Research

23 pages, 1501 KiB

Open AccessArticle

A Comparative Analysis of Fractional-Order Kaup–Kupershmidt Equation within Different Operators

by Nehad Ali Shah, Yasser S. Hamed, Khadijah M. Abualnaja, Jae-Dong Chung, Rasool Shah and Adnan Khan

Symmetry 2022, 14(5), 986; https://doi.org/10.3390/sym14050986 - 11 May 2022

Cited by 63 | Viewed by 4101

Abstract

In this paper, we find the solution of the fractional-order Kaup–Kupershmidt (KK) equation by implementing the natural decomposition method with the aid of two different fractional derivatives, namely the Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF). When investigating capillary gravity waves [...] Read more.

In this paper, we find the solution of the fractional-order Kaup–Kupershmidt (KK) equation by implementing the natural decomposition method with the aid of two different fractional derivatives, namely the Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF). When investigating capillary gravity waves and nonlinear dispersive waves, the KK equation is extremely important. To demonstrate the accuracy and efficiency of the proposed technique, we study the nonlinear fractional KK equation in three distinct cases. The results are given in the form of a series, which converges quickly. The numerical simulations are presented through tables to illustrate the validity of the suggested technique. Numerical simulations in terms of absolute error are performed to ensure that the proposed methodologies are trustworthy and accurate. The resulting solutions are graphically shown to ensure the applicability and validity of the algorithms under consideration. The results that we obtain confirm that the proposed method is the best tool for handling any nonlinear problems arising in science and technology. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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27 pages, 2086 KiB

Open AccessArticle

Bessel Collocation Method for Solving Fredholm–Volterra Integro-Fractional Differential Equations of Multi-High Order in the Caputo Sense

by Shazad Shawki Ahmed and Shabaz Jalil MohammedFaeq

Symmetry 2021, 13(12), 2354; https://doi.org/10.3390/sym13122354 - 07 Dec 2021

Cited by 5 | Viewed by 2157

Abstract

The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on [...] Read more.

The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on transforming the linear equation and conditions into the matrix relations (some time symmetry matrix), which results in resolving a linear algebraic equation with unknown generalized Bessel coefficients. Numerical examples are given to show the technique’s validity and application, and comparisons are made with existing results by applying this process in order to express these solutions, most general programs are written in Python V.3.8.8 (2021). Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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22 pages, 347 KiB

Open AccessArticle

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

by Malgorzata Klimek

Symmetry 2021, 13(12), 2265; https://doi.org/10.3390/sym13122265 - 28 Nov 2021

Cited by 7 | Viewed by 1334

Abstract

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both [...] Read more.

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval

(1 / 2, 1]

. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

18 pages, 315 KiB

Open AccessArticle

Analysis and Optimal Control of φ-Hilfer Fractional Semilinear Equations Involving Nonlocal Impulsive Conditions

by Sarra Guechi, Rajesh Dhayal, Amar Debbouche and Muslim Malik

Symmetry 2021, 13(11), 2084; https://doi.org/10.3390/sym13112084 - 03 Nov 2021

Cited by 12 | Viewed by 1486

Abstract

The goal of this paper is to consider a new class of

φ

-Hilfer fractional differential equations with impulses and nonlocal conditions. By using fractional calculus, semigroup theory, and with the help of the fixed point theorem, the existence and uniqueness of mild [...] Read more.

The goal of this paper is to consider a new class of

φ

-Hilfer fractional differential equations with impulses and nonlocal conditions. By using fractional calculus, semigroup theory, and with the help of the fixed point theorem, the existence and uniqueness of mild solutions are obtained for the proposed fractional system. Symmetrically, we discuss the existence of optimal controls for the

φ

-Hilfer fractional control system. Our main results are well supported by an illustrative example. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

19 pages, 5798 KiB

Open AccessArticle

Aboodh Transform Iterative Method for Solving Fractional Partial Differential Equation with Mittag–Leffler Kernel

by Michael A. Awuya and Dervis Subasi

Symmetry 2021, 13(11), 2055; https://doi.org/10.3390/sym13112055 - 01 Nov 2021

Cited by 15 | Viewed by 1609

Abstract

The major aim of this paper is the presentation of Aboodh transform of the Atangana–Baleanu fractional differential operator both in Caputo and Riemann–Liouville sense by using the connection between the Laplace transform and the Aboodh transform. Moreover, we aim to obtain the approximate [...] Read more.

