Research

22 pages, 2805 KiB

Open AccessArticle

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

by Mahmoud A. Zaky, Ahmed S. Hendy and Rob H. De Staelen

Mathematics 2021, 9(2), 183; https://doi.org/10.3390/math9020183 - 18 Jan 2021

Cited by 18 | Viewed by 2495

Abstract

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov

L 2

-

1_{σ}

difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used [...] Read more.

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov

L 2

-

1_{σ}

difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

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14 pages, 424 KiB

Open AccessArticle

Solutions of Sturm-Liouville Problems

by Upeksha Perera and Christine Böckmann

Mathematics 2020, 8(11), 2074; https://doi.org/10.3390/math8112074 - 20 Nov 2020

Cited by 6 | Viewed by 3470

Abstract

This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm–Liouville problems. Next, a concrete implementation to the inverse Sturm–Liouville problem algorithm proposed by Barcilon (1974) is [...] Read more.

This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm–Liouville problems. Next, a concrete implementation to the inverse Sturm–Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm–Liouville problems of higher order (for

n = 2, 4

) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm–Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

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17 pages, 341 KiB

Open AccessArticle

Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝ^N

by Yun-Ho Kim

Mathematics 2020, 8(10), 1792; https://doi.org/10.3390/math8101792 - 15 Oct 2020

Cited by 4 | Viewed by 1586

Abstract

We are concerned with the following elliptic equations: [...] Read more.

We are concerned with the following elliptic equations:

{(- Δ)}_{p}^{s} v + {V (x) | v |}^{p - 2} v = λ a (x) {| v |}^{r - 2} v + g (x, v) i n R^{N},

where

{(- Δ)}_{p}^{s}

is the fractional p-Laplacian operator with

0 < s < 1 < r < p < + \infty

,

s p < N

, the potential function

V : R^{N} \to (0, \infty)

is a continuous potential function, and

g : R^{N} \times R \to R

satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

16 pages, 1353 KiB

Open AccessArticle

A Closed-Form Solution for the Boundary Value Problem of Gas Pressurized Circular Membranes in Contact with Frictionless Rigid Plates

by Dong Mei, Jun-Yi Sun, Zhi-Hang Zhao and Xiao-Ting He

Mathematics 2020, 8(6), 1017; https://doi.org/10.3390/math8061017 - 22 Jun 2020

Cited by 5 | Viewed by 2437

Abstract

In this paper, the static problem of equilibrium of contact between an axisymmetric deflected circular membrane and a frictionless rigid plate was analytically solved, where an initially flat circular membrane is fixed on its periphery and pressurized on one side by gas such [...] Read more.

In this paper, the static problem of equilibrium of contact between an axisymmetric deflected circular membrane and a frictionless rigid plate was analytically solved, where an initially flat circular membrane is fixed on its periphery and pressurized on one side by gas such that it comes into contact with a frictionless rigid plate, resulting in a restriction on the maximum deflection of the deflected circular membrane. The power series method was employed to solve the boundary value problem of the resulting nonlinear differential equation, and a closed-form solution of the problem addressed here was presented. The difference between the axisymmetric deformation caused by gas pressure loading and that caused by gravity loading was investigated. In order to compare the presented solution applying to gas pressure loading with the existing solution applying to gravity loading, a numerical example was conducted. The result of the conducted numerical example shows that the two solutions agree basically closely for membranes lightly loaded and diverge as the external loads intensify. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

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13 pages, 270 KiB

Open AccessArticle

Nontrivial Solutions for a System of Fractional q-Difference Equations Involving q-Integral Boundary Conditions

by Yaohong Li, Jie Liu, Donal O’Regan and Jiafa Xu

Mathematics 2020, 8(5), 828; https://doi.org/10.3390/math8050828 - 20 May 2020

Cited by 12 | Viewed by 1877

Abstract

In this paper, we study the existence of nontrivial solutions for a system of fractional q-difference equations involving q-integral boundary conditions, and we use the topological degree to establish our main results by considering the first eigenvalue of some associated linear [...] Read more.

In this paper, we study the existence of nontrivial solutions for a system of fractional q-difference equations involving q-integral boundary conditions, and we use the topological degree to establish our main results by considering the first eigenvalue of some associated linear integral operators. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

18 pages, 354 KiB

Open AccessArticle

Existence of Positive Solutions to Singular φ-Laplacian Nonlocal Boundary Value Problems when φ is a Sup-multiplicative-like Function

by Jeongmi Jeong and Chan-Gyun Kim

Mathematics 2020, 8(3), 420; https://doi.org/10.3390/math8030420 - 14 Mar 2020

Cited by 3 | Viewed by 1952

Abstract

In this paper, using a fixed point index theorem on a cone, we present some existence results for one or multiple positive solutions to

φ

-Laplacian nonlocal boundary value problems when

φ

is a sup-multiplicative-like function and the nonlinearity may not satisfy the [...] Read more.

In this paper, using a fixed point index theorem on a cone, we present some existence results for one or multiple positive solutions to

φ

-Laplacian nonlocal boundary value problems when

φ

is a sup-multiplicative-like function and the nonlinearity may not satisfy the

L^{1}

-Carath

\overset{´}{e}

odory condition. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

13 pages, 616 KiB

Open AccessArticle

Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions

by Alberto Cabada and Om Kalthoum Wanassi

Mathematics 2020, 8(2), 255; https://doi.org/10.3390/math8020255 - 14 Feb 2020

Cited by 7 | Viewed by 2390

Abstract

This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the

α

-Riemann-Liouville fractional derivative, with

α \in (1, 2]

. [...] Read more.

