Symmetries in Differential Equation and Application

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

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 19208

Special Issue Editor

Department of Mathematics, Hannam University, Daejeon 34430, Republic of Korea
Interests: numerical verification method; scientific computing; differential equations; dynamical systems; quantum calculus and special functions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The study of differential equations is a broad field in pure and applied mathematics.  All of these fields relate to the properties of different types of differential equations.

Pure mathematics investigates the existence and uniqueness of solutions, while applied mathematics enforces a strict justification of how to approximate solutions

Differential equations play a significant role in modeling virtually every physical, technical and biological process, etc. These areas are still at the center of advanced mathematical research.  Differential equations, such as those used to solve real problems, are not necessarily directly solvable.  Instead, solutions can be approximated using numerical methods.  These methods are central to studies in advanced mathematics, physics, and engineering with many potential applications. Recently, differential equations have been closely related to several areas in mathematics, applied mathematics, physics, chemistry, biological sciences, and engineering, and have been used to share the recent knowledge and research in pure as well as applied mathematical sciences.

This Special Issue aims to publish major research papers and latest trends in pure and applied mathematical sciences including the area of differential equations.

Prof. Dr. Cheon-Seoung Ryoo
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • differential equation
  • symmetry
  • pseudo-differential operator
  • numerical analysis
  • approximation
  • a priori estimates
  • stability
  • asymptotic properties
  • numerical verification method
  • dynamical systems
  • quantum calculus
  • special functions

Published Papers (14 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

12 pages, 293 KiB  
Article
Delay Differential Equations with Several Sublinear Neutral Terms: Investigation of Oscillatory Behavior
by Waed Muhsin, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani and Elmetwally M. Elabbasy
Symmetry 2023, 15(12), 2105; https://doi.org/10.3390/sym15122105 - 23 Nov 2023
Viewed by 675
Abstract
In this work, new oscillation criteria are established for a second-order differential equation with several sublinear neutral terms and in the canonical case. To determine the oscillation conditions, we followed the Riccati approach and also compared the studied equation with a first-order delay [...] Read more.
In this work, new oscillation criteria are established for a second-order differential equation with several sublinear neutral terms and in the canonical case. To determine the oscillation conditions, we followed the Riccati approach and also compared the studied equation with a first-order delay equation. Obtaining the oscillation conditions required deducing some new relationships linking the solution to the corresponding function as well as its derivatives. The paper addresses some interesting analytical points in the study of the oscillation of equations with several sublinear neutral terms. These new findings complement some well-known findings in the literature. Furthermore, an example is provided to show the importance of the results. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
26 pages, 2543 KiB  
Article
Differential Evolution Using Enhanced Mutation Strategy Based on Random Neighbor Selection
by Muhammad Hassan Baig, Qamar Abbas, Jamil Ahmad, Khalid Mahmood, Sultan Alfarhood, Mejdl Safran and Imran Ashraf
Symmetry 2023, 15(10), 1916; https://doi.org/10.3390/sym15101916 - 14 Oct 2023
Viewed by 706
Abstract
Symmetry in a differential evolution (DE) transforms a solution without impacting the family of solutions. For symmetrical problems in differential equations, DE is a strong evolutionary algorithm that provides a powerful solution to resolve global optimization problems. DE/best/1 and DE/rand/1 are the two [...] Read more.
Symmetry in a differential evolution (DE) transforms a solution without impacting the family of solutions. For symmetrical problems in differential equations, DE is a strong evolutionary algorithm that provides a powerful solution to resolve global optimization problems. DE/best/1 and DE/rand/1 are the two most commonly used mutation strategies in DE. The former provides better exploitation while the latter ensures better exploration. DE/Neighbor/1 is an improved form of DE/rand/1 to maintain a balance between exploration and exploitation which was used with a random neighbor-based differential evolution (RNDE) algorithm. However, this mutation strategy slows down convergence. It should achieve a global minimum by using 1000 × D, where D is the dimension, but due to exploration and exploitation balancing trade-offs, it can not achieve a global minimum within the range of 1000 × D in some of the objective functions. To overcome this issue, a new and enhanced mutation strategy and algorithm have been introduced in this paper, called DE/Neighbor/2, as well as an improved random neighbor-based differential evolution algorithm. The new DE/Neighbor/2 mutation strategy also uses neighbor information such as DE/Neighbor/1; however, in addition, we add weighted differences after various tests. The DE/Neighbor/2 and IRNDE algorithm has also been tested on the same 27 commonly used benchmark functions on which the DE/Neighbor/1 mutation strategy and RNDE were tested. Experimental results demonstrate that the DE/Neighbor/2 mutation strategy and IRNDE algorithm show overall better and faster convergence than the DE/Neighbor/1 mutation strategy and RNDE algorithm. The parametric significance test shows that there is a significance difference in the performance of RNDE and IRNDE algorithms at the 0.05 level of significance. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