The major aim of this paper is the presentation of Aboodh transform of the Atangana–Baleanu fractional differential operator both in Caputo and Riemann–Liouville sense by using the connection between the Laplace transform and the Aboodh transform. Moreover, we aim to obtain the approximate series solutions for the time-fractional differential equations with an Atangana–Baleanu fractional differential operator in the Caputo sense using the Aboodh transform iterative method, which is the modification of the Aboodh transform by combining it with the new iterative method. The relation between the Laplace transform and the Aboodh transform is symmetrical. Some graphical illustrations are presented to describe the effect of the fractional order. The outcome reveals that Aboodh transform iterative method is easy to implement and adequately captures the behavior and the fractional effect of the fractional differential equation. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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12 pages, 2382 KiB

Open AccessArticle

λ-Interval of Triple Positive Solutions for the Perturbed Gelfand Problem

by Shugui Kang, Youmin Lu and Wenying Feng

Symmetry 2021, 13(9), 1606; https://doi.org/10.3390/sym13091606 - 01 Sep 2021

Cited by 1 | Viewed by 1191

Abstract

We study a two-point Boundary Value Problem depending on two parameters that represents a mathematical model arising from the combustion theory. Applying fixed point theorems for concave operators, we prove uniqueness, existence, upper, and lower bounds of positive solutions. In addition, we give [...] Read more.

We study a two-point Boundary Value Problem depending on two parameters that represents a mathematical model arising from the combustion theory. Applying fixed point theorems for concave operators, we prove uniqueness, existence, upper, and lower bounds of positive solutions. In addition, we give an estimation for the value of

λ_{*}

such that, for the parameter

λ \in [λ_{*}, λ^{*}]

, there exist exactly three positive solutions. Numerical examples are presented to illustrate various cases. The results complement previous work on this problem. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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26 pages, 433 KiB

Open AccessArticle

Approximate Solutions of an Extended Multi-Order Boundary Value Problem by Implementing Two Numerical Algorithms

by Surang Sitho, Sina Etemad, Brahim Tellab, Shahram Rezapour, Sotiris K. Ntouyas and Jessada Tariboon

Symmetry 2021, 13(8), 1341; https://doi.org/10.3390/sym13081341 - 25 Jul 2021

Cited by 5 | Viewed by 1650

Abstract

In this paper, we establish several necessary conditions to confirm the uniqueness-existence of solutions to an extended multi-order finite-term fractional differential equation with double-order integral boundary conditions with respect to asymmetric operators by relying on the Banach’s fixed-point criterion. We validate our study [...] Read more.

In this paper, we establish several necessary conditions to confirm the uniqueness-existence of solutions to an extended multi-order finite-term fractional differential equation with double-order integral boundary conditions with respect to asymmetric operators by relying on the Banach’s fixed-point criterion. We validate our study by implementing two numerical schemes to handle some Riemann–Liouville fractional boundary value problems and obtain approximate series solutions that converge to the exact ones. In particular, we present several examples that illustrate the closeness of the approximate solutions to the exact solutions. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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31 pages, 2255 KiB

Open AccessArticle

A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

by Saima Rashid, Aasma Khalid, Sobia Sultana, Zakia Hammouch, Rasool Shah and Abdullah M. Alsharif

Symmetry 2021, 13(7), 1254; https://doi.org/10.3390/sym13071254 - 13 Jul 2021

Cited by 39 | Viewed by 2841

Abstract

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of [...] Read more.

