Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges

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

Deadline for manuscript submissions: closed (30 June 2021) | Viewed by 17153

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Istituto per le Applicazioni del Calcolo M. Picone, CNR, Via Amendola 122, I-70126 Bari, Italy
Interests: numerical methods for dynamical systems; ordinary and partial differential equations; geometric numerical integration with applications in ecology, health, biology, chemistry, public heritage, and economy
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Special Issue Information

Dear colleagues,

Models of differential equations (DEs) describe a wide range of complex issues of ecology, health, biology, chemistry, cultural heritage conservation, engineering, physical sciences, economics, and finance. Differential modelling and difference equations are tools to understand the dynamics and to do forecasting and scenario analysis; in addition, they allow for the detection of optimal solutions according to selected criteria. 

This issue focuses on modeling through differential equations (both ODE and PDE) and aims to highlight old and new challenges in the formulation, solution, understanding, and interpretation of differential models in different real world applications. 

Classical formulations or more recent approaches based on compartmental models, dynamic systems on networks, multiscale problems, fractional differential equations, and Hamiltonian dynamic evolutions are all welcome. The covered technical topics range from analytical methods including phase plane analysis, linearization of non-linear systems, bifurcations, general theory of existence and approximation of non-linear solutions of DEs, to explicit, implicit, positive, non-standard, geometric numerical methods for initial and boundary valued DE problems. Classical research questions are faced and new challenges, such as the numerical treatment of uncertainty, will be addressed.

Dr. Fasma Diele
Guest Editor

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Keywords

  • Differential modelling
  • Real world applications
  • Analytical tools
  • Numerical schemes

Published Papers (8 papers)

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Editorial

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3 pages, 157 KiB  
Editorial
Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges
by Fasma Diele
Mathematics 2022, 10(2), 249; https://doi.org/10.3390/math10020249 - 14 Jan 2022
Viewed by 1087
Abstract
The articles published in the Special Issue “Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges” of the MDPI Mathematics journal are here collected [...] Full article

