Ordinary and Partial Differential Equations: Theory and Applications II

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (1 November 2022) | Viewed by 33685

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Section of Mathematics, International Telematic University, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
Interests: special functions; orthogonal polynomials; differential equations; operator theory; multivariate approximation theory; Lie algebra
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Special Issue Information

Dear Colleagues,

The study of differential equations is useful for understanding natural phenomena. In this Special Issue, we aim to present the latest research on the properties of ODE (Ordinary Differential Equations) and PDE (Partial Differential Equations) related to different techniques for finding solutions and methods describing the nature of these solutions or their related approximations.

In addition, we welcome papers on numerical aspects using classical or non-standard approaches, for example, the concepts and related formalism of special functions. Furthermore, articles on fractional differential equations are of interest, as are contributions related to the symmetry approach to problems of integrability in the field of differential equations.

Prof. Dr. Clemente Cesarano
Guest Editor

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Keywords

  • symmetries
  • ODE
  • PDE
  • numerical methods
  • fractional calculus

Published Papers (21 papers)

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Research

13 pages, 293 KiB  
Article
Oscillation Test for Second-Order Differential Equations with Several Delays
by Aml Abdelnaser, Osama Moaaz, Clemente Cesarano, Sameh Askar and Elmetwally M. Elabbasy
Symmetry 2023, 15(2), 452; https://doi.org/10.3390/sym15020452 - 8 Feb 2023
Cited by 1 | Viewed by 1116
Abstract
In this paper, the oscillatory properties of certain second-order differential equations of neutral type are investigated. We obtain new oscillation criteria, which guarantee that every solution of these equations oscillates. Further, we get conditions of an iterative nature. These results complement and extend [...] Read more.
In this paper, the oscillatory properties of certain second-order differential equations of neutral type are investigated. We obtain new oscillation criteria, which guarantee that every solution of these equations oscillates. Further, we get conditions of an iterative nature. These results complement and extend some beforehand results obtained in the literature. In order to illustrate the results we present an example. Full article
27 pages, 3044 KiB  
Article
Numerical Modeling of Pollutant Transport: Results and Optimal Parameters
by Olaoluwa Ayodeji Jejeniwa, Hagos Hailu Gidey and Appanah Rao Appadu
Symmetry 2022, 14(12), 2616; https://doi.org/10.3390/sym14122616 - 9 Dec 2022
Cited by 6 | Viewed by 1082
Abstract
In this work, we used three finite difference schemes to solve 1D and 2D convective diffusion equations. The three methods are the Kowalic–Murty scheme, Lax–Wendroff scheme, and nonstandard finite difference (NSFD) scheme. We considered a total of four numerical experiments; in all of [...] Read more.
In this work, we used three finite difference schemes to solve 1D and 2D convective diffusion equations. The three methods are the Kowalic–Murty scheme, Lax–Wendroff scheme, and nonstandard finite difference (NSFD) scheme. We considered a total of four numerical experiments; in all of these cases, the initial conditions consisted of symmetrical profiles. We looked at cases when the advection velocity was much greater than the diffusion of the coefficient and cases when the coefficient of diffusion was much greater than the advection velocity. The dispersion analysis of the three methods was studied for one of the cases and the optimal value of the time step size k, minimizing the dispersion error at a given value of the spatial step size. From our findings, we conclude that Lax–Wendroff is the most efficient scheme for all four cases. We also show that the optimal value of k computed by minimizing the dispersion error at a given value of a spacial step size gave the lowest l2 and l errors. Full article
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12 pages, 1139 KiB  
Article
Multiplicative Brownian Motion Stabilizes the Exact Stochastic Solutions of the Davey–Stewartson Equations
by Farah M. Al-Askar, Clemente Cesarano and Wael W. Mohammed
Symmetry 2022, 14(10), 2176; https://doi.org/10.3390/sym14102176 - 17 Oct 2022
Cited by 17 | Viewed by 1162
Abstract
In this article, the stochastic Davey–Stewartson equations (SDSEs) forced by multiplicative noise are addressed. We use the mapping method to find new rational, elliptic, hyperbolic and trigonometric functions. In addition, we generalize some previously obtained results. Due to the significance of the Davey–Stewartson [...] Read more.