14 pages, 631 KiB  
Article
Properties of Differential Equations Related to Degenerate q-Tangent Numbers and Polynomials
by Jung-Yoog Kang
Symmetry 2023, 15(4), 874; https://doi.org/10.3390/sym15040874 - 06 Apr 2023
Cited by 2 | Viewed by 811
Abstract
In this paper, we construct degenerate q-tangent numbers and polynomials and determine their related properties. Based on these numbers and polynomials, we also confirm that the structure of the approximate root changes according to changes in q and h. We find [...] Read more.
In this paper, we construct degenerate q-tangent numbers and polynomials and determine their related properties. Based on these numbers and polynomials, we also confirm that the structure of the approximate root changes according to changes in q and h. We find differential equations that have degenerate q-tangent polynomials as solutions and also find differential equations that have other polynomials as coefficients, confirming the relationships among these. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

12 pages, 304 KiB  
Article
Application of Fixed-Point Results to Integral Equation through F-Khan Contraction
by Arul Joseph Gnanaprakasam, Gunaseelan Mani, Rajagopalan Ramaswamy, Ola A. Ashour Abdelnaby, Khizar Hyatt Khan and Stojan Radenović
Symmetry 2023, 15(3), 773; https://doi.org/10.3390/sym15030773 - 22 Mar 2023
Cited by 1 | Viewed by 980
Abstract
In this article, we establish fixed point results by defining the concept of F-Khan contraction of an orthogonal set by modifying the symmetry of usual contractive conditions. We also provide illustrative examples to support our results. The derived results have been applied [...] Read more.
In this article, we establish fixed point results by defining the concept of F-Khan contraction of an orthogonal set by modifying the symmetry of usual contractive conditions. We also provide illustrative examples to support our results. The derived results have been applied to find analytical solutions to integral equations. The analytical solutions are verified with numerical simulation. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

11 pages, 2591 KiB  
Article
Analytical Solutions for a New Form of the Generalized q-Deformed Sinh–Gordon Equation: 2uzζ=eαu[sinhq(uγ)]pδ
by Khalid K. Ali, Haifa I. Alrebdi, Norah A. M. Alsaif, Abdel-Haleem Abdel-Aty and Hichem Eleuch
Symmetry 2023, 15(2), 470; https://doi.org/10.3390/sym15020470 - 10 Feb 2023
Cited by 2 | Viewed by 1080
Abstract
In this article, a new version of the generalized q-deformed Sinh–Gordon equation is presented, and analytical solutions are developed for specific parameter sets using those equations. There is a possibility that the new equation can be used to model physical systems that [...] Read more.
In this article, a new version of the generalized q-deformed Sinh–Gordon equation is presented, and analytical solutions are developed for specific parameter sets using those equations. There is a possibility that the new equation can be used to model physical systems that have broken symmetries and include also effects related to amplification or dissipation. In addition, we have include some illustrations that depict the varied patterns of soliton propagation. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

13 pages, 345 KiB  
Article
Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations
by Hamid Faraji, Nikola Mirkov, Zoran D. Mitrović, Rajagopalan Ramaswamy, Ola A. Ashour Abdelnaby and Stojan Radenović
Symmetry 2022, 14(11), 2429; https://doi.org/10.3390/sym14112429 - 16 Nov 2022
Cited by 3 | Viewed by 1055
Abstract
In this paper, we consider and extend some fixed point results in F-complete F-metric spaces by relaxing the symmetry of complete metric spaces. We generalize α,β-admissible mappings in the setting of F-metric spaces. The derived results are [...] Read more.
In this paper, we consider and extend some fixed point results in F-complete F-metric spaces by relaxing the symmetry of complete metric spaces. We generalize α,β-admissible mappings in the setting of F-metric spaces. The derived results are supplemented with suitable examples, and the obtained results are applied to find the existence of the solution to the integral equation. The analytical results are compared through numerical simulation. We pose certain open problems for extending and applying our results in the future. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

