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Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges-II
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Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges-II

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

Deadline for manuscript submissions: closed (15 July 2023) | Viewed by 6728

Special Issue Editor

Dr. Fasma Diele

E-Mail Website
Guest Editor
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
Special Issues, Collections and Topics in MDPI journals

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

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

  • differential modelling
  • real world applications
  • analytical tools
  • numerical schemes

Published Papers (5 papers)

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Research

25 pages, 1239 KiB  
Article
Z-Type Control Methods on a Three-Species Model with an Invasive Prey
by Fabiana Camattari, Francesca Acotto and Ezio Venturino
Mathematics 2023, 11(19), 4182; https://doi.org/10.3390/math11194182 - 6 Oct 2023
Viewed by 819
Abstract
The process by which nature and evolution discriminate between species bound to survive or become extinct has always been an interesting phenomenon. One example of this selection is represented by biological invasions, when alien species spread into new environments causing ecological disruptions, alteration [...] Read more.
The process by which nature and evolution discriminate between species bound to survive or become extinct has always been an interesting phenomenon. One example of this selection is represented by biological invasions, when alien species spread into new environments causing ecological disruptions, alteration of native population dynamics and ecosystem dysfunctioning. In this paper, we concentrate on the introduction of the Eastern cottontail (Sylvilagus floridanus) into the regions of northern and central Italy. This has influenced the local predator–prey dynamics of the indigenous European hares (Lepus europaeus) and red foxes (Vulpes vulpes), consequently producing a “hyperpredation” effect on hares. In this framework, we investigate this scenario with the objective of mitigating the negative effects of the invasion. Specifically, we devise and apply a Z-control technique in order to drive chosen populations toward a desired state that at least theoretically can ensure the survival of native species. The purpose of such a study is twofold: on one hand, this approach is employed with the aim of reducing the invasive population; secondly, it aims to avoid the extinction of native prey, i.e., the hare population. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges-II)
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20 pages, 10781 KiB  
Article
An Innate Moving Frame on Parametric Surfaces: The Dynamics of Principal Singular Curves
by Moody T. Chu and Zhenyue Zhang
Mathematics 2023, 11(15), 3306; https://doi.org/10.3390/math11153306 - 27 Jul 2023
Cited by 1 | Viewed by 752
Abstract
This article reports an experimental work that unveils some interesting yet unknown phenomena underneath all smooth nonlinear maps. The findings are based on the fact that, generalizing the conventional gradient dynamics, the right singular vectors of the Jacobian matrix of any differentiable map [...] Read more.
This article reports an experimental work that unveils some interesting yet unknown phenomena underneath all smooth nonlinear maps. The findings are based on the fact that, generalizing the conventional gradient dynamics, the right singular vectors of the Jacobian matrix of any differentiable map point in directions that are most pertinent to the infinitesimal deformation of the underlying function and that the singular values measure the rate of deformation in the corresponding directions. A continuous adaption of these singular vectors, therefore, constitutes a natural moving frame that carries indwelling information of the variation. This structure exists in any dimensional space, but the development of the fundamental theory and algorithm for surface exploration is an important first step for immediate application and further generalization. In this case, trajectories of these singular vectors, referred to as singular curves, unveil some intriguing patterns per the given function. At points where singular values coalesce, curious and complex behaviors occur, manifesting specific landmarks for the function. Upon analyzing the dynamics, it is discovered that there is a remarkably simple and universal structure underneath all smooth two-parameter maps. This work delineates graphs with this interesting dynamical system and the possible new discovery that, analogous to the double helix with two base parings in DNA, two strands of critical curves and eight base pairings could encode properties of a generic and arbitrary surface. This innate structure suggests that this approach could provide a unifying paradigm in functional genetics, where all smooth surfaces could be genome-sequenced and classified accordingly. Such a concept has sparked curiosity and warrants further investigation. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges-II)
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17 pages, 3348 KiB  
Article
Splines Parameterization of Planar Domains by Physics-Informed Neural Networks
by Antonella Falini, Giuseppe Alessio D’Inverno, Maria Lucia Sampoli and Francesca Mazzia
Mathematics 2023, 11(10), 2406; https://doi.org/10.3390/math11102406 - 22 May 2023
