Special Issue "Computational Mathematics and Neural Systems"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational Mathematics".

Deadline for manuscript submissions: 31 January 2021.

Special Issue Editors

Dr. Stefania Tomasiello

Guest Editor
1. DISA-MIS, Università degli Studi di Salerno, via Giovanni Paolo II, 132, Fisciano 84084, Italy
2. Institute of Computer Science, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia (from August 2019)
Interests: scientific and soft computing; fuzzy mathematics; machine learning
Dr. Carla M.A. Pinto

Guest Editor
School of Engineering, Polytechnic of Porto, Rua Dr António Bernardino de Almeida, 431, 4249-015 Porto, Portugal
Interests: epidemiological and within-host models; dynamical systems; fractional order models
Special Issues and Collections in MDPI journals
Dr. Ivanka Stamova
Website
Guest Editor
Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA
Interests: nonlinear analysis; mathematical modeling; fractional order systems
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

The Special Issue, Computational Mathematics and Neural Systems, of MDPI is devoted to exploring the latest advancements in the field of computational techniques for solving forward and inverse problems. The real world offers numerous application problems of interest in several fields. Mathematical modelling is the first step to handle real-world problems. Translating application problems into suitable mathematical formulations entails some issues. The derived models are often complex, and they require numerical techniques able to provide the originating application with insightful answers, especially in the presence of uncertainty. Another important aspect that cannot be neglected nowadays is the explosive growth of available data. This has motivated research on artificial neural systems, in several versions, over the last decades. Such kinds of computing systems seem to be able to address the modelling and numerical issues, at least to some extent, in some contexts.

This Special Issue aims at bringing together the above-mentioned areas of investigation from a broader perspective. Topics of interest include, but are not restricted to fractional systems, uncertain systems, biological neural network modelling, computational learning theory, and analysis of network dynamics applications.

Dr. Stefania Tomasiello
Dr. Carla M.A. Pinto
Dr. Ivanka Stamova
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 papers will be 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 monthly 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 1200 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

  • Analysis of network dynamics
  • Neural networks
  • Uncertain systems
  • Computational learning theory
  • Fractional differential equations
  • Complex systems
  • Dynamical systems
  • Delayed systems
  • Computational methods for systems of fractional order
  • Applications in bioengineering medicine, mechanical, engineering, finances economics, biology, epidemiology, mathematical physics, and so on.

Published Papers (4 papers)