36 pages, 384 KiB  
Article
Solution of Integral Equation with Neutrosophic Rectangular Triple Controlled Metric Spaces
by Gunaseelan Mani, Rajagopalan Ramaswamy, Arul Joseph Gnanaprakasam, Ola A Ashour Abdelnaby, Slobodan Radojevic and Stojan Radenović
Symmetry 2022, 14(10), 2074; https://doi.org/10.3390/sym14102074 - 06 Oct 2022
Viewed by 822
Abstract
In this paper, we introduce the notion of neutrosophic rectangular triple-controlled metric space, relaxing the symmetry requirement of neutrosophic metric spaces, by replacing triangular inequalities with rectangular inequalities, and prove fixed point theorems. We have derived several interesting results for contraction mappings supplemented [...] Read more.
In this paper, we introduce the notion of neutrosophic rectangular triple-controlled metric space, relaxing the symmetry requirement of neutrosophic metric spaces, by replacing triangular inequalities with rectangular inequalities, and prove fixed point theorems. We have derived several interesting results for contraction mappings supplemented with non-trivial examples. The derived results have been applied to prove the existence of a unique analytical solution as well as a closed form of the unique solution to the integral equation. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

19 pages, 929 KiB  
Article
Analytical Formulas for Conditional Mixed Moments of Generalized Stochastic Correlation Process
by Ampol Duangpan, Ratinan Boonklurb, Kittisak Chumpong and Phiraphat Sutthimat
Symmetry 2022, 14(5), 897; https://doi.org/10.3390/sym14050897 - 27 Apr 2022
Cited by 4 | Viewed by 1420
Abstract
This paper proposes a simple and novel approach based on solving a partial differential equation (PDE) to establish the concise analytical formulas for a conditional moment and mixed moment of the Jacobi process with constant parameters, accomplished by including random fluctuations with an [...] Read more.
This paper proposes a simple and novel approach based on solving a partial differential equation (PDE) to establish the concise analytical formulas for a conditional moment and mixed moment of the Jacobi process with constant parameters, accomplished by including random fluctuations with an asymmetric Wiener process and without any knowledge of the transition probability density function. Our idea involves a system with a recurrence differential equation which leads to the PDE by involving an asymmetric matrix. Then, by using Itô’s lemma, all formulas for the Jacobi process with constant parameters as well as time-dependent parameters are extended to the generalized stochastic correlation processes. In addition, their statistical properties are provided in closed forms. Finally, to illustrate applications of the proposed formulas in practice, estimations of parametric methods based on the moments are mentioned, particularly in the method of moments estimators. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

11 pages, 277 KiB  
Article
A q-Difference Equation and Fourier Series Expansions of q-Lidstone Polynomials
by Maryam Al-Towailb
Symmetry 2022, 14(4), 782; https://doi.org/10.3390/sym14040782 - 09 Apr 2022
Cited by 3 | Viewed by 997
Abstract
In this paper, we present the q-Lidstone polynomials which are q-Bernoulli polynomials generated by the third Jackson q-Bessel function, based on the Green’s function of a certain q-difference equation. Also, we provide the q-Fourier series expansions of these [...] Read more.
In this paper, we present the q-Lidstone polynomials which are q-Bernoulli polynomials generated by the third Jackson q-Bessel function, based on the Green’s function of a certain q-difference equation. Also, we provide the q-Fourier series expansions of these polynomials and derive some results related to them. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
15 pages, 947 KiB  
Article
Some Identities Involving Degenerate q-Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros
by Cheon-Seoung Ryoo and Jung-Yoog Kang
Symmetry 2022, 14(4), 706; https://doi.org/10.3390/sym14040706 - 31 Mar 2022
Cited by 4 | Viewed by 1230
Abstract
This paper intends to define degenerate q-Hermite polynomials, namely degenerate q-Hermite polynomials by means of generating function. Some significant properties of degenerate q-Hermite polynomials such as recurrence relations, explicit identities and differential equations are established. Many mathematicians have been studying [...] Read more.
This paper intends to define degenerate q-Hermite polynomials, namely degenerate q-Hermite polynomials by means of generating function. Some significant properties of degenerate q-Hermite polynomials such as recurrence relations, explicit identities and differential equations are established. Many mathematicians have been studying the differential equations arising from the generating functions of special numbers and polynomials. Based on the results so far, we find the differential equations for the degenerate q-Hermite polynomials. We also provide some identities for the degenerate q-Hermite polynomials using the coefficients of this differential equation. Finally, we use a computer to view the location of the zeros in degenerate q-Hermite equations. Numerical experiments have confirmed that the roots of the degenerate q-Hermit equations are not symmetric with respect to the imaginary axis. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

