Mathematics

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13 pages, 298 KiB

Open AccessArticle

Mixed Hilfer and Caputo Fractional Riemann–Stieltjes Integro-Differential Equations with Non-Separated Boundary Conditions

by Ayub Samadi, Sotiris K. Ntouyas and Jessada Tariboon

Mathematics 2024, 12(9), 1361; https://doi.org/10.3390/math12091361 - 30 Apr 2024

Cited by 1 | Viewed by 953

Abstract

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach’s fixed point theorem, while by applying Leray–Schauder’s [...] Read more.

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach’s fixed point theorem, while by applying Leray–Schauder’s nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

16 pages, 320 KiB

Open AccessArticle

Analysis on Controllability Results for Impulsive Neutral Hilfer Fractional Differential Equations with Nonlocal Conditions

by Thitiporn Linitda, Kulandhaivel Karthikeyan, Palanisamy Raja Sekar and Thanin Sitthiwirattham

Mathematics 2023, 11(5), 1071; https://doi.org/10.3390/math11051071 - 21 Feb 2023

Cited by 6 | Viewed by 1552

Abstract

In this paper, we investigate the controllability of the system with non-local conditions. The existence of a mild solution is established. We obtain the results by using resolvent operators functions, the Hausdorff measure of non-compactness, and Monch’s fixed point theorem. We also present [...] Read more.

In this paper, we investigate the controllability of the system with non-local conditions. The existence of a mild solution is established. We obtain the results by using resolvent operators functions, the Hausdorff measure of non-compactness, and Monch’s fixed point theorem. We also present an example, in order to elucidate one of the results discussed. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

11 pages, 2933 KiB

Open AccessArticle

Self-Similar Solutions of a Gravitating Dark Fluid

by Imre Ferenc Barna, Mihály András Pocsai and Gergely Gábor Barnaföldi

Mathematics 2022, 10(18), 3220; https://doi.org/10.3390/math10183220 - 6 Sep 2022

Cited by 2 | Viewed by 1470

Abstract

In this paper, a fluid model is presented which contains the general linear equation of state including the gravitation term. The obtained spherical symmetric Euler equation and the continuity equations were investigated with the Sedov-type time-dependent self-similar ansatz which is capable of describing [...] Read more.

In this paper, a fluid model is presented which contains the general linear equation of state including the gravitation term. The obtained spherical symmetric Euler equation and the continuity equations were investigated with the Sedov-type time-dependent self-similar ansatz which is capable of describing physically relevant diffusive and disperse solutions. The result of the space and time-dependent fluid density and radial velocity fields are presented and analyzed. Additionally, the role of the initial velocity on the kinetic and total energy densities of the fluid is discussed. This leads to a model, which can be considered as a simple model for a dark-fluid. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

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16 pages, 719 KiB

Open AccessArticle

Time-Dependent Analytic Solutions for Water Waves above Sea of Varying Depths

by Imre Ferenc Barna, Mihály András Pocsai and László Mátyás

Mathematics 2022, 10(13), 2311; https://doi.org/10.3390/math10132311 - 1 Jul 2022

Cited by 2 | Viewed by 1404

Abstract

We investigate a hydrodynamic equation system which—with some approximation—is capable of describing the tsunami propagation in the open ocean with the time-dependent self-similar Ansatz. We found analytic solutions of how the wave height and velocity behave in time and space for constant and [...] Read more.

We investigate a hydrodynamic equation system which—with some approximation—is capable of describing the tsunami propagation in the open ocean with the time-dependent self-similar Ansatz. We found analytic solutions of how the wave height and velocity behave in time and space for constant and linear seabed functions. First, we study waves on open water, where the seabed can be considered relatively constant, sufficiently far from the shore. We found original shape functions for the ocean waves. In the second part of the study, we also consider a seabed which is oblique. Most of the solutions can be expressed with special functions. Finally, we apply the most common traveling wave Ansatz and present relative simple, although instructive solutions as well. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

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17 pages, 348 KiB

Open AccessArticle

Inverse Spectral Problems for Arbitrary-Order Differential Operators with Distribution Coefficients

by Natalia P. Bondarenko

Mathematics 2021, 9(22), 2989; https://doi.org/10.3390/math9222989 - 22 Nov 2021

Cited by 11 | Viewed by 2171

Abstract

In this paper, we propose an approach to inverse spectral problems for the n-th order

(n \geq 2)

ordinary differential operators with distribution coefficients. The inverse problems which consist in the reconstruction of the differential expression coefficients by the Weyl [...] Read more.

In this paper, we propose an approach to inverse spectral problems for the n-th order

(n \geq 2)

ordinary differential operators with distribution coefficients. The inverse problems which consist in the reconstruction of the differential expression coefficients by the Weyl matrix and by several spectra are studied. We prove the uniqueness of solution for these inverse problems, by developing the method of spectral mappings. The results of this paper generalize the previously known results for the second-order differential operators with singular potentials and for the higher-order differential operators with regular coefficients. In the future, the approach of this paper can be used for constructive solution and for investigation of solvability of the considered inverse problems. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

18 pages, 315 KiB

Open AccessArticle

Fixed Point Results via α-Admissibility in Extended Fuzzy Rectangular b-Metric Spaces with Applications to Integral Equations

by Badshah-e-Rome, Muhammad Sarwar and Rosana Rodríguez-López

Mathematics 2021, 9(16), 2009; https://doi.org/10.3390/math9162009 - 22 Aug 2021

Cited by 8 | Viewed by 1941

Abstract

In this article, the concept of extended fuzzy rectangular b-metric space (EFR_bMS, for short) is initiated, and some fixed point results frequently used in the literature are generalized via α-admissibility in the setting of EFR_bMS. For the [...] Read more.