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Research

Open AccessFeature PaperArticle
Stability of Sets Criteria for Impulsive Cohen-Grossberg Delayed Neural Networks with Reaction-Diffusion Terms
Mathematics 2020, 8(1), 27; https://doi.org/10.3390/math8010027 - 21 Dec 2019
Abstract
The paper proposes an extension of stability analysis methods for a class of impulsive reaction-diffusion Cohen-Grossberg delayed neural networks by addressing a challenge namely stability of sets. Such extended concept is of considerable interest to numerous systems capable of approaching not only one [...] Read more.
The paper proposes an extension of stability analysis methods for a class of impulsive reaction-diffusion Cohen-Grossberg delayed neural networks by addressing a challenge namely stability of sets. Such extended concept is of considerable interest to numerous systems capable of approaching not only one equilibrium state. Results on uniform global asymptotic stability and uniform global exponential stability with respect to sets for the model under consideration are established. The main tools are expansions of the Lyapunov method and the comparison principle. In addition, the obtained results for the uncertain case contributed to the development of the stability theory of uncertain reaction-diffusion Cohen-Grossberg delayed neural networks and their applications. Moreover, examples are given to demonstrate the feasibility of our results. Full article
(This article belongs to the Special Issue Computational Mathematics and Neural Systems)
Open AccessFeature PaperArticle
Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion
Mathematics 2019, 7(12), 1172; https://doi.org/10.3390/math7121172 - 03 Dec 2019
Cited by 1
Abstract
In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis–Menten-type reactions. For the sake of generality, we consider an extended [...] Read more.
In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis–Menten-type reactions. For the sake of generality, we consider an extended form of that system in various spatial dimensions and propose two finite-difference methods to approximate its solutions. Both methodologies are presented in alternative forms to facilitate their analyses and computer implementations. We show that both schemes are structure-preserving techniques, in the sense that they can keep the positive and bounded character of the computational approximations. This is in agreement with the relevant solutions of the original population model. Moreover, we prove rigorously that the schemes are consistent discretizations of the generalized continuous model and that they are stable and convergent. The methodologies were implemented efficiently using MATLAB. Some computer simulations are provided for illustration purposes. In particular, we use our schemes in the investigation of complex patterns in some two- and three-dimensional predator–prey systems with anomalous diffusion. Full article
(This article belongs to the Special Issue Computational Mathematics and Neural Systems)
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Open AccessArticle
Analysis and Nonstandard Numerical Design of a Discrete Three-Dimensional Hepatitis B Epidemic Model
Mathematics 2019, 7(12), 1157; https://doi.org/10.3390/math7121157 - 01 Dec 2019
Cited by 2
Abstract
In this work, we numerically investigate a three-dimensional nonlinear reaction-diffusion susceptible-infected-recovered hepatitis B epidemic model. To that end, the stability and bifurcation analyses of the mathematical model are rigorously discussed using the Routh–Hurwitz condition. Numerically, an efficient structure-preserving nonstandard finite-difference time-splitting method is [...] Read more.
In this work, we numerically investigate a three-dimensional nonlinear reaction-diffusion susceptible-infected-recovered hepatitis B epidemic model. To that end, the stability and bifurcation analyses of the mathematical model are rigorously discussed using the Routh–Hurwitz condition. Numerically, an efficient structure-preserving nonstandard finite-difference time-splitting method is proposed to approximate the solutions of the hepatitis B model. The dynamical consistency of the splitting method is verified mathematically and graphically. Moreover, we perform a mathematical study of the stability of the proposed scheme. The properties of consistency, stability and convergence of our technique are thoroughly analyzed in this work. Some comparisons are provided against existing standard techniques in order to validate the efficacy of our scheme. Our computational results show a superior performance of the present approach when compared against existing methods available in the literature. Full article
(This article belongs to the Special Issue Computational Mathematics and Neural Systems)
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Open AccessArticle
Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme
Mathematics 2019, 7(11), 1095; https://doi.org/10.3390/math7111095 - 13 Nov 2019
Abstract
In this work, we investigate numerically a one-dimensional wave equation in generalized form. The system considers the presence of constant damping and functional anomalous diffusion of the Riesz type. Reaction terms are also considered, in such way that the mathematical model can be [...] Read more.
In this work, we investigate numerically a one-dimensional wave equation in generalized form. The system considers the presence of constant damping and functional anomalous diffusion of the Riesz type. Reaction terms are also considered, in such way that the mathematical model can be presented in variational form when damping is not present. As opposed to previous efforts available in the literature, the reaction terms are not only functions of the solution. Instead, we consider the presence of smooth functions that depend on fractional derivatives of the solution function. Using a finite-difference approach, we propose a numerical scheme to approximate the solutions of the fractional wave equation. Along with this integrator, we propose discrete forms of the local and the total energy operators. In a first stage, we show rigorously that the energy properties of the continuous system are mimicked by our discrete methodology. In particular, we prove that the discrete system is dissipative (respectively, conservative) when damping is present (respectively, absent), in agreement with the continuous model. The theoretical numerical analysis of this system is more complicated in light of the presence of the functional form of the anomalous diffusion. To solve this problem, some novel technical lemmas are proved and used to establish the stability and the quadratic convergence of the scheme. Finally, we provide some computer simulations to show the capability of the scheme to conserve/dissipate the energy. Various fractional problems with functional forms of the anomalous diffusion of the solution are considered to that effect. Full article
(This article belongs to the Special Issue Computational Mathematics and Neural Systems)
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