15 pages, 1002 KiB  
Article
Mathematical Modeling and Analysis of Tumor Chemotherapy
by Ge Song, Guizhen Liang, Tianhai Tian and Xinan Zhang
Symmetry 2022, 14(4), 704; https://doi.org/10.3390/sym14040704 - 31 Mar 2022
Cited by 6 | Viewed by 3463
Abstract
Cancer diseases lead to the second-highest death rate all over the world. For treating tumors, one of the most common schemes is chemotherapy, which can decrease the tumor size and control the progression of cancer diseases. To better understand the mechanisms of chemotherapy, [...] Read more.
Cancer diseases lead to the second-highest death rate all over the world. For treating tumors, one of the most common schemes is chemotherapy, which can decrease the tumor size and control the progression of cancer diseases. To better understand the mechanisms of chemotherapy, we developed a mathematical model of tumor growth under chemotherapy. This model includes both immune system response and drug therapy. We characterize the symmetrical properties and dynamics of this differential equation model by finding the equilibrium points and exploring the stability and symmetry properties in a range of model parameters. Sensitivity analyses suggest that the chemotherapy drug-induced tumor mortality rate and the drug decay rate contribute significantly to the determination of treatment outcomes. Numerical simulations highlight the importance of CTL activation in tumor chemotherapy. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

8 pages, 226 KiB  
Article
Estimates of Eigenvalues of a Semiperiodic Dirichlet Problem for a Class of Degenerate Elliptic Equations
by Mussakan Muratbekov and Sabit Igissinov
Symmetry 2022, 14(4), 692; https://doi.org/10.3390/sym14040692 - 28 Mar 2022
Cited by 2 | Viewed by 1334
Abstract
In this paper, we consider a class of degenerate elliptic equations with arbitrary power degeneration. The issues about the existence, uniqueness, and smoothness of solutions of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration are studied. [...] Read more.
In this paper, we consider a class of degenerate elliptic equations with arbitrary power degeneration. The issues about the existence, uniqueness, and smoothness of solutions of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration are studied. The two-sided estimates for singular numbers (s-numbers) are obtained. Note that estimates of singular numbers (s-numbers) show the rate of approximation of the found solutions by finite-dimensional subspaces. Here, we also obtain estimates for the eigenvalues. We note that, in this paper, apparently, two-sided estimates of singular numbers (s-numbers) for degenerate elliptic operators are obtained for the first time. At the end of the paper, a symmetric operator is considered, i.e., a self-adjoint case. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
14 pages, 345 KiB  
Article
Lie Symmetry Analysis, Conservation Laws, Power Series Solutions, and Convergence Analysis of Time Fractional Generalized Drinfeld-Sokolov Systems
by Selahattin Gülşen, Shao-Wen Yao and Mustafa Inc
Symmetry 2021, 13(5), 874; https://doi.org/10.3390/sym13050874 - 14 May 2021
Cited by 11 | Viewed by 1373
Abstract
In this work, we investigate invariance analysis, conservation laws, and exact power series solutions of time fractional generalized Drinfeld–Sokolov systems (GDSS) using Lie group analysis. Using Lie point symmetries and the Erdelyi–Kober (EK) fractional differential operator, the time fractional GDSS equation is reduced [...] Read more.
In this work, we investigate invariance analysis, conservation laws, and exact power series solutions of time fractional generalized Drinfeld–Sokolov systems (GDSS) using Lie group analysis. Using Lie point symmetries and the Erdelyi–Kober (EK) fractional differential operator, the time fractional GDSS equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. Moreover, we have constructed conservation laws for time fractional GDSS and obtained explicit power series solutions of the reduced nonlinear ODEs that converge. Lastly, some figures are presented for explicit solutions. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

22 pages, 1243 KiB  
Article
Explicit Identities for 3-Variable Degenerate Hermite Kampé de Fériet Polynomials and Differential Equation Derived from Generating Function
by Kyung-Won Hwang, Young-Soo Seol and Cheon-Seoung Ryoo
Symmetry 2021, 13(1), 7; https://doi.org/10.3390/sym13010007 - 22 Dec 2020
Cited by 5 | Viewed by 1227
Abstract
We get the 3-variable degenerate Hermite Kampé de Fériet polynomials and get symmetric identities for 3-variable degenerate Hermite Kampé de Fériet polynomials. We make differential equations coming from the generating functions of degenerate Hermite Kampé de Fériet polynomials to get some identities for [...] Read more.
We get the 3-variable degenerate Hermite Kampé de Fériet polynomials and get symmetric identities for 3-variable degenerate Hermite Kampé de Fériet polynomials. We make differential equations coming from the generating functions of degenerate Hermite Kampé de Fériet polynomials to get some identities for 3-variable degenerate Hermite Kampé de Fériet polynomials,. Finally, we study the structure and symmetry of pattern about the zeros of the 3-variable degenerate Hermite Kampé de Fériet equations. Full article
(This article belongs to the Special Issue Symmetries in Differential Equation and Application)
Show Figures

Figure 1

Back to TopTop