Cited by 3 | Viewed by 1289
Abstract
The generation of structured grids on bounded domains is a crucial issue in the development of numerical models for solving differential problems. In particular, the representation of the given computational domain through a regular parameterization allows us to define a univalent mapping, which [...] Read more.
The generation of structured grids on bounded domains is a crucial issue in the development of numerical models for solving differential problems. In particular, the representation of the given computational domain through a regular parameterization allows us to define a univalent mapping, which can be computed as the solution of an elliptic problem, equipped with suitable Dirichlet boundary conditions. In recent years, Physics-Informed Neural Networks (PINNs) have been proved to be a powerful tool to compute the solution of Partial Differential Equations (PDEs) replacing standard numerical models, based on Finite Element Methods and Finite Differences, with deep neural networks; PINNs can be used for predicting the values on simulation grids of different resolutions without the need to be retrained. In this work, we exploit the PINN model in order to solve the PDE associated to the differential problem of the parameterization on both convex and non-convex planar domains, for which the describing PDE is known. The final continuous model is then provided by applying a Hermite type quasi-interpolation operator, which can guarantee the desired smoothness of the sought parameterization. Finally, some numerical examples are presented, which show that the PINNs-based approach is robust. Indeed, the produced mapping does not exhibit folding or self-intersection at the interior of the domain and, also, for highly non convex shapes, despite few faulty points near the boundaries, has better shape-measures, e.g., lower values of the Winslow functional. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges-II)
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16 pages, 1170 KiB  
Article
A Novel Fractional-Order RothC Model
by Vsevolod Bohaienko, Fasma Diele, Carmela Marangi, Cristiano Tamborrino, Sebastian Aleksandrowicz and Edyta Woźniak
Mathematics 2023, 11(7), 1677; https://doi.org/10.3390/math11071677 - 31 Mar 2023
Viewed by 1478
Abstract
A new fractional q-order variation of the RothC model for the dynamics of soil organic carbon is introduced. A computational method based on the discretization of the analytic solution along with the finite-difference technique are suggested and the stability results for the [...] Read more.
A new fractional q-order variation of the RothC model for the dynamics of soil organic carbon is introduced. A computational method based on the discretization of the analytic solution along with the finite-difference technique are suggested and the stability results for the latter are given. The accuracy of the scheme, in terms of the temporal step size h, is confirmed through numerical testing of a constructed analytic solution. The effectiveness of the proposed discrete method is compared with that of the classical discrete RothC model. Results from real-world experiments show that, by adjusting the fractional order q and the multiplier term ζ(t,q), a better match between simulated and actual data can be achieved compared to the traditional integer-order model. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges-II)
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19 pages, 1058 KiB  
Article
A Dynamical Model with Time Delay for Risk Contagion
by Mauro Aliano, Lucianna Cananà, Greta Cestari and Stefania Ragni
Mathematics 2023, 11(2), 425; https://doi.org/10.3390/math11020425 - 13 Jan 2023
Cited by 3 | Viewed by 1677
Abstract
The explanation of risk contagion among economic players—not only in financial crises—and how they spread across the world has fascinated scholars and scientists in the last few decades. Inspired by the literature dealing with the analogy between financial systems and ecosystems, we model [...] Read more.
The explanation of risk contagion among economic players—not only in financial crises—and how they spread across the world has fascinated scholars and scientists in the last few decades. Inspired by the literature dealing with the analogy between financial systems and ecosystems, we model risk contagion by revisiting the mathematical approach of epidemiological models for infectious disease spread in a new paradigm. We propose a time delay differential system describing risk diffusion among companies inside an economic sector by means of a SIR dynamics. Contagion is modelled in terms of credit and financial risks with low and high levels. A complete theoretical analysis of the problem is carried out: well-posedness and solution positivity are proven. The existence of a risk-free steady state together with an endemic equilibrium is verified. Global asymptotic stability is investigated for both equilibria by the classical Lyapunov functional theory. The model is tested on a case study of some companies operating in the food economic sector in a specific Italian region. The analysis allows for understanding the crucial role of both incubation time and financial immunity period in the asymptotic behaviour of any solution in terms of endemic permanence of risk rather than its disappearance. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges-II)
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