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 (31 March 2020).

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. 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
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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

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Published Papers (11 papers)

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Research

Article
Analysis of a Model for Coronavirus Spread
Mathematics 2020, 8(5), 820; https://doi.org/10.3390/math8050820 - 19 May 2020
Cited by 4 | Viewed by 1436
Abstract
The spread of epidemics has always threatened humanity. In the present circumstance of the Coronavirus pandemic, a mathematical model is considered. It is formulated via a compartmental dynamical system. Its equilibria are investigated for local stability. Global stability is established for the disease-free [...] Read more.
The spread of epidemics has always threatened humanity. In the present circumstance of the Coronavirus pandemic, a mathematical model is considered. It is formulated via a compartmental dynamical system. Its equilibria are investigated for local stability. Global stability is established for the disease-free point. The allowed steady states are an unlikely symptomatic-infected-free point, which must still be considered endemic due to the presence of asymptomatic individuals; and the disease-free and the full endemic equilibria. A transcritical bifurcation is shown to exist among them, preventing bistability. The disease basic reproduction number is calculated. Simulations show that contact restrictive measures are able to delay the epidemic’s outbreak, if taken at a very early stage. However, if lifted too early, they could become ineffective. In particular, an intermittent lock-down policy could be implemented, with the advantage of spreading the epidemics over a longer timespan, thereby reducing the sudden burden on hospitals. Full article
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Article
Fixed-Point Results for a Generalized Almost (s, q)—Jaggi F-Contraction-Type on b—Metric-Like Spaces
Mathematics 2020, 8(1), 63; https://doi.org/10.3390/math8010063 - 02 Jan 2020
Cited by 2 | Viewed by 782
Abstract
The purpose of this article is to present a new generalized almost ( s , q ) Jaggi F contraction-type and a generalized almost ( s , q ) Jaggi F Suzuki contraction-type and some results in related fixed point on it in the context of b metric-like spaces are discussed. Also, we support our theoretical results with non-trivial examples. Finally, applications to find a solution for the electric circuit equation and second-order differential equations are presented and an strong example is given here to support the first application. Full article
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Article
Mathematical Analysis of a Thermostatted Equation with a Discrete Real Activity Variable
Mathematics 2020, 8(1), 57; https://doi.org/10.3390/math8010057 - 02 Jan 2020
Cited by 2 | Viewed by 508
Abstract
This paper deals with the mathematical analysis of a thermostatted kinetic theory equation. Specifically, the assumption on the domain of the activity variable is relaxed allowing for the discrete activity to attain real values. The existence and uniqueness of the solution of the [...] Read more.
This paper deals with the mathematical analysis of a thermostatted kinetic theory equation. Specifically, the assumption on the domain of the activity variable is relaxed allowing for the discrete activity to attain real values. The existence and uniqueness of the solution of the related Cauchy problem and of the related non-equilibrium stationary state are established, generalizing the existing results. Full article
Article
Fractional Cauchy Problems for Infinite Interval Case-II
Mathematics 2019, 7(12), 1165; https://doi.org/10.3390/math7121165 - 02 Dec 2019
Viewed by 601
Abstract
We consider fractional abstract Cauchy problems on infinite intervals. A fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the [...] Read more.
We consider fractional abstract Cauchy problems on infinite intervals. A fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Required conditions on spaces are also given, guaranteeing the existence and uniqueness of solutions. The fractional powers of the involved operator B X have been investigated in the space which consists of continuous functions u on [ 0 , ) without assuming u ( 0 ) = 0 . This enables us to refine some previous results and obtain the required abstract results when the operator B X is not necessarily densely defined. Full article
Article
Mathematical Analysis of an Autoimmune Diseases Model: Kinetic Approach
Mathematics 2019, 7(11), 1024; https://doi.org/10.3390/math7111024 - 30 Oct 2019
Cited by 5 | Viewed by 744
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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Article
Direct and Inverse Fractional Abstract Cauchy Problems
Mathematics 2019, 7(11), 1016; https://doi.org/10.3390/math7111016 - 25 Oct 2019
Cited by 2 | Viewed by 675
Abstract
We are concerned with a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Moreover, we succeeded in handling [...] Read more.
We are concerned with a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Moreover, we succeeded in handling related inverse problems, extending the treatment given by Alfredo Lorenzi. Some basic assumptions on the involved operators are also introduced allowing application of the real interpolation theory of Lions and Peetre. Our abstract approach improves previous results given by Favini–Yagi by using more general real interpolation spaces with indices θ , p, p ( 0 , ] instead of the indices θ , . As a possible application of the abstract theorems, some examples of partial differential equations are given. Full article
Article
Exact Solutions for a Modified Schrödinger Equation
Mathematics 2019, 7(10), 908; https://doi.org/10.3390/math7100908 - 29 Sep 2019
Cited by 14 | Viewed by 928
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 and here present exact solutions of the equation. Full article
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Article
A Distributed Control Problem for a Fractional Tumor Growth Model
Mathematics 2019, 7(9), 792; https://doi.org/10.3390/math7090792 - 31 Aug 2019
Cited by 8 | Viewed by 808
Abstract
In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn–Hilliard type phase field system modeling tumor [...] Read more.
In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn–Hilliard type phase field system modeling tumor growth that has been proposed by Hawkins–Daarud, van der Zee and Oden. The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in a recent work by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type. Full article
Article
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
Cited by 1 | Viewed by 1183
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
Article
Dependence on the Initial Data for the Continuous Thermostatted Framework
Mathematics 2019, 7(7), 602; https://doi.org/10.3390/math7070602 - 06 Jul 2019
Cited by 3 | Viewed by 632
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
Article
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 10 | Viewed by 794
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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