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Mathematics
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Functional Differential Equations and Applications 2020
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Functional Differential Equations and Applications 2020

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 September 2021) | Viewed by 19612

Special Issue Editors

Dr. F. Adrián F. Tojo

E-Mail Website
Guest Editor
Instituto de Matemáticas, Universidade de Santiago de Compostela, 15705 Santiago de Compostela, Spain
Interests: functional differential equations; equations with involutions
Prof. Dr. Marlène Frigon

E-Mail Website
Guest Editor
Département de Mathématiques et de Statistique, Université de Montréal, Montreal, Canada
Interests: nonlinear analysis; differential equations; variational methods; fixed points; analysis and probability

Special Issue Information

Dear Colleagues

Functional differential equations (FDEs) have attracted much attention from researchers for decades. This is due to the fact that many real-life problems present situations where the framework provided by ordinary differential equations (ODEs) and partial differential equations (PDEs) is not enough to provide an accurate model.

Whereas ODEs and PDEs are local in nature, relating the derivative of the solution at a point with some transformation of the known values at the point, FDEs convey that relevant information may lie beyond what is observable here and now. In this globalist spirit, FDEs connect the derivative of the solution at a point with the value of a functional evaluated on the solution. This scheme is, of course, too broad in general, so it is usual that the studies concerning FDEs fall under some of the following categories:

  • Differential equations with delays;
  • Differential equations with deviations;
  • Differential equations with involutions (in particular reflections);
  • Integro-differential equations;
  • Measureable differential equations;
  • Neutral differential equations;
  • Reducible differential equations.

The list is not exhaustive and the categories are not mutually exclusive, but it illustrates the richness of the field.

The purpose of this Special Issue is to gather the latest contributions, both theoretical and applied, to this expanding field, providing connections with other research topics in differential and difference equations.

Dr. F. Adrián F. Tojo
Prof. Dr. Marlène Frigon
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

  • Delay
  • Involutions
  • Topological methods
  • Existence and uniqueness
  • Green’s functions

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Published Papers (9 papers)

