Special Issue "New Trends in Differential and Difference Equations and Applications"

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (31 January 2019)

Special Issue Editors

Guest Editor
Prof. Dr. Feliz Manuel Minhós

Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação Avançada, Universidade de Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
Website | E-Mail
Interests: differential and difference equations; boundary value problems; topological and variational methods
Guest Editor
Prof. Dr. João Fialho

British University Vietnam, CDC Building, 25-27 Le Dai Hanh, Le Dai Hanh Ward, Hai Ba Trung District, 10000 Ha Noi, Vietnam
Website | E-Mail
Interests: differential and difference equations; boundary value problems; topological and variational methods, mathematical modelling

Special Issue Information

Dear Colleagues,

Differential and difference equations, their methods, techniques and the huge variety of applications have had an increasing interest in different fields of science, in the last few years. Not only their solvability and the study of qualitative properties have been the aim of many research papers, but, also, their role on different types of boundary value problems have allowed the study of a large number of real world phenomena.

This Special Issue will accept high-quality articles containing original research results and survey articles of exceptional merit on new developments in the following topics:

Ordinary and Partial Differential Equations

Difference equations

Integral and integrodifferential equations

Boundary value problems

Fractional Calculus and Applications

Calculus on Time Scales and Applications

Existence, uniqueness and multiplicity results by variational and/or topological methods

Qualitative, asymptotic and oscillation properties.

Fixed point theory

Applications to real world phenomena.

Before submission authors should carefully read over the journal's instructions for Authors, in https://www.mdpi.com/journal/axioms/instructions.

Prof. Dr. Feliz Manuel Minhós
Prof. Dr. João Fialho
Guest Editors

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. Axioms is an international peer-reviewed open access quarterly 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 350 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 and Difference equations
  • Integral equations
  • Boundary Value Problems
  • Variational and topological methods
  • Qualitative theory
  • Mathematical modelling

Published Papers (9 papers)

