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

Deadline for manuscript submissions: closed (31 January 2021) | Viewed by 23369

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Dr. Stefania Tomasiello

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Institute of Computer Science, University of Tartu, Narva mnt 18, 50090 Tartu, Estonia
Interests: soft computing; machine learning; dynamical systems and control
Special Issues, Collections and Topics in MDPI journals

Dr. Carla M. A. Pinto

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Instituto Superior de Engenharia do 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, Collections and Topics in MDPI journals

Dr. Ivanka Stamova

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Guest Editor

Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA
Interests: nonlinear analysis; mathematical modeling; fractional order systems
Special Issues, Collections and Topics 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 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

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 (11 papers)

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Editorial

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3 pages, 168 KiB

Open AccessEditorial

Computational Mathematics and Neural Systems

by Stefania Tomasiello, Carla M.A. Pinto and Ivanka Stamova

Mathematics 2021, 9(7), 754; https://doi.org/10.3390/math9070754 - 01 Apr 2021

Viewed by 1369

Abstract

This special issue was conceived to explore the latest advancements in the field of computational techniques for solving forward and inverse problems [...] Full article

(This article belongs to the Special Issue Computational Mathematics and Neural Systems)

Research

Jump to: Editorial

16 pages, 3821 KiB

Open AccessArticle

Fuzzy Numerical Solution via Finite Difference Scheme of Wave Equation in Double Parametrical Fuzzy Number Form

by Maryam Almutairi, Hamzeh Zureigat, Ahmad Izani Ismail and Ali Fareed Jameel

Mathematics 2021, 9(6), 667; https://doi.org/10.3390/math9060667 - 21 Mar 2021

Cited by 13 | Viewed by 2116

Abstract

The use of fuzzy partial differential equations has become an important tool in which uncertainty or vagueness exists to model real-life problems. In this article, two numerical techniques based on finite difference schemes that are the centered time center space and implicit schemes to solve fuzzy wave equations were used. The core of the article is to formulate a new form of centered time center space and implicit schemes to obtain numerical solutions fuzzy wave equations in the double parametric fuzzy number approach. Convex normalized triangular fuzzy numbers are represented by fuzziness, based on a double parametric fuzzy number form. The properties of fuzzy set theory are used for the fuzzy analysis and formulation of the proposed numerical schemes followed by the new proof stability thermos under in the double parametric form of fuzzy numbers approach. The consistency and the convergence of the proposed scheme are discussed. Two test examples are carried out to illustrate the feasibility of the numerical schemes and the new results are displayed in the forms of tables and figures where the results show that the schemes have not only been effective for accuracy but also for reducing computational cost. Full article

(This article belongs to the Special Issue Computational Mathematics and Neural Systems)

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15 pages, 729 KiB

Open AccessArticle

An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

by Ahmad Sami Bataineh, Osman Rasit Isik, Moa’ath Oqielat and Ishak Hashim

Mathematics 2021, 9(4), 425; https://doi.org/10.3390/math9040425 - 21 Feb 2021

Cited by 7 | Viewed by 1790

Abstract

In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure. Full article

(This article belongs to the Special Issue Computational Mathematics and Neural Systems)

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26 pages, 13584 KiB

Open AccessArticle

Double Parametric Fuzzy Numbers Approximate Scheme for Solving One-Dimensional Fuzzy Heat-Like and Wave-Like Equations

by Ali Fareed Jameel, Sarmad A. Jameel Altaie, Sardar Gul Amen Aljabbari, Abbas AlZubaidi and Noraziah Haji Man

Mathematics 2020, 8(10), 1737; https://doi.org/10.3390/math8101737 - 10 Oct 2020

Cited by 6 | Viewed by 1957

Abstract

This article discusses an approximate scheme for solving one-dimensional heat-like and wave-like equations in fuzzy environment based on the homotopy perturbation method (HPM). The concept of topology in homotopy is used to create a convergent series solution of the fuzzy equations. The objective of the study is to formulate the double parametric fuzzy HPM to obtain approximate solutions of fuzzy heat-like and fuzzy wave-like equations. The fuzzification and the defuzzification analysis for the double parametric form of fuzzy numbers of the fuzzy heat-like and the fuzzy wave-like equations is carried out. The proof of convergence of the solution under the developed approximate scheme is provided. The effectiveness of the proposed method is tested by numerically solving examples of fuzzy heat-like and wave-like equations where results indicate that the approach is efficient not only in terms of accuracy but also with respect to CPU time consumption. Full article

(This article belongs to the Special Issue Computational Mathematics and Neural Systems)