In this article, the stochastic Davey–Stewartson equations (SDSEs) forced by multiplicative noise are addressed. We use the mapping method to find new rational, elliptic, hyperbolic and trigonometric functions. In addition, we generalize some previously obtained results. Due to the significance of the Davey–Stewartson equations in plasma physics, nonlinear optics, hydrodynamics and other fields, the discovered solutions are useful in explaining a number of intriguing physical phenomena. By using MATLAB tools to simulate our results and display some of 3D graphs, we show how the multiplicative Brownian motion impacts the analytical solutions of the SDSEs. Finally, we demonstrate the effect of multiplicative Brownian motion on the stability and the symmetry of the achieved solutions of the SDSEs. Full article
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12 pages, 1137 KiB  
Article
The Analytical Solutions of Stochastic-Fractional Drinfel’d-Sokolov-Wilson Equations via (G′/G)-Expansion Method
by Farah M. Al-Askar, Clemente Cesarano and Wael W. Mohammed
Symmetry 2022, 14(10), 2105; https://doi.org/10.3390/sym14102105 - 11 Oct 2022
Cited by 31 | Viewed by 1490
Abstract
Fractional–stochastic Drinfel’d–Sokolov–Wilson equations (FSDSWEs) forced by multiplicative Brownian motion are assumed. This equation is employed in mathematical physics, plasma physics, surface physics, applied sciences, and population dynamics. The (G/G)-expansion method is utilized to find rational, hyperbolic, and [...] Read more.
Fractional–stochastic Drinfel’d–Sokolov–Wilson equations (FSDSWEs) forced by multiplicative Brownian motion are assumed. This equation is employed in mathematical physics, plasma physics, surface physics, applied sciences, and population dynamics. The (G/G)-expansion method is utilized to find rational, hyperbolic, and trigonometric stochastic solutions for FSDSWEs. Because of the priority of FSDSWEs, the derived solutions are more useful and effective in understanding various important physical phenomena. Furthermore, we used the MATLAB package to create 3D graphs for specific solutions in order to investigate the effect of fractional-order and Brownian motions on the solutions of FSDSWEs. Full article
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15 pages, 321 KiB  
Article
(Δ∇)-Pachpatte Dynamic Inequalities Associated with Leibniz Integral Rule on Time Scales with Applications
by Ahmed A. El-Deeb, Dumitru Baleanu and Jan Awrejcewicz
Symmetry 2022, 14(9), 1867; https://doi.org/10.3390/sym14091867 - 7 Sep 2022
Cited by 3 | Viewed by 1119
Abstract
We prove some new dynamic inequalities of the Gronwall–Bellman–Pachpatte type on time scales. Our results can be used in analyses as useful tools for some types of partial dynamic equations on time scales and in their applications in environmental phenomena and physical and [...] Read more.
We prove some new dynamic inequalities of the Gronwall–Bellman–Pachpatte type on time scales. Our results can be used in analyses as useful tools for some types of partial dynamic equations on time scales and in their applications in environmental phenomena and physical and engineering sciences that are described by partial differential equations. Full article
13 pages, 295 KiB  
Article
Δ–Gronwall–Bellman–Pachpatte Dynamic Inequalities and Their Applications on Time Scales
by Ahmed A. El-Deeb, Dumitru Baleanu and Jan Awrejcewicz
Symmetry 2022, 14(9), 1804; https://doi.org/10.3390/sym14091804 - 31 Aug 2022
Viewed by 1137
Abstract
In this article, with the help of Leibniz integral rule on time scales, we prove some new dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales. These inequalities can be used as handy tools to study the qualitative and quantitative properties of solutions of the [...] Read more.