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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20 pages, 379 KiB

Open AccessArticle

Existence of Solutions for a Singular Fractional q-Differential Equations under Riemann–Liouville Integral Boundary Condition

by Mohammad Esmael Samei, Rezvan Ghaffari, Shao-Wen Yao, Mohammed K. A. Kaabar, Francisco Martínez and Mustafa Inc

Symmetry 2021, 13(7), 1235; https://doi.org/10.3390/sym13071235 - 09 Jul 2021

Cited by 20 | Viewed by 1673

Abstract

We investigate the existence of solutions for a system of m-singular sum fractional q-differential equations in this work under some integral boundary conditions in the sense of Caputo fractional q-derivatives. By means of a fixed point Arzelá–Ascoli theorem, the existence [...] Read more.

We investigate the existence of solutions for a system of m-singular sum fractional q-differential equations in this work under some integral boundary conditions in the sense of Caputo fractional q-derivatives. By means of a fixed point Arzelá–Ascoli theorem, the existence of positive solutions is obtained. By providing examples involving graphs, tables, and algorithms, our fundamental result about the endpoint is illustrated with some given computational results. In general, symmetry and q-difference equations have a common correlation between each other. In Lie algebra, q-deformations can be constructed with the help of the symmetry concept. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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15 pages, 334 KiB

Open AccessArticle

Non-Trivial Solutions of Non-Autonomous Nabla Fractional Difference Boundary Value Problems

by Alberto Cabada, Nikolay D. Dimitrov and Jagan Mohan Jonnalagadda

Symmetry 2021, 13(6), 1101; https://doi.org/10.3390/sym13061101 - 21 Jun 2021

Cited by 3 | Viewed by 1552

Abstract

In this article, we present a two-point boundary value problem with separated boundary conditions for a finite nabla fractional difference equation. First, we construct an associated Green’s function as a series of functions with the help of spectral theory, and obtain some of [...] Read more.

In this article, we present a two-point boundary value problem with separated boundary conditions for a finite nabla fractional difference equation. First, we construct an associated Green’s function as a series of functions with the help of spectral theory, and obtain some of its properties. Under suitable conditions on the nonlinear part of the nabla fractional difference equation, we deduce two existence results of the considered nonlinear problem by means of two Leray–Schauder fixed point theorems. We provide a couple of examples to illustrate the applicability of the established results. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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11 pages, 256 KiB

Open AccessArticle

Uniqueness of Abel’s Integral Equations of the Second Kind with Variable Coefficients

by Chenkuan Li and Joshua Beaudin

Symmetry 2021, 13(6), 1064; https://doi.org/10.3390/sym13061064 - 13 Jun 2021

Cited by 4 | Viewed by 1886

Abstract

This paper studies the uniqueness of the solutions of several of Abel’s integral equations of the second kind with variable coefficients as well as an in-symmetry system in Banach spaces

L (Ω)

and [...] Read more.

This paper studies the uniqueness of the solutions of several of Abel’s integral equations of the second kind with variable coefficients as well as an in-symmetry system in Banach spaces

L (Ω)

and

L (Ω) \times L (Ω)

, respectively. The results derived are new and original, and can be applied to solve the generalized Abel’s integral equations and obtain convergent series as solutions. We also provide a few examples to demonstrate the use of our main theorems based on convolutions, the gamma function and the Mittag–Leffler function. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

13 pages, 301 KiB

Open AccessArticle

Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces

by Vladimir E. Fedorov, Marina V. Plekhanova and Elizaveta M. Izhberdeeva

Symmetry 2021, 13(6), 1058; https://doi.org/10.3390/sym13061058 - 11 Jun 2021

Cited by 16 | Viewed by 1673

Abstract

Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such [...] Read more.

Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan–Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan–Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

29 pages, 706 KiB

Open AccessArticle

Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order

by Tinggang Zhao and Yujiang Wu

Symmetry 2021, 13(5), 872; https://doi.org/10.3390/sym13050872 - 13 May 2021

Cited by 4 | Viewed by 2225

Abstract

In this paper, we develop a Hermite cubic spline collocation method (HCSCM) for solving variable-order nonlinear fractional differential equations, which apply

C^{1}

-continuous nodal basis functions to an approximate problem. We also verify that the order of convergence of the HCSCM is [...] Read more.

