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23 pages, 364 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math9172020 - 24 Aug 2021

Cited by 5 | Viewed by 1815

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

Open AccessFeature PaperArticle

A Liouville’s Formula for Systems with Reflection

by Santiago Codesido and F. Adrián F. Tojo

Mathematics 2021, 9(8), 866; https://0-doi-org.brum.beds.ac.uk/10.3390/math9080866 - 15 Apr 2021

Cited by 1 | Viewed by 1748

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

Open AccessArticle

Higher-Order Functional Discontinuous Boundary Value Problems on the Half-Line

by Feliz Minhós and Infeliz Coxe

Mathematics 2021, 9(5), 499; https://0-doi-org.brum.beds.ac.uk/10.3390/math9050499 - 01 Mar 2021

Cited by 1 | Viewed by 1206

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) t h

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) t h

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

Open AccessArticle

Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

by Gennaro Infante

Mathematics 2021, 9(4), 330; https://0-doi-org.brum.beds.ac.uk/10.3390/math9040330 - 07 Feb 2021

Cited by 5 | Viewed by 1364

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

Open AccessArticle

Permanence for Nonautonomous Differential Systems with Delays in the Linear and Nonlinear Terms

by Teresa Faria

Mathematics 2021, 9(3), 263; https://0-doi-org.brum.beds.ac.uk/10.3390/math9030263 - 28 Jan 2021

Cited by 6 | Viewed by 1499

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

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math9030214 - 21 Jan 2021

Cited by 9 | Viewed by 2186

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

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math8122249 - 20 Dec 2020

Cited by 6 | Viewed by 2291

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

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math8122241 - 18 Dec 2020

Cited by 12 | Viewed by 1676

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

Open AccessArticle

Riccati Technique and Asymptotic Behavior of Fourth-Order Advanced Differential Equations

by Omar Bazighifan and Ioannis Dassios

Mathematics 2020, 8(4), 590; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040590 - 15 Apr 2020

Cited by 30 | Viewed by 1817

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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