In this article, with the help of Leibniz integral rule on time scales, we prove some new dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales. These inequalities can be used as handy tools to study the qualitative and quantitative properties of solutions of the initial boundary value problem for partial delay dynamic equation. Full article
15 pages, 660 KiB  
Article
New Analytical Solutions for Coupled Stochastic Korteweg–de Vries Equations via Generalized Derivatives
by Abd-Allah Hyder, Mohamed A. Barakat, Ahmed H. Soliman, Areej A. Almoneef and Clemente Cesarano
Symmetry 2022, 14(9), 1770; https://doi.org/10.3390/sym14091770 - 25 Aug 2022
Cited by 3 | Viewed by 1154
Abstract
In this paper, the coupled nonlinear KdV (CNKdV) equations are solved in a stochastic environment. Hermite transforms, generalized conformable derivative, and an algorithm that merges the white noise instruments and the (G/G2)-expansion technique are utilized to [...] Read more.
In this paper, the coupled nonlinear KdV (CNKdV) equations are solved in a stochastic environment. Hermite transforms, generalized conformable derivative, and an algorithm that merges the white noise instruments and the (G/G2)-expansion technique are utilized to obtain white noise functional conformable solutions for these equations. New stochastic kinds of periodic and soliton solutions for these equations under conformable generalized derivatives are produced. Moreover, three-dimensional (3D) depictions are shown to illustrate that the monotonicity and symmetry of the obtained solutions can be controlled by giving a value of the conformable parameter. Furthermore, some remarks are presented to give a comparison between the obtained wave solutions and the wave solutions constructed under the conformable derivatives and Newton’s derivatives. Full article
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14 pages, 309 KiB  
Article
(γ,a)-Nabla Reverse Hardy–Hilbert-Type Inequalities on Time Scales
by Ahmed A. El-Deeb, Dumitru Baleanu and Jan Awrejcewicz
Symmetry 2022, 14(8), 1714; https://doi.org/10.3390/sym14081714 - 17 Aug 2022
Viewed by 1152
Abstract
In this article, using a (γ,a)-nabla conformable integral on time scales, we study several novel Hilbert-type dynamic inequalities via nabla time scales calculus. Our results generalize various inequalities on time scales, unifying and extending several discrete inequalities and their corresponding continuous [...] Read more.
In this article, using a (γ,a)-nabla conformable integral on time scales, we study several novel Hilbert-type dynamic inequalities via nabla time scales calculus. Our results generalize various inequalities on time scales, unifying and extending several discrete inequalities and their corresponding continuous analogues. We say that symmetry plays an essential role in determining the correct methods with which to solve dynamic inequalities. Full article
16 pages, 314 KiB  
Article
On Some Important Dynamic Inequalities of Hardy–Hilbert-Type on Timescales
by Ahmed A. El-Deeb, Dumitru Baleanu, Clemente Cesarano and Ahmed Abdeldaim
Symmetry 2022, 14(7), 1421; https://doi.org/10.3390/sym14071421 - 11 Jul 2022
Cited by 1 | Viewed by 1013
Abstract
In this article, by using some algebraic inequalities, nabla Hölder inequalities, and nabla Jensen’s inequalities on timescales, we proved some new nabla Hilbert-type dynamic inequalities on timescales. These inequalities extend some known dynamic inequalities on timescales and unify some continuous inequalities and their [...] Read more.