In this paper, we develop a Hermite cubic spline collocation method (HCSCM) for solving variable-order nonlinear fractional differential equations, which apply

C^{1}

-continuous nodal basis functions to an approximate problem. We also verify that the order of convergence of the HCSCM is about

O (h^{min {4 - α, p}})

while the interpolating function belongs to

C^{p} (p \geq 1)

, where h is the mesh size and

α

the order of the fractional derivative. Many numerical tests are performed to confirm the effectiveness of the HCSCM for fractional differential equations, which include Helmholtz equations and the fractional Burgers equation of constant-order and variable-order with Riemann-Liouville, Caputo and Patie-Simon sense as well as two-sided cases. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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14 pages, 533 KiB

Open AccessArticle

Bilateral Tempered Fractional Derivatives

by Manuel Duarte Ortigueira and Gabriel Bengochea

Symmetry 2021, 13(5), 823; https://doi.org/10.3390/sym13050823 - 08 May 2021

Cited by 8 | Viewed by 1663

Abstract

The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and shows that it cannot be considered as a derivative. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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18 pages, 980 KiB

Open AccessArticle

The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation

by Jehad Alzabut, A. George Maria Selvam, R. Dhineshbabu and Mohammed K. A. Kaabar

Symmetry 2021, 13(5), 789; https://doi.org/10.3390/sym13050789 - 02 May 2021

Cited by 28 | Viewed by 3072

Abstract

An elastic beam equation (EBEq) described by a fourth-order fractional difference equation is proposed in this work with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. New sufficient conditions ensuring the solutions’ existence and uniqueness of the proposed problem are established. The [...] Read more.

An elastic beam equation (EBEq) described by a fourth-order fractional difference equation is proposed in this work with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. New sufficient conditions ensuring the solutions’ existence and uniqueness of the proposed problem are established. The findings are obtained by employing properties of discrete fractional equations, Banach contraction, and Brouwer fixed-point theorems. Further, we discuss our problem’s results concerning

H

yers–

U

lam (

HU

), generalized

H

yers–

U

lam (

GHU

),

H

yers–

U

lam–

R

assias (

HUR

), and generalized

H

yers–

U

lam–

R

assias (

GHUR

) stability. Specific examples with graphs and numerical experiment are presented to demonstrate the effectiveness of our results. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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15 pages, 783 KiB

Open AccessArticle

The Comparative Study for Solving Fractional-Order Fornberg–Whitham Equation via ρ-Laplace Transform

by Pongsakorn Sunthrayuth, Ahmed M. Zidan, Shao-Wen Yao, Rasool Shah and Mustafa Inc

Symmetry 2021, 13(5), 784; https://doi.org/10.3390/sym13050784 - 01 May 2021

Cited by 35 | Viewed by 2719

Abstract

In this article, we also introduced two well-known computational techniques for solving the time-fractional Fornberg–Whitham equations. The methods suggested are the modified form of the variational iteration and Adomian decomposition techniques by

ρ

-Laplace. Furthermore, an illustrative scheme is introduced to verify the [...] Read more.

In this article, we also introduced two well-known computational techniques for solving the time-fractional Fornberg–Whitham equations. The methods suggested are the modified form of the variational iteration and Adomian decomposition techniques by

ρ

-Laplace. Furthermore, an illustrative scheme is introduced to verify the accuracy of the available methods. The graphical representation of the exact and derived results is presented to show the suggested approaches reliability. The comparative solution analysis via graphs also represented the higher reliability and accuracy of the current techniques. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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14 pages, 261 KiB

Open AccessEditor’s ChoiceArticle

General Fractional Integrals and Derivatives of Arbitrary Order

by Yuri Luchko

Symmetry 2021, 13(5), 755; https://doi.org/10.3390/sym13050755 - 27 Apr 2021

Cited by 54 | Viewed by 3267

Abstract

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy [...] Read more.

