and the initial condition

g (t)

are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space

L^{p}

of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an

L^{p}

-solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay

τ

tends to 0, the random delay equation tends in

L^{p}

to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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13 pages, 1148 KiB

Open AccessArticle

Second-Order Dual Phase Lag Equation. Modeling of Melting and Resolidification of Thin Metal Film Subjected to a Laser Pulse

by Ewa Majchrzak and Bohdan Mochnacki

Mathematics 2020, 8(6), 999; https://doi.org/10.3390/math8060999 - 18 Jun 2020

Cited by 8 | Viewed by 2281

Abstract

The process of partial melting and resolidification of a thin metal film subjected to a high-power laser beam is considered. The mathematical model of the process is based on the second-order dual phase lag equation (DPLE). Until now, this equation has not been used for the modeling of phase changes associated with heating and cooling of thin metal films and the considerations regarding this issue are the most important part of the article. In the basic energy equation, the internal heat sources associated with the laser action and the evolution of phase change latent heat are taken into account. Thermal processes in the domain of pure metal (chromium) are analyzed and it is assumed that the evolution of latent heat occurs at a certain interval of temperature to which the solidification point was conventionally extended. This approach allows one to introduce the continuous function corresponding to the volumetric fraction of solid or liquid state at the neighborhood of the point considered, which significantly simplifies the phase changes modeling. At the stage of numerical computations, the authorial program based on the implicit scheme of the finite difference method (FDM) was used. In the final part of the paper, the examples of numerical computations (including the results of simulations for different laser intensities and different characteristic times of laser pulse) are presented and the conclusions are formulated. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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12 pages, 259 KiB

Open AccessArticle

Two New Strategies for Pricing Freight Options by Means of a Valuation PDE and by Functional Bounds

by Lourdes Gómez-Valle, Miguel Angel López-Marcos and Julia Martínez-Rodríguez

Mathematics 2020, 8(4), 620; https://doi.org/10.3390/math8040620 - 17 Apr 2020

Cited by 4 | Viewed by 2150

Abstract

Freight derivative prices have been modeled assuming that the spot freight follows a particular stochastic process in order to manage them, like freight futures, forwards and options. However, an explicit formula for pricing freight options is not known, not even for simple spot freight processes. This is partly due to the fact that there is no valuation equation for pricing freight options. In this paper, we deal with this problem from two independent points of view. On the one hand, we provide a novel theoretical framework for pricing these Asian-style options. In this way, we build a partial differential equation whose solution is the freight option price obtained from stochastic delay differential equations. On the other hand, we prove lower and upper bounds for those freight options which enables us to estimate the option price. In this work, we consider that the spot freight rate follows a general stochastic diffusion process without restrictions in the drift and volatility functions. Finally, using recent data from the Baltic Exchange, we compare the described bounds with the freight option prices. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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17 pages, 403 KiB

Open AccessArticle

Stability Analysis of an Age-Structured SEIRS Model with Time Delay

by Zhe Yin, Yongguang Yu and Zhenzhen Lu

Mathematics 2020, 8(3), 455; https://doi.org/10.3390/math8030455 - 23 Mar 2020

Cited by 11 | Viewed by 3465

Abstract

This paper is concerned with the stability of an age-structured susceptible–exposed– infective–recovered–susceptible (SEIRS) model with time delay. Firstly, the traveling wave solution of system can be obtained by using the method of characteristic. The existence and uniqueness of the continuous traveling wave solution is investigated under some hypotheses. Moreover, the age-structured SEIRS system is reduced to the nonlinear autonomous system of delay ODE using some insignificant simplifications. It is studied that the dimensionless indexes for the existence of one disease-free equilibrium point and one endemic equilibrium point of the model. Furthermore, the local stability for the disease-free equilibrium point and the endemic equilibrium point of the infection-induced disease model is established. Finally, some numerical simulations were carried out to illustrate our theoretical results. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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38 pages, 455 KiB

Open AccessArticle

Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

by Allaberen Ashyralyev and Deniz Agirseven

Mathematics 2019, 7(12), 1163; https://doi.org/10.3390/math7121163 - 2 Dec 2019

Cited by 11 | Viewed by 2204

Abstract

In this paper, we study the initial value problem for a semilinear delay hyperbolic equation in Hilbert spaces with a self-adjoint positive definite operator. The mean theorem on the existence and uniqueness of a bounded solution of this differential problem for a semilinear hyperbolic equation with unbounded time delay term is established. In applications, the existence and uniqueness of bounded solutions of four problems for semilinear hyperbolic equations with time delay in unbounded term are obtained. For the approximate solution of this abstract differential problem, the two-step difference scheme of a first order of accuracy is presented. The mean theorem on the existence and uniqueness of a uniformly bounded solution of this difference scheme with respect to time stepsize is established. In applications, the existence and uniqueness of a uniformly bounded solutions with respect to time and space stepsizes of difference schemes for four semilinear partial differential equations with time delay in unbounded term are obtained. In general, it is not possible to get the exact solution of semilinear hyperbolic equations with unbounded time delay term. Therefore, numerical results for the solution of difference schemes for one and two dimensional semilinear hyperbolic equation with time delay are presented. Finally, some numerical examples are given to confirm the theoretical analysis. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

14 pages, 3464 KiB

Open AccessArticle

Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems

by María Ángeles Castro, Miguel Antonio García, José Antonio Martín and Francisco Rodríguez

Mathematics 2019, 7(11), 1038; https://doi.org/10.3390/math7111038 - 3 Nov 2019

Cited by 13 | Viewed by 2419

Abstract

In recent works, exact and nonstandard finite difference schemes for scalar first order linear delay differential equations have been proposed. The aim of the present work is to extend these previous results to systems of coupled delay differential equations [...] Read more.

X^{'} (t) = A X (t) + B X (t - τ),

where X is a vector, and A and B are commuting real matrices, in general not simultaneously diagonalizable. Based on a constructive expression for the exact solution of the vector equation, an exact scheme is obtained, and different nonstandard numerical schemes of increasing order are proposed. Dynamic consistency properties of the new nonstandard schemes are illustrated with numerical examples, and proved for a class of methods. Full article

(This article belongs to the Special Issue Models of Delay Differential Equations)

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