In this article, by using some algebraic inequalities, nabla Hölder inequalities, and nabla Jensen’s inequalities on timescales, we proved some new nabla Hilbert-type dynamic inequalities on timescales. These inequalities extend some known dynamic inequalities on timescales and unify some continuous inequalities and their corresponding discrete analogues. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article
12 pages, 299 KiB  
Article
New Conditions for Testing the Oscillation of Fourth-Order Differential Equations with Several Delays
by Ali Muhib, Osama Moaaz, Clemente Cesarano and Sameh S. Askar
Symmetry 2022, 14(5), 1068; https://doi.org/10.3390/sym14051068 - 23 May 2022
Cited by 3 | Viewed by 1438
Abstract
In this paper, we establish oscillation theorems for all solutions to fourth-order neutral differential equations using the Riccati transformation approach and some inequalities. Some new criteria are established that can be used in cases where known theorems fail to apply. The approach followed [...] Read more.
In this paper, we establish oscillation theorems for all solutions to fourth-order neutral differential equations using the Riccati transformation approach and some inequalities. Some new criteria are established that can be used in cases where known theorems fail to apply. The approach followed depends on finding conditions that guarantee the exclusion of positive solutions, and as a result of the symmetry between the positive and negative solutions of the studied equation, we therefore exclude negative solutions. An illustrative example is given. Full article
12 pages, 251 KiB  
Article
A Method for the Solution of Coupled System of Emden–Fowler–Type Equations
by Aishah A. Alsulami, Mariam AL-Mazmumy, Huda O. Bakodah and Nawal Alzaid
Symmetry 2022, 14(5), 843; https://doi.org/10.3390/sym14050843 - 19 Apr 2022
Cited by 3 | Viewed by 1219
Abstract
A dependable semi-analytical method via the application of a modified Adomian Decomposition Method (ADM) to tackle the coupled system of Emden–Fowler-type equations has been proposed. More precisely, an effective differential operator together with its corresponding inverse is successfully constructed. Moreover, this operator is [...] Read more.
A dependable semi-analytical method via the application of a modified Adomian Decomposition Method (ADM) to tackle the coupled system of Emden–Fowler-type equations has been proposed. More precisely, an effective differential operator together with its corresponding inverse is successfully constructed. Moreover, this operator is able to navigate to the closed-form solution easily without resorting to converting the coupled system to a system of Volterra integral equations; as in the case of a well-known reference in the literature. Lastly, the effectiveness of the method is demonstrated on some coupled systems of the governing model, and a speedier convergence rate was noted. Full article
19 pages, 6391 KiB  
Article
On the Analytical Solution of the Kuwabara-Type Particle-in-Cell Model for the Non-Axisymmetric Spheroidal Stokes Flow via the Papkovich–Neuber Representation
by Panayiotis Vafeas, Eleftherios Protopapas and Maria Hadjinicolaou
Symmetry 2022, 14(1), 170; https://doi.org/10.3390/sym14010170 - 15 Jan 2022
Cited by 1 | Viewed by 1622
Abstract
Modern engineering technology often involves the physical application of heat and mass transfer. These processes are associated with the creeping motion of a relatively homogeneous swarm of small particles, where the spheroidal geometry represents the shape of the embedded particles within such aggregates. [...] Read more.