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

10 pages, 290 KiB

Open AccessEditor’s ChoiceArticle

Regularity Criteria for the 3D Magneto-Hydrodynamics Equations in Anisotropic Lorentz Spaces

by Maria Alessandra Ragusa and Fan Wu

Symmetry 2021, 13(4), 625; https://doi.org/10.3390/sym13040625 - 08 Apr 2021

Cited by 5 | Viewed by 1849

Abstract

In this paper, we investigate the regularity of weak solutions to the 3D incompressible MHD equations. We provide a regularity criterion for weak solutions involving any two groups functions

(\partial_{1} u_{1}, \partial_{1} b_{1})

, [...] Read more.

In this paper, we investigate the regularity of weak solutions to the 3D incompressible MHD equations. We provide a regularity criterion for weak solutions involving any two groups functions

(\partial_{1} u_{1}, \partial_{1} b_{1})

,

(\partial_{2} u_{2}, \partial_{2} b_{2})

and

(\partial_{3} u_{3}, \partial_{3} b_{3})

in anisotropic Lorentz space. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

17 pages, 362 KiB

Open AccessArticle

New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter

by Ricardo Almeida and Natália Martins

Symmetry 2021, 13(4), 592; https://doi.org/10.3390/sym13040592 - 02 Apr 2021

Viewed by 1434

Abstract

This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives [...] Read more.

This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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22 pages, 375 KiB

Open AccessArticle

Midpoint Inequalities in Fractional Calculus Defined Using Positive Weighted Symmetry Function Kernels

by Pshtiwan Othman Mohammed, Hassen Aydi, Artion Kashuri, Y. S. Hamed and Khadijah M. Abualnaja

Symmetry 2021, 13(4), 550; https://doi.org/10.3390/sym13040550 - 26 Mar 2021

Cited by 35 | Viewed by 2100

Abstract

The aim of our study is to establish, for convex functions on an interval, a midpoint version of the fractional HHF type inequality. The corresponding fractional integral has a symmetric weight function composed with an increasing function as integral kernel. We also consider [...] Read more.

The aim of our study is to establish, for convex functions on an interval, a midpoint version of the fractional HHF type inequality. The corresponding fractional integral has a symmetric weight function composed with an increasing function as integral kernel. We also consider a midpoint identity and establish some related inequalities based on this identity. Some special cases can be considered from our main results. These results confirm the generality of our attempt. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

15 pages, 308 KiB

Open AccessArticle

Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System

by Sachin Kumar, Baljinder Kour, Shao-Wen Yao, Mustafa Inc and Mohamed S. Osman

Symmetry 2021, 13(3), 477; https://doi.org/10.3390/sym13030477 - 15 Mar 2021

Cited by 22 | Viewed by 2426

Abstract

In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the [...] Read more.

In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether’s operator, the conservation laws of the KP system are obtained. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

22 pages, 1657 KiB

Open AccessArticle

Condensing Functions and Approximate Endpoint Criterion for the Existence Analysis of Quantum Integro-Difference FBVPs

by Shahram Rezapour, Atika Imran, Azhar Hussain, Francisco Martínez, Sina Etemad and Mohammed K. A. Kaabar

Symmetry 2021, 13(3), 469; https://doi.org/10.3390/sym13030469 - 12 Mar 2021

Cited by 59 | Viewed by 3415

Abstract

A nonlinear quantum boundary value problem (q-FBVP) formulated in the sense of quantum Caputo derivative, with fractional q-integro-difference conditions along with its fractional quantum-difference inclusion q-BVP are investigated in this research. To prove the solutions’ existence for these quantum systems, we rely on [...] Read more.

A nonlinear quantum boundary value problem (q-FBVP) formulated in the sense of quantum Caputo derivative, with fractional q-integro-difference conditions along with its fractional quantum-difference inclusion q-BVP are investigated in this research. To prove the solutions’ existence for these quantum systems, we rely on the notions such as the condensing functions and approximate endpoint criterion (AEPC). Two numerical examples are provided to apply and validate our main results in this research work. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

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