Research

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29 pages, 550 KiB  
Article
BVPs Codes for Solving Optimal Control Problems
by Francesca Mazzia and Giuseppina Settanni
Mathematics 2021, 9(20), 2618; https://doi.org/10.3390/math9202618 - 17 Oct 2021
Cited by 4 | Viewed by 3800
Abstract
Optimal control problems arise in many applications and need suitable numerical methods to obtain a solution. The indirect methods are an interesting class of methods based on the Pontryagin’s minimum principle that generates Hamiltonian Boundary Value Problems (BVPs). In this paper, we review [...] Read more.
Optimal control problems arise in many applications and need suitable numerical methods to obtain a solution. The indirect methods are an interesting class of methods based on the Pontryagin’s minimum principle that generates Hamiltonian Boundary Value Problems (BVPs). In this paper, we review some general-purpose codes for the solution of BVPs and we show their efficiency in solving some challenging optimal control problems. Full article
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9 pages, 255 KiB  
Article
Important Criteria for Asymptotic Properties of Nonlinear Differential Equations
by Ahmed AlGhamdi, Omar Bazighifan and Rami Ahmad El-Nabulsi
Mathematics 2021, 9(14), 1659; https://doi.org/10.3390/math9141659 - 14 Jul 2021
Cited by 5 | Viewed by 1269
Abstract
In this article, we prove some new oscillation theorems for fourth-order differential equations. New oscillation results are established that complement related contributions to the subject. We use the Riccati technique and the integral averaging technique to prove our results. As proof of the [...] Read more.
In this article, we prove some new oscillation theorems for fourth-order differential equations. New oscillation results are established that complement related contributions to the subject. We use the Riccati technique and the integral averaging technique to prove our results. As proof of the effectiveness of the new criteria, we offer more than one practical example. Full article
13 pages, 288 KiB  
Article
Inverse Problem for the Sobolev Type Equation of Higher Order
by Alyona Zamyshlyaeva and Aleksandr Lut
Mathematics 2021, 9(14), 1647; https://doi.org/10.3390/math9141647 - 13 Jul 2021
Cited by 6 | Viewed by 1293
Abstract
The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular and singular. For these problems, the theory of polynomially bounded [...] Read more.
The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular and singular. For these problems, the theory of polynomially bounded operator pencils is used. The unknown coefficient of the original equation is restored using the method of successive approximations. The main result of this work is a theorem on the unique solvability of the original problem. This study continues and generalizes the authors’ previous research in this area. All the obtained results can be applied to the mathematical modeling of various processes and phenomena that fit the problem under study. Full article
11 pages, 6046 KiB  
Article
On-Off Intermittency in a Three-Species Food Chain
by Gabriele Vissio and Antonello Provenzale
Mathematics 2021, 9(14), 1641; https://doi.org/10.3390/math9141641 - 12 Jul 2021
Cited by 4 | Viewed by 1527
Abstract
The environment affects population dynamics through multiple drivers. Here we explore a simplified version of such influence in a three-species food chain, making use of the Hastings–Powell model. This represents an idealized resource–consumer–predator chain, or equivalently, a vegetation–host–parasitoid system. By stochastically perturbing the [...] Read more.
The environment affects population dynamics through multiple drivers. Here we explore a simplified version of such influence in a three-species food chain, making use of the Hastings–Powell model. This represents an idealized resource–consumer–predator chain, or equivalently, a vegetation–host–parasitoid system. By stochastically perturbing the value of some parameters in this dynamical system, we observe dramatic modifications in the system behavior. In particular, we show the emergence of on–off intermittency, i.e., an irregular alternation between stable phases and sudden bursts in population size, which hints towards a possible conceptual description of population outbursts grounded into an environment-driven mechanism. Full article
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34 pages, 1852 KiB  
Article
Mass-Preserving Approximation of a Chemotaxis Multi-Domain Transmission Model for Microfluidic Chips
by Elishan Christian Braun, Gabriella Bretti and Roberto Natalini
Mathematics 2021, 9(6), 688; https://doi.org/10.3390/math9060688 - 23 Mar 2021
Cited by 7 | Viewed by 1911
Abstract
The present work is inspired by the recent developments in laboratory experiments made on chips, where the culturing of multiple cell species was possible. The model is based on coupled reaction-diffusion-transport equations with chemotaxis and takes into account the interactions among cell populations [...] Read more.
The present work is inspired by the recent developments in laboratory experiments made on chips, where the culturing of multiple cell species was possible. The model is based on coupled reaction-diffusion-transport equations with chemotaxis and takes into account the interactions among cell populations and the possibility of drug administration for drug testing effects. Our effort is devoted to the development of a simulation tool that is able to reproduce the chemotactic movement and the interactions between different cell species (immune and cancer cells) living in a microfluidic chip environment. The main issues faced in this work are the introduction of mass-preserving and positivity-preserving conditions, involving the balancing of incoming and outgoing fluxes passing through interfaces between 2D and 1D domains of the chip and the development of mass-preserving and positivity preserving numerical conditions at the external boundaries and at the interfaces between 2D and 1D domains. Full article
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17 pages, 429 KiB  
Article
Handling Hysteresis in a Referral Marketing Campaign with Self-Information. Hints from Epidemics
by Deborah Lacitignola
Mathematics 2021, 9(6), 680; https://doi.org/10.3390/math9060680 - 22 Mar 2021
Cited by 5 | Viewed by 1625
Abstract
In this study we show that concept of backward bifurcation, borrowed from epidemics, can be fruitfully exploited to shed light on the mechanism underlying the occurrence of hysteresis in marketing and for the strategic planning of adequate tools for its control. We enrich [...] Read more.
In this study we show that concept of backward bifurcation, borrowed from epidemics, can be fruitfully exploited to shed light on the mechanism underlying the occurrence of hysteresis in marketing and for the strategic planning of adequate tools for its control. We enrich the model introduced in (Gaurav et al., 2019) with the mechanism of self-information that accounts for information about the product performance basing on consumers’ experience on the recent past. We obtain conditions for which the model exhibits a forward or a backward phenomenology and evaluate the impact of self-information on both these scenarios. Our analysis suggests that, even if hysteretic dynamics in referral campaigns is intimately linked to the mechanism of referrals, an adequate level of self-information and a fairly high level of customer-satisfaction can act as strategic tools to manage hysteresis and allow the campaign to spread in more controllable conditions. Full article
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Review

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20 pages, 3812 KiB  
Review
Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives
by Oana Brandibur, Roberto Garrappa and Eva Kaslik
Mathematics 2021, 9(8), 914; https://doi.org/10.3390/math9080914 - 20 Apr 2021
Cited by 19 | Viewed by 3163
Abstract
Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting [...] Read more.
Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results. Full article
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