Modern engineering technology often involves the physical application of heat and mass transfer. These processes are associated with the creeping motion of a relatively homogeneous swarm of small particles, where the spheroidal geometry represents the shape of the embedded particles within such aggregates. Here, the steady Stokes flow of an incompressible, viscous fluid through an assemblage of particles, at low Reynolds numbers, is studied by employing a particle-in-cell model. The mathematical formulation adopts the Kuwabara-type assumption, according to which each spheroidal particle is stationary and it is surrounded by a confocal spheroid that creates a fluid envelope, in which the Newtonian fluid moves with a constant velocity of arbitrary orientation. The boundary value problem in the fluid envelope is solved by imposing non-slip conditions on the surface of the spheroid, which is also considered as non-penetrable, while zero vorticity is assumed on the fictitious spheroidal boundary along with a uniform approaching velocity. The three-dimensional flow fields are calculated analytically for the first time, in the spheroidal geometry, by virtue of the Papkovich–Neuber representation. Through this, the velocity and the total pressure fields are provided in terms of a vector and the scalar spheroidal harmonic potentials, which enables the thorough study of the relevant physical characteristics of the flow fields. The newly obtained analytical expressions generalize to any direction with the existing results holding for the asymmetrical case, which were obtained with the aid of a stream function. These can be employed for the calculation of quantities of physical or engineering interest. Numerical implementation reveals the flow behavior within the fluid envelope for different geometrical cell characteristics and for the arbitrarily-assumed velocity field, thus reflecting the different flow/porous media situations. Sample calculations show the excellent agreement of the obtained results with those available for special geometrical cases. All of these findings demonstrate the usefulness of the proposed method and the powerfulness of the obtained analytical expansions. Full article
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11 pages, 271 KiB  
Article
An Extension of Caputo Fractional Derivative Operator by Use of Wiman’s Function
by Rahul Goyal, Praveen Agarwal, Alexandra Parmentier and Clemente Cesarano
Symmetry 2021, 13(12), 2238; https://doi.org/10.3390/sym13122238 - 23 Nov 2021
Cited by 7 | Viewed by 1445
Abstract
The main aim of this work is to study an extension of the Caputo fractional derivative operator by use of the two-parameter Mittag–Leffler function given by Wiman. We have studied some generating relations, Mellin transforms and other relationships with extended hypergeometric functions in [...] Read more.
The main aim of this work is to study an extension of the Caputo fractional derivative operator by use of the two-parameter Mittag–Leffler function given by Wiman. We have studied some generating relations, Mellin transforms and other relationships with extended hypergeometric functions in order to derive this extended operator. Due to symmetry in the family of special functions, it is easy to study their various properties with the extended fractional derivative operators. Full article
12 pages, 324 KiB  
Article
Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation
by Haji Gul, Sajjad Ali, Kamal Shah, Shakoor Muhammad, Thanin Sitthiwirattham and Saowaluck Chasreechai
Symmetry 2021, 13(11), 2215; https://doi.org/10.3390/sym13112215 - 19 Nov 2021
Cited by 6 | Viewed by 1533
Abstract
In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the [...] Read more.
In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit. Full article
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14 pages, 4711 KiB  
Article
Phase Diagram and Order Reconstruction Modeling for Nematics in Asymmetric π-Cells
by Antonino Amoddeo and Riccardo Barberi
Symmetry 2021, 13(11), 2156; https://doi.org/10.3390/sym13112156 - 11 Nov 2021
Cited by 1 | Viewed by 1518
Abstract
Intense electric fields applied to an asymmetric π-cell containing a nematic liquid crystal subjected to strong mechanical stresses induce distortions that are relaxed through a fast-switching mechanism: the order reconstruction transition. Topologically different nematic textures are connected by such a mechanism that is [...] Read more.
Intense electric fields applied to an asymmetric π-cell containing a nematic liquid crystal subjected to strong mechanical stresses induce distortions that are relaxed through a fast-switching mechanism: the order reconstruction transition. Topologically different nematic textures are connected by such a mechanism that is spatially driven by the intensity of the applied electric fields and by the anchoring angles of the nematic molecules on the confining plates of the cell. Using the finite element method, we implemented the moving mesh partial differential equation numerical technique, and we simulated the nematic evolution inside the cell in the context of the Landau–de Gennes order tensor theory. The order dynamics have been well captured, putting in evidence the possible existence of a metastable biaxial state, and a phase diagram of the nematic texture has been built, therefore confirming the appropriateness of the used technique for the study of this type of problem. Full article
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14 pages, 2611 KiB  
Article
Traveling Wave Solutions to the Nonlinear Evolution Equation Using Expansion Method and Addendum to Kudryashov’s Method
by Hammad Alotaibi
Symmetry 2021, 13(11), 2126; https://doi.org/10.3390/sym13112126 - 8 Nov 2021
Cited by 22 | Viewed by 3459
Abstract
The inspection of wave motion and propagation of diffusion, convection, dispersion, and dissipation is a key research area in mathematics, physics, engineering, and real-time application fields. This article addresses the generalized dimensional Hirota–Maccari equation by using two different methods: the [...] Read more.