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Research

Open AccessArticle
On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices
Received: 31 January 2019 / Revised: 26 March 2019 / Accepted: 10 April 2019 / Published: 21 April 2019
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Abstract
In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, [...] Read more.
In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or q-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the “left” inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution—non polynomial solution—of a second-order divided-difference equation of hypergeometric type. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle
Bäcklund Transformations for Nonlinear Differential Equations and Systems
Received: 5 February 2019 / Revised: 4 April 2019 / Accepted: 7 April 2019 / Published: 11 April 2019
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Abstract
In this work, new Bäcklund transformations (BTs) for generalized Liouville equations were obtained. Special cases of Liouville equations with exponential nonlinearity that have a multiplier that depends on the independent variables and first-order derivatives from the function were considered. Two- and three-dimensional cases [...] Read more.
In this work, new Bäcklund transformations (BTs) for generalized Liouville equations were obtained. Special cases of Liouville equations with exponential nonlinearity that have a multiplier that depends on the independent variables and first-order derivatives from the function were considered. Two- and three-dimensional cases were considered. The BTs construction is based on the method proposed by Clairin. The solutions of the considered equations have been found using the BTs, with a unified algorithm. In addition, the work develops the Clairin’s method for the system of two third-order equations related to the integrable perturbation and complexification of the Korteweg-de Vries (KdV) equation. Among the constructed BTs an analog of the Miura transformations was found. The Miura transformations transfer the initial system to that of perturbed modified KdV (mKdV) equations. It could be shown on this way that, considering the system as a link between the real and imaginary parts of a complex function, it is possible to go to the complexified KdV (cKdV) and here the analog of the Miura transformations transforms it into the complexification of the mKdV. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle
The 3D Navier–Stokes Equations: Invariants, Local and Global Solutions
Received: 31 January 2019 / Revised: 7 March 2019 / Accepted: 1 April 2019 / Published: 7 April 2019
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Abstract
In this article, I consider local solutions of the 3D Navier–Stokes equations and its properties such as an existence of global and smooth solution, uniform boundedness. The basic role is assigned to a special invariant class of solenoidal vector fields and three parameters [...] Read more.
In this article, I consider local solutions of the 3D Navier–Stokes equations and its properties such as an existence of global and smooth solution, uniform boundedness. The basic role is assigned to a special invariant class of solenoidal vector fields and three parameters that are invariant with respect to the scaling procedure. Since in spaces of even dimensions the scaling procedure is a conformal mapping on the Heisenberg group, then an application of invariant parameters can be considered as the application of conformal invariants. It gives the possibility to prove the sufficient and necessary conditions for existence of a global regular solution. This is the main result and one among some new statements. With some compliments, the rest improves well-known classical results. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle
First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model
Received: 6 January 2019 / Revised: 13 February 2019 / Accepted: 13 February 2019 / Published: 16 February 2019
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Abstract
The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. [...] Read more.
The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arzèla Ascoli theorem and Schauder’s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
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Open AccessArticle
Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations
Received: 28 December 2018 / Revised: 28 January 2019 / Accepted: 11 February 2019 / Published: 15 February 2019
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Abstract
In this paper, we consider the second order discontinuous differential equation in the real line, at,uϕu=ft,u,u,a.e.tR,u( [...] Read more.
In this paper, we consider the second order discontinuous differential equation in the real line, a t , u ϕ u = f t , u , u , a . e . t R , u ( ) = ν , u ( + ) = ν + , with ϕ an increasing homeomorphism such that ϕ ( 0 ) = 0 and ϕ ( R ) = R , a C ( R 2 , R ) with a ( t , x ) > 0 for ( t , x ) R 2 , f : R 3 R a L 1 -Carathéodory function and ν , ν + R such that ν < ν + . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities ϕ and f . To the best of our knowledge, this result is even new when ϕ ( y ) = y , that is for equation a t , u ( t ) u ( t ) = f t , u ( t ) , u ( t ) , a . e . t R . Moreover, these results can be applied to classical and singular ϕ -Laplacian equations and to the mean curvature operator. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle
Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations
Received: 15 January 2019 / Revised: 1 February 2019 / Accepted: 2 February 2019 / Published: 6 February 2019
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Abstract
The paper is devoted to the discrete Lyapunov equation XA*XA=C, where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that [...] Read more.
The paper is devoted to the discrete Lyapunov equation X A * X A = C , where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle
Note on Limit-Periodic Solutions of the Difference Equation xt + 1 − [h(xt) + λ]xt = rt, λ > 1
Received: 7 January 2019 / Revised: 31 January 2019 / Accepted: 1 February 2019 / Published: 5 February 2019
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Abstract
As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar [...] Read more.
As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle
Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays
Received: 21 November 2018 / Revised: 24 December 2018 / Accepted: 25 December 2018 / Published: 29 December 2018
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Abstract
In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. [...] Read more.
In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We consider the case of impulses that start abruptly at some points and their actions continue on given finite intervals. The study of Lipschitz stability by Lyapunov functions requires appropriate derivatives among fractional differential equations. A brief overview of different types of derivative known in the literature is given. Some sufficient conditions for uniform Lipschitz stability and uniform global Lipschitz stability are obtained by an application of several types of derivatives of Lyapunov functions. Examples are given to illustrate the results. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
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Open AccessArticle
A New Efficient Method for the Numerical Solution of Linear Time-Dependent Partial Differential Equations
Received: 5 August 2018 / Revised: 21 September 2018 / Accepted: 28 September 2018 / Published: 1 October 2018
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Abstract
This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order [...] Read more.
This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order accurate in time. A comparative study between the proposed method and the one-step wavelet collocation method is provided. In order to verify the stability of these methods, asymptotic stability analysis is employed. Numerical illustrations are investigated to show the reliability and efficiency of the proposed method. An important property of the presented method is that unlike the one-step wavelet collocation method, it is not necessary to choose a small time step to achieve stability. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
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