Special Issue "Mathematical and Numerical Analysis of Nonlinear Evolution Equations : Advances and Perspectives"

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 November 2019).

Special Issue Editor

Prof. Carlo Bianca
E-Mail Website
Guest Editor
Laboratoire Quartz, ECAM-EPMI, 13 Boulevard de l’Hautil 95092 Cergy Pontoise, France
Interests: mathematical modeling and analysis of complex systems, kinetic equations, numerical methods for PDE

Special Issue Information

Dear Colleagues,

Recently, interactions between researchers working in the field of mathematical physics and in the field of applied sciences have gained much attention, and new challenges have been raised including the possibility to derive evolution differential equations that are able to describe most phenomena arising in real-world systems. On the one hand, mathematical analysis allows one to obtain information on the qualitative behaviors of the system including the existence of solutions, asymptotic behaviors, and nonlinear dynamics. On the other hand, numerical and computational analysis furnishes methods to obtain quantitative information about  solutions and the possibility to compare the time evolution of solutions to differential equations with empirical data.

This Special Issue is devoted to researchers working in the fields of pure and applied mathematical physics, specifically to researchers who are involved in the mathematical and numerical analysis of nonlinear evolution equations and their applications. Original research articles as well as review articles are welcome.

The topics include, but are not limited to, the following:

  • Prey–predator models;
  • Kinetic-type models;
  • Multiscale models;
  • Computational models;
  • Fractional models;
  • Asymptotic analysis and methods;
  • Approximative methods;
  • Bifurcation analysis;
  • Chaos and synchronization analysis;
  • Nonlinear dynamics;
  • Complex dynamics;
  • Far-from-equilibrium dynamics;
  • Blow-up of solutions;
  • Fractional calculus.

Prof. Carlo Bianca 
Guest Editor

Manuscript Submission Information

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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.

Published Papers (5 papers)

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Research

Open AccessArticle
Mathematical Analysis of an Autoimmune Diseases Model: Kinetic Approach
Mathematics 2019, 7(11), 1024; https://doi.org/10.3390/math7111024 - 30 Oct 2019
Abstract
A new mathematical model of a general autoimmune disease is presented. Basic information about autoimmune diseases is given and illustrated with examples. The model is developed by using ideas from the kinetic theory describing individuals expressing certain functions. The modeled problem is formulated [...] Read more.
A new mathematical model of a general autoimmune disease is presented. Basic information about autoimmune diseases is given and illustrated with examples. The model is developed by using ideas from the kinetic theory describing individuals expressing certain functions. The modeled problem is formulated by ordinary and partial equations involving a variable for a functional state. Numerical results are presented and discussed from a medical view point. Full article
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Open AccessArticle
Exact Solutions for a Modified Schrödinger Equation
Mathematics 2019, 7(10), 908; https://doi.org/10.3390/math7100908 - 29 Sep 2019
Abstract
The aim of this paper was to propose a systematic study of a ( 1 + 1 ) -dimensional higher order nonlinear Schrödinger equation, arising in two different contexts regarding the biological science and the nonlinear optics. We performed a Lie symmetry analysis [...] Read more.
The aim of this paper was to propose a systematic study of a ( 1 + 1 ) -dimensional higher order nonlinear Schrödinger equation, arising in two different contexts regarding the biological science and the nonlinear optics. We performed a Lie symmetry analysis and here present exact solutions of the equation. Full article
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Open AccessArticle
A Convergence Theorem for the Nonequilibrium States in the Discrete Thermostatted Kinetic Theory
Mathematics 2019, 7(8), 673; https://doi.org/10.3390/math7080673 - 28 Jul 2019
Abstract
The existence and reaching of nonequilibrium stationary states are important issues that need to be taken into account in the development of mathematical modeling frameworks for far off equilibrium complex systems. The main result of this paper is the rigorous proof that the [...] Read more.
The existence and reaching of nonequilibrium stationary states are important issues that need to be taken into account in the development of mathematical modeling frameworks for far off equilibrium complex systems. The main result of this paper is the rigorous proof that the solution of the discrete thermostatted kinetic model catches the stationary solutions as time goes to infinity. The approach towards nonequilibrium stationary states is ensured by the presence of a dissipative term (thermostat) that counterbalances the action of an external force field. The main result is obtained by employing the Discrete Fourier Transform (DFT). Full article
Open AccessArticle
Dependence on the Initial Data for the Continuous Thermostatted Framework
Mathematics 2019, 7(7), 602; https://doi.org/10.3390/math7070602 - 06 Jul 2019
Abstract
The paper deals with the problem of continuous dependence on initial data of solutions to the equation describing the evolution of a complex system in the presence of an external force acting on the system and of a thermostat, simply identified with the [...] Read more.
The paper deals with the problem of continuous dependence on initial data of solutions to the equation describing the evolution of a complex system in the presence of an external force acting on the system and of a thermostat, simply identified with the condition that the second order moment of the activity variable (see Section 1) is a constant. We are able to prove that these solutions are stable with respect to the initial conditions in the Hadamard’s sense. In this connection, two remarks spontaneously arise and must be carefully considered: first, one could complain the lack of information about the “distance” between solutions at any time t [ 0 , + ) ; next, one cannot expect any more complete information without taking into account the possible distribution of the transition probabiliy densities and the interaction rates (see Section 1 again). This work must be viewed as a first step of a research which will require many more steps to give a sufficiently complete picture of the relations between solutions (see Section 5). Full article
Open AccessArticle
A Delayed Epidemic Model for Propagation of Malicious Codes in Wireless Sensor Network
Mathematics 2019, 7(5), 396; https://doi.org/10.3390/math7050396 - 01 May 2019
Cited by 1
Abstract
In this paper, we investigate a delayed SEIQRS-V epidemic model for propagation of malicious codes in a wireless sensor network. The communication radius and distributed density of nodes is considered in the proposed model. With this model, first we find a feasible region [...] Read more.
In this paper, we investigate a delayed SEIQRS-V epidemic model for propagation of malicious codes in a wireless sensor network. The communication radius and distributed density of nodes is considered in the proposed model. With this model, first we find a feasible region which is invariant and where the solutions of our model are positive. To show that the system is locally asymptotically stable, a Lyapunov function is constructed. After that, sufficient conditions for local stability and existence of Hopf bifurcation are derived by analyzing the distribution of the roots of the corresponding characteristic equation. Finally, numerical simulations are presented to verify the obtained theoretical results and to analyze the effects of some parameters on the dynamical behavior of the proposed model in the paper. Full article
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