The inspection of wave motion and propagation of diffusion, convection, dispersion, and dissipation is a key research area in mathematics, physics, engineering, and real-time application fields. This article addresses the generalized dimensional Hirota–Maccari equation by using two different methods: the exp(φ(ζ)) expansion method and Addendum to Kudryashov’s method to obtain the optical traveling wave solutions. By utilizing suitable transformations, the nonlinear pdes are transformed into odes. The traveling wave solutions are expressed in terms of rational functions. For certain parameter values, the obtained optical solutions are described graphically with the aid of Maple 15 software. Full article
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14 pages, 305 KiB  
Article
Refinements of Ostrowski Type Integral Inequalities Involving Atangana–Baleanu Fractional Integral Operator
by Hijaz Ahmad, Muhammad Tariq, Soubhagya Kumar Sahoo, Sameh Askar, Ahmed E. Abouelregal and Khaled Mohamed Khedher
Symmetry 2021, 13(11), 2059; https://doi.org/10.3390/sym13112059 - 1 Nov 2021
Cited by 18 | Viewed by 1633
Abstract
In this article, first, we deduce an equality involving the Atangana–Baleanu (AB)-fractional integral operator. Next, employing this equality, we present some novel generalization of Ostrowski type inequality using the Hölder inequality, the power-mean inequality, Young’s inequality, and the Jensen integral inequality [...] Read more.
In this article, first, we deduce an equality involving the Atangana–Baleanu (AB)-fractional integral operator. Next, employing this equality, we present some novel generalization of Ostrowski type inequality using the Hölder inequality, the power-mean inequality, Young’s inequality, and the Jensen integral inequality for the convexity of |Υ|. We also deduced some new special cases from the main results. There exists a solid connection between fractional operators and convexity because of their fascinating properties in the mathematical sciences. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. It is assumed that the results presented in this article will show new directions in the field of fractional calculus. Full article
16 pages, 805 KiB  
Article
Implicit Hybrid Fractional Boundary Value Problem via Generalized Hilfer Derivative
by Abdellatif ‬Boutiara, Mohammed S. ‬Abdo, Mohammed A. ‬Almalahi, Hijaz Ahmad and Amira Ishan
Symmetry 2021, 13(10), 1937; https://doi.org/10.3390/sym13101937 - 15 Oct 2021
Cited by 3 | Viewed by 1199
Abstract
This research paper is dedicated to the study of a class of boundary value problems for a nonlinear, implicit, hybrid, fractional, differential equation, supplemented with boundary conditions involving general fractional derivatives, known as the ϑ-Hilfer and ϑ-Riemann–Liouville fractional operators. The existence [...] Read more.
This research paper is dedicated to the study of a class of boundary value problems for a nonlinear, implicit, hybrid, fractional, differential equation, supplemented with boundary conditions involving general fractional derivatives, known as the ϑ-Hilfer and ϑ-Riemann–Liouville fractional operators. The existence of solutions to the mentioned problem is obtained by some auxiliary conditions and applied Dhage’s fixed point theorem on Banach algebras. The considered problem covers some symmetry cases, with respect to a ϑ function. Moreover, we present a pertinent example to corroborate the reported results. Full article
12 pages, 312 KiB  
Article
Several Integral Inequalities of Hermite–Hadamard Type Related to k-Fractional Conformable Integral Operators
by Muhammad Tariq, Soubhagya Kumar Sahoo, Hijaz Ahmad, Thanin Sitthiwirattham and Jarunee Soontharanon
Symmetry 2021, 13(10), 1880; https://doi.org/10.3390/sym13101880 - 5 Oct 2021
Cited by 4 | Viewed by 1345
Abstract
In this paper, we present some ideas and concepts related to the k-fractional conformable integral operator for convex functions. First, we present a new integral identity correlated with the k-fractional conformable operator for the first-order derivative of a given function. Employing [...] Read more.