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19 pages, 2715 KiB

Open AccessArticle

A Node Embedding-Based Influential Spreaders Identification Approach

by Dongming Chen, Panpan Du, Bo Fang, Dongqi Wang and Xinyu Huang

Mathematics 2020, 8(9), 1554; https://doi.org/10.3390/math8091554 - 10 Sep 2020

Cited by 8 | Viewed by 2385

Abstract

Node embedding is a representation learning technique that maps network nodes into lower-dimensional vector space. Embedding nodes into vector space can benefit network analysis tasks, such as community detection, link prediction, and influential node identification, in both calculation and richer application scope. In this paper, we propose a two-step node embedding-based solution for the social influence maximization problem (IMP). The solution employs a revised network-embedding algorithm to map input nodes into vector space in the first step. In the second step, the solution clusters the vector space nodes into subgroups and chooses the subgroups’ centers to be the influential spreaders. The proposed approach is a simple but effective IMP solution because it takes both the social reinforcement and homophily characteristics of the social network into consideration in node embedding and seed spreaders selection operation separately. The information propagation simulation experiment of single-point contact susceptible-infected-recovered (SIR) and full-contact SIR models on six different types of real network data sets proved that the proposed social influence maximization (SIM) solution exhibits significant propagation capability. Full article

(This article belongs to the Special Issue Computational Mathematics and Neural Systems)

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15 pages, 335 KiB

Open AccessArticle

A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation

by Ahmed S. Hendy and Jorge E. Macías-Díaz

Mathematics 2020, 8(9), 1539; https://doi.org/10.3390/math8091539 - 09 Sep 2020

Cited by 10 | Viewed by 2030

Abstract

In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. 16: 892–910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order

O (τ^{2 - α} + h^{4})

, in the case that

0 < α < 1

satisfies

3^{α} \geq \frac{3}{2}

, which means that

0.369 ⪅ α \leq 1

. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order

0 < α < 1

used for that scheme at

t_{k + 1 / 2}

. Full article

(This article belongs to the Special Issue Computational Mathematics and Neural Systems)

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17 pages, 3031 KiB

Open AccessArticle

RNA: A Reject Neighbors Algorithm for Influence Maximization in Complex Networks

by Dongqi Wang, Jiarui Yan, Dongming Chen, Bo Fang and Xinyu Huang

Mathematics 2020, 8(8), 1313; https://doi.org/10.3390/math8081313 - 07 Aug 2020

Cited by 6 | Viewed by 1790

Abstract

The influence maximization problem (IMP) in complex networks is to address finding a set of key nodes that play vital roles in the information diffusion process, and when these nodes are employed as ”seed nodes”, the diffusion effect is maximized. First, this paper presents a refined network centrality measure, a refined shell (RS) index for node ranking, and then proposes an algorithm for identifying key node sets, namely the reject neighbors algorithm (RNA), which consists of two main sequential parts, i.e., node ranking and node selection. The RNA refuses to select multiple-order neighbors of the seed nodes, scatters the selected nodes from each other, and results in the maximum influence of the identified node set on the whole network. Experimental results on real-world network datasets show that the key node set identified by the RNA exhibits significant propagation capability. Full article

(This article belongs to the Special Issue Computational Mathematics and Neural Systems)

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20 pages, 334 KiB

Open AccessFeature PaperArticle

Stability of Sets Criteria for Impulsive Cohen-Grossberg Delayed Neural Networks with Reaction-Diffusion Terms

by Gani Stamov, Stefania Tomasiello, Ivanka Stamova and Cvetelina Spirova

Mathematics 2020, 8(1), 27; https://doi.org/10.3390/math8010027 - 21 Dec 2019

Cited by 6 | Viewed by 1816

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 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)

31 pages, 6528 KiB

Open AccessFeature PaperArticle

Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion

by Joel Alba-Pérez and Jorge E. Macías-Díaz

Mathematics 2019, 7(12), 1172; https://doi.org/10.3390/math7121172 - 03 Dec 2019

Cited by 6 | Viewed by 2379

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 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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16 pages, 2387 KiB

Open AccessArticle

Analysis and Nonstandard Numerical Design of a Discrete Three-Dimensional Hepatitis B Epidemic Model

by Jorge E. Macías-Díaz, Nauman Ahmed and Muhammad Rafiq

Mathematics 2019, 7(12), 1157; https://doi.org/10.3390/math7121157 - 01 Dec 2019

Cited by 25 | Viewed by 2199

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 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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27 pages, 5836 KiB

Open AccessArticle

Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme

by Jorge E. Macías-Díaz

Mathematics 2019, 7(11), 1095; https://doi.org/10.3390/math7111095 - 13 Nov 2019

Cited by 2 | Viewed by 2028

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 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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