This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the

α

-Riemann-Liouville fractional derivative, with

α \in (1, 2]

. To deduce the existence and non-existence results, we first study the linear equation, by deducing the main properties of the related Green functions. We obtain the optimal set of parameters where the Green function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

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17 pages, 334 KiB

Open AccessArticle

Existence of Solutions for Kirchhoff-Type Fractional Dirichlet Problem with p-Laplacian

by Danyang Kang, Cuiling Liu and Xingyong Zhang

Mathematics 2020, 8(1), 106; https://doi.org/10.3390/math8010106 - 08 Jan 2020

Cited by 3 | Viewed by 2472

Abstract

In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann–Liouville fractional derivatives and a parameter

λ

. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution [...] Read more.

In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann–Liouville fractional derivatives and a parameter

λ

. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution

u_{λ}

under some local conditions for each given large parameter

λ

. We get a concrete lower bound of the parameter

λ

, and then obtain two estimates of weak solutions

u_{λ}

. We also obtain that

u_{λ} \to 0

if

λ

tends to ∞. Finally, we present an example as an application of our results. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

15 pages, 290 KiB

Open AccessArticle

Nonlinear Impulsive Multi-Order Caputo-Type Generalized Fractional Differential Equations with Infinite Delay

by Bashir Ahmad, Madeaha Alghanmi, Ahmed Alsaedi and Ravi P. Agarwal

Mathematics 2019, 7(11), 1108; https://doi.org/10.3390/math7111108 - 15 Nov 2019

Cited by 5 | Viewed by 1798

Abstract

We establish sufficient conditions for the existence of solutions for a nonlinear impulsive multi-order Caputo-type generalized fractional differential equation with infinite delay and nonlocal generalized integro-initial value conditions. The existence result is proved by means of Krasnoselskii’s fixed point theorem, while the contraction [...] Read more.

We establish sufficient conditions for the existence of solutions for a nonlinear impulsive multi-order Caputo-type generalized fractional differential equation with infinite delay and nonlocal generalized integro-initial value conditions. The existence result is proved by means of Krasnoselskii’s fixed point theorem, while the contraction mapping principle is employed to obtain the uniqueness of solutions for the problem at hand. The paper concludes with illustrative examples. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

15 pages, 290 KiB

Open AccessFeature PaperArticle

Variational Methods for an Impulsive Fractional Differential Equations with Derivative Term

by Yulin Zhao, Jiafa Xu and Haibo Chen

Mathematics 2019, 7(10), 880; https://doi.org/10.3390/math7100880 - 21 Sep 2019

Cited by 3 | Viewed by 2019

Abstract

This paper is devoted to studying the existence of solutions to a class of impulsive fractional differential equations with derivative dependence. The used technical approach is based on variational methods and iterative methods. In addition, an example is given to demonstrate the main [...] Read more.

This paper is devoted to studying the existence of solutions to a class of impulsive fractional differential equations with derivative dependence. The used technical approach is based on variational methods and iterative methods. In addition, an example is given to demonstrate the main results. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

10 pages, 270 KiB

Open AccessArticle

A Lyapunov-Type Inequality for a Laplacian System on a Rectangular Domain with Zero Dirichlet Boundary Conditions

by Mohamed Jleli and Bessem Samet

Mathematics 2019, 7(9), 850; https://doi.org/10.3390/math7090850 - 14 Sep 2019

Cited by 2 | Viewed by 1981

Abstract

We consider a coupled system of partial differential equations involving Laplacian operator, on a rectangular domain with zero Dirichlet boundary conditions. A Lyapunov-type inequality related to this problem is derived. This inequality provides a necessary condition for the existence of nontrivial positive solutions. [...] Read more.

We consider a coupled system of partial differential equations involving Laplacian operator, on a rectangular domain with zero Dirichlet boundary conditions. A Lyapunov-type inequality related to this problem is derived. This inequality provides a necessary condition for the existence of nontrivial positive solutions. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

20 pages, 402 KiB

Open AccessArticle

Numerical Solution of the Boundary Value Problems Arising in Magnetic Fields and Cylindrical Shells

by Aasma Khalid, Muhammad Nawaz Naeem, Zafar Ullah, Abdul Ghaffar, Dumitru Baleanu, Kottakkaran Sooppy Nisar and Maysaa M. Al-Qurashi

Mathematics 2019, 7(6), 508; https://doi.org/10.3390/math7060508 - 03 Jun 2019

Cited by 22 | Viewed by 3249

Abstract

This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied [...] Read more.

This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied physics, fluid dynamics, and applied mathematics. The recommended method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 8th order BVPs using Cubic-B spline but it also describes the estimated derivatives of 1st order to 8th order of the analytic solution. The strategy is effectively applied to numerical examples and the outcomes are compared with the existing results. The method proposed in this paper provides better approximations to the exact solution. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

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10 pages, 255 KiB

Open AccessArticle

Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

by Guotao Wang, Jianfang Qin, Huanhe Dong and Tingting Guan

Mathematics 2019, 7(6), 500; https://doi.org/10.3390/math7060500 - 02 Jun 2019

Cited by 6 | Viewed by 2378

Abstract

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer [...] Read more.

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

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