In this article, the concept of extended fuzzy rectangular b-metric space (EFR_bMS, for short) is initiated, and some fixed point results frequently used in the literature are generalized via α-admissibility in the setting of EFR_bMS. For the illustration of the work presented, some supporting examples and an application to the existence of solutions for a class of integral equations are also discussed. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

25 pages, 401 KiB

Open AccessArticle

An Exterior Neumann Boundary-Value Problem for the Div-Curl System and Applications

by Briceyda B. Delgado and Jorge Eduardo Macías-Díaz

Mathematics 2021, 9(14), 1609; https://doi.org/10.3390/math9141609 - 8 Jul 2021

Cited by 7 | Viewed by 2416

Abstract

We investigate a generalization of the equation

curl \vec{w} = \vec{g}

to an arbitrary number n of dimensions, which is based on the well-known Moisil–Teodorescu differential operator. Explicit solutions are derived for a particular problem in bounded domains of

R^{n}

[...] Read more.

We investigate a generalization of the equation

curl \vec{w} = \vec{g}

to an arbitrary number n of dimensions, which is based on the well-known Moisil–Teodorescu differential operator. Explicit solutions are derived for a particular problem in bounded domains of

R^{n}

using classical operators from Clifford analysis. In the physically significant case

n = 3

, two explicit solutions to the div-curl system in exterior domains of

R^{3}

are obtained following different constructions of hyper-conjugate harmonic pairs. One of the constructions hinges on the use of a radial integral operator introduced recently in the literature. An exterior Neumann boundary-value problem is considered for the div-curl system. That system is conveniently reduced to a Neumann boundary-value problem for the Laplace equation in exterior domains. Some results on its uniqueness and regularity are derived. Finally, some applications to the construction of solutions of the inhomogeneous Lamé–Navier equation in bounded and unbounded domains are discussed. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

18 pages, 916 KiB

Open AccessArticle

First Solution of Fractional Bioconvection with Power Law Kernel for a Vertical Surface

by Muhammad Imran Asjad, Saif Ur Rehman, Ali Ahmadian, Soheil Salahshour and Mehdi Salimi

Mathematics 2021, 9(12), 1366; https://doi.org/10.3390/math9121366 - 12 Jun 2021

Cited by 16 | Viewed by 2488

Abstract

The present study provides the heat transfer analysis of a viscous fluid in the presence of bioconvection with a Caputo fractional derivative. The unsteady governing equations are solved by Laplace after using a dimensional analysis approach subject to the given constraints on the [...] Read more.

The present study provides the heat transfer analysis of a viscous fluid in the presence of bioconvection with a Caputo fractional derivative. The unsteady governing equations are solved by Laplace after using a dimensional analysis approach subject to the given constraints on the boundary. The impact of physical parameters can be seen through a graphical illustration. It is observed that the maximum decline in bioconvection and velocity can be attained for smaller values of the fractional parameter. The fractional approach can be very helpful in controlling the boundary layers of the fluid properties for different values of time. Additionally, it is observed that the model obtained with generalized constitutive laws predicts better memory than the model obtained with artificial replacement. Further, these results are compared with the existing literature to verify the validity of the present results. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

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25 pages, 435 KiB

Open AccessArticle

Analysis and Computation of Solutions for a Class of Nonlinear SBVPs Arising in Epitaxial Growth

by Amit K Verma, Biswajit Pandit and Ravi P. Agarwal

Mathematics 2021, 9(7), 774; https://doi.org/10.3390/math9070774 - 2 Apr 2021

Cited by 4 | Viewed by 1885

Abstract

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is [...] Read more.

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is non-self adjoint, it does not have exact solutions, and it admits multiple solutions. Here,

λ \in R

measures the intensity of the flux and G is stationary flux. The solution depends on the size of the parameter

λ

. We use a monotone iterative technique and integral equations along with upper and lower solutions to prove that solutions exist. We establish the qualitative properties of the solutions and provide bounds for the values of the parameter

λ

, which help us to separate existence from nonexistence. These results complement some existing results in the literature. To verify the analytical results, we also propose a new computational iterative technique and use it to verify the bounds on

λ

and the dependence of solutions for these computed bounds on

λ

. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

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19 pages, 330 KiB

Open AccessArticle

Nonlocal Sequential Boundary Value Problems for Hilfer Type Fractional Integro-Differential Equations and Inclusions

by Nawapol Phuangthong, Sotiris K. Ntouyas, Jessada Tariboon and Kamsing Nonlaopon

Mathematics 2021, 9(6), 615; https://doi.org/10.3390/math9060615 - 15 Mar 2021

Cited by 15 | Viewed by 1741

Abstract

In the present research, we study boundary value problems for fractional integro-differential equations and inclusions involving the Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray–Schauder in the single-valued case, while [...] Read more.

In the present research, we study boundary value problems for fractional integro-differential equations and inclusions involving the Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray–Schauder in the single-valued case, while Martelli’s fixed point theorem, a nonlinear alternative for multivalued maps, and the Covitz–Nadler fixed point theorem are used in the inclusion case. Examples are presented to illustrate our results. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

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