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Research

23 pages, 364 KiB  
Article
On a Boundary Value Problem for the Biharmonic Equation with Multiple Involutions
by Batirkhan Turmetov, Valery Karachik and Moldir Muratbekova
Mathematics 2021, 9(17), 2020; https://doi.org/10.3390/math9172020 - 24 Aug 2021
Cited by 7 | Viewed by 2246
Abstract
A nonlocal analogue of the biharmonic operator with involution-type transformations was considered. For the corresponding biharmonic equation with involution, we investigated the solvability of boundary value problems with a fractional-order boundary operator having a derivative of the Hadamard-type. First, transformations of the involution [...] Read more.
A nonlocal analogue of the biharmonic operator with involution-type transformations was considered. For the corresponding biharmonic equation with involution, we investigated the solvability of boundary value problems with a fractional-order boundary operator having a derivative of the Hadamard-type. First, transformations of the involution type were considered. The properties of the matrices of these transformations were investigated. As applications of the considered transformations, the questions about the solvability of a boundary value problem for a nonlocal biharmonic equation were studied. Modified Hadamard derivatives were considered as the boundary operator. The considered problems covered the Dirichlet and Neumann-type boundary conditions. Theorems on the existence and uniqueness of solutions to the studied problems were proven. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications 2020)
7 pages, 241 KiB  
Article
A Liouville’s Formula for Systems with Reflection
by Santiago Codesido and F. Adrián F. Tojo
Mathematics 2021, 9(8), 866; https://doi.org/10.3390/math9080866 - 15 Apr 2021
Cited by 1 | Viewed by 2075
Abstract
In this work, we derived an Abel–Jacobi–Liouville identity for the case of two-dimensional linear systems of ODEs (ordinary differential equations) with reflection. We also present a conjecture for the general case and an application to coupled harmonic oscillators. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications 2020)
16 pages, 309 KiB  
Article
Higher-Order Functional Discontinuous Boundary Value Problems on the Half-Line
by Feliz Minhós and Infeliz Coxe
Mathematics 2021, 9(5), 499; https://doi.org/10.3390/math9050499 - 1 Mar 2021
Cited by 2 | Viewed by 1438
Abstract
In this paper, we consider a discontinuous, fully nonlinear, higher-order equation on the half-line, together with functional boundary conditions, given by general continuous functions with dependence on the several derivatives and asymptotic information on the (n−1)th derivative [...] Read more.
In this paper, we consider a discontinuous, fully nonlinear, higher-order equation on the half-line, together with functional boundary conditions, given by general continuous functions with dependence on the several derivatives and asymptotic information on the (n−1)th derivative of the unknown function. These functional conditions generalize the usual boundary data and allow other types of global assumptions on the unknown function and its derivatives, such as nonlocal, integro-differential, infinite multipoint, with maximum or minimum arguments, among others. Considering the half-line as the domain carries on a lack of compactness, which is overcome with the definition of a space of weighted functions and norms, and the equiconvergence at ∞. In the last section, an example illustrates the applicability of our main result. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications 2020)
9 pages, 257 KiB  
Article
Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms
by Gennaro Infante
Mathematics 2021, 9(4), 330; https://doi.org/10.3390/math9040330 - 7 Feb 2021
Cited by 6 | Viewed by 1666
Abstract
We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The [...] Read more.
We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications 2020)
20 pages, 398 KiB  
Article
Permanence for Nonautonomous Differential Systems with Delays in the Linear and Nonlinear Terms
by Teresa Faria
Mathematics 2021, 9(3), 263; https://doi.org/10.3390/math9030263 - 28 Jan 2021
Cited by 6 | Viewed by 1771
Abstract
In this paper, we obtain sufficient conditions for the persistence and permanence of a family of nonautonomous systems of delay differential equations. This family includes structured models from mathematical biology, with either discrete or distributed delays in both the linear and nonlinear terms, [...] Read more.
In this paper, we obtain sufficient conditions for the persistence and permanence of a family of nonautonomous systems of delay differential equations. This family includes structured models from mathematical biology, with either discrete or distributed delays in both the linear and nonlinear terms, and where typically the nonlinear terms are nonmonotone. Applications to systems inspired by mathematical biology models are given. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications 2020)
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15 pages, 398 KiB  
Article
The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative
by Sivaporn Ampun and Panumart Sawangtong
Mathematics 2021, 9(3), 214; https://doi.org/10.3390/math9030214 - 21 Jan 2021
Cited by 11 | Viewed by 2574
Abstract
In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral [...] Read more.
In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications 2020)
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15 pages, 937 KiB  
Article
Asymptotics of Solutions of Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely Distant Point
by Maria Korovina
Mathematics 2020, 8(12), 2249; https://doi.org/10.3390/math8122249 - 20 Dec 2020
Cited by 8 | Viewed by 2570
Abstract
This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an [...] Read more.
This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications 2020)
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32 pages, 427 KiB  
Article
Asymptotics and Uniqueness of Solutions of the Elasticity System with the Mixed Dirichlet–Robin Boundary Conditions
by Hovik A. Matevossian
Mathematics 2020, 8(12), 2241; https://doi.org/10.3390/math8122241 - 18 Dec 2020
Cited by 14 | Viewed by 2004
Abstract
We study properties of generalized solutions of the Dirichlet–Robin problem for an elasticity system in the exterior of a compact, as well as the asymptotic behavior of solutions of this mixed problem at infinity, with the condition that the energy integral with the [...] Read more.
We study properties of generalized solutions of the Dirichlet–Robin problem for an elasticity system in the exterior of a compact, as well as the asymptotic behavior of solutions of this mixed problem at infinity, with the condition that the energy integral with the weight |x|a is finite. Depending on the value of the parameter a, we have proved uniqueness (or non-uniqueness) theorems for the mixed Dirichlet–Robin problem, and also given exact formulas for the dimension of the space of solutions. The main method for studying the problem under consideration is the variational principle, which assumes the minimization of the corresponding functional in the class of admissible functions. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications 2020)
11 pages, 736 KiB  
Article
Riccati Technique and Asymptotic Behavior of Fourth-Order Advanced Differential Equations
by Omar Bazighifan and Ioannis Dassios
Mathematics 2020, 8(4), 590; https://doi.org/10.3390/math8040590 - 15 Apr 2020
Cited by 31 | Viewed by 2157
Abstract
In this paper, we deal with the oscillation of fourth-order nonlinear advanced differential equations of the form r t y ‴ t α ′ + p t f y ‴ t + q t g y σ t = 0 . We provide [...] Read more.
In this paper, we deal with the oscillation of fourth-order nonlinear advanced differential equations of the form r t y ‴ t α ′ + p t f y ‴ t + q t g y σ t = 0 . We provide oscillation criteria for this type of equations, and examples to illustrate the criteria. Full article
(This article belongs to the Special Issue Functional Differential Equations and Applications 2020)
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