In this paper, we present some ideas and concepts related to the k-fractional conformable integral operator for convex functions. First, we present a new integral identity correlated with the k-fractional conformable operator for the first-order derivative of a given function. Employing this new identity, the authors have proved some generalized inequalities of Hermite–Hadamard type via Hölder’s inequality and the power mean inequality. Inequalities have a strong correlation with convex and symmetric convex functions. There exist expansive properties and strong correlations between the symmetric function and various areas of convexity, including convex functions, probability theory, and convex geometry on convex sets because of their fascinating properties in the mathematical sciences. The results of this paper show that the methodology can be directly applied and is computationally easy to use and exact. Full article
13 pages, 1432 KiB  
Article
Analytical Analysis of Fractional-Order Multi-Dimensional Dispersive Partial Differential Equations
by Shuang-Shuang Zhou, Mounirah Areshi, Praveen Agarwal, Nehad Ali Shah, Jae Dong Chung and Kamsing Nonlaopon
Symmetry 2021, 13(6), 939; https://doi.org/10.3390/sym13060939 - 26 May 2021
Cited by 11 | Viewed by 2203
Abstract
In this paper, a novel technique called the Elzaki decomposition method has been using to solve fractional-order multi-dimensional dispersive partial differential equations. Elzaki decomposition method results for both integer and fractional orders are achieved in series form, providing a higher convergence rate to [...] Read more.
In this paper, a novel technique called the Elzaki decomposition method has been using to solve fractional-order multi-dimensional dispersive partial differential equations. Elzaki decomposition method results for both integer and fractional orders are achieved in series form, providing a higher convergence rate to the suggested technique. Illustrative problems are defined to confirm the validity of the current technique. It is also researched that the conclusions of the fractional-order are convergent to an integer-order result. Moreover, the proposed method results are compared with the exact solution of the problems, which has confirmed that approximate solutions are convergent to the exact solution of each problem as the terms of the series increase. The accuracy of the method is examined with the help of some examples. It is shown that the proposed method is found to be reliable, efficient and easy to use for various related problems of applied science. Full article
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21 pages, 354 KiB  
Article
Numerical Solutions Caused by DGJIM and ADM Methods for Multi-Term Fractional BVP Involving the Generalized ψ-RL-Operators
by Shahram Rezapour, Sina Etemad, Brahim Tellab, Praveen Agarwal and Juan Luis Garcia Guirao
Symmetry 2021, 13(4), 532; https://doi.org/10.3390/sym13040532 - 25 Mar 2021
Cited by 29 | Viewed by 2359
Abstract
In this research study, we establish some necessary conditions to check the uniqueness-existence of solutions for a general multi-term ψ-fractional differential equation via generalized ψ-integral boundary conditions with respect to the generalized asymmetric operators. To arrive at such purpose, we utilize [...] Read more.
In this research study, we establish some necessary conditions to check the uniqueness-existence of solutions for a general multi-term ψ-fractional differential equation via generalized ψ-integral boundary conditions with respect to the generalized asymmetric operators. To arrive at such purpose, we utilize a procedure based on the fixed-point theory. We follow our study by suggesting two numerical algorithms called the Dafterdar-Gejji and Jafari method (DGJIM) and the Adomian decomposition method (ADM) techniques in which a series of approximate solutions converge to the exact ones of the given ψ-RLFBVP and the equivalent ψ-integral equation. To emphasize for the compatibility and the effectiveness of these numerical algorithms, we end this investigation by providing some examples showing the behavior of the exact solution of the existing ψ-RLFBVP compared with the approximate ones caused by DGJIM and ADM techniques graphically. Full article
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