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Symmetry
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Differential Equations and Applied Mathematics
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Differential Equations and Applied Mathematics

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

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 15421

Special Issue Editors

Dr. Manuel Duarte Ortigueira

E-Mail Website
Guest Editor
Centre of Technology and Systems-UNINOVA, NOVA School of Science and Technology, NOVA University of Lisbon, Quinta da Torre, 2829-516 Caparica, Portugal
Interests: signal processing; fractional signals and systems; EEG and ECG processing
Special Issues, Collections and Topics in MDPI journals
Dr. Jose M. Carvalho

E-Mail Website
Guest Editor
LAQV-REQUIMTE, Chemistry Department, NOVA School of Science and Technology of NOVA University of Lisbon, Caparica, Portugal
Interests: bioengineering; mathematical chemistry; chemical kinetics; queueing theory; transient waiting lines; master equations; exactly solved models; functional equations; difference-differential equations; special functions; differential equations with deviating Arguments

Special Issue Information

Dear Colleagues,

Complexity may arise in all areas of Applied Sciences where the most suitable models are described by nonlinear equations, usually very difficult to solve, and often nearly intractable. However, distinct areas may lead to similar models. Consequently, we are led to search for similar formalisms in other areas so that we can find resolution methods that we can adapt to our model. As an example, the analysis of a model in Life Sciences can lead to master equations of complexity equivalent to the systems of differential-difference equations, found in the analysis of transient waiting lines or in complicated chemical network kinetics (for instance, in polymerisation and other aggregation processes kinetics).  However, these similar procedures are found in dispersed literature and it is very common that they are unknown outside their original scientific area. One purpose of the current Special Issue is to gather such algorithms, compare them, perform their symmetry analysis and look for other alternative methods.

Exact explicit closed-form solutions, if possible, allow us to know everything about a system, and starting from a guessed solution, seem to be a promising and challenging method. Therefore, exactly solved models, usually difficult to obtain, are also another objective of this Special Issue.

Prof. Dr. Manuel Ortigueira
Dr. Jose M. Carvalho
Guest Editors

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

  • master equations
  • exactly solved models
  • symmetry analysis
  • functional equations
  • nonlinear difference equations
  • delay-differential equations
  • stochastic differential equations
  • differential-difference equations
  • differential equations with deviating arguments

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Published Papers (11 papers)

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Research

21 pages, 2437 KiB  
Article
On a Hierarchy of Vector Derivative Nonlinear Schrödinger Equations
by Aleksandr O. Smirnov, Eugene A. Frolov and Lada L. Dmitrieva
Symmetry 2024, 16(1), 60; https://doi.org/10.3390/sym16010060 - 2 Jan 2024
Viewed by 1049
Abstract
We propose a new hierarchy of the vector derivative nonlinear Schrödinger equations and consider the simplest multiphase solutions of this hierarchy. The study of the simplest solutions of these equations led to the following results. First, the three-leaf spectral curves [...] Read more.
We propose a new hierarchy of the vector derivative nonlinear Schrödinger equations and consider the simplest multiphase solutions of this hierarchy. The study of the simplest solutions of these equations led to the following results. First, the three-leaf spectral curves Γ={(μ,λ)} of the simplest multiphase solutions have a quite simple symmetry. They are invariant with respect to holomorphic involution τ. The type of this involution depends on the genus of the spectral curve. Or the involution has the form τ:(μ,λ)→(μ,−λ), or τ:(μ,λ)→(−μ,−λ). The presence of symmetry leads to the fact that the dynamics of the solution is determined not by the entire spectral curve Γ, but by its factor Γ/τ, which has a smaller genus. Secondly, it turned out that the dynamics of the two-component vector p=(p1,p2)t is determined, first of all, by the dynamics of its length |p|. Independent equations determine the dependence of the direction of the vector p from its length. In cases where the direction of the vector p is fixed, the corresponding spectral curve splits into separate components. In conclusion, we note that, as in the case of the Manakov system, the equation of the spectral curve is invariant with respect to the orthogonal transformation of the vector solutions. I.e., the solution can be found from the spectral curve up to the orthogonal transformation. This fact indicates that the spectral curve does not depend on the individual components of the solution, but on their symmetric functions. Thus, the spectral data of multiphase solutions have two symmetries. These symmetries make it difficult to reconstruct signals from their spectral data. The work contains examples illustrating these statements. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
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15 pages, 312 KiB  
Article
New Results on the Oscillation of Solutions of Third-Order Differential Equations with Multiple Delays
by Najiyah Omar, Osama Moaaz, Ghada AlNemer and Elmetwally M. Elabbasy
Symmetry 2023, 15(10), 1920; https://doi.org/10.3390/sym15101920 - 15 Oct 2023
Cited by 1 | Viewed by 1116
Abstract
This study aims to examine the oscillatory behavior of third-order differential equations involving various delays within the context of functional differential equations of the neutral type. The oscillation criteria for the solutions of our equation have been obtained in this study to extend [...] Read more.
This study aims to examine the oscillatory behavior of third-order differential equations involving various delays within the context of functional differential equations of the neutral type. The oscillation criteria for the solutions of our equation have been obtained in this study to extend and supplement existing findings in the literature. In this study, a technique that relies on repeatedly improving monotonic properties was used in order to exclude positive solutions to the studied equation. Negative solutions are excluded based on the symmetry between the positive and negative solutions. Our results are important because they become sharper when applied to a Euler-type equation as compared to previous studies of the same equation. The significance of the findings was illustrated through the application of these findings to specific cases of the investigated equation. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
23 pages, 338 KiB  
Article
Existence and General Decay of Solutions for a Weakly Coupled System of Viscoelastic Kirchhoff Plate and Wave Equations
by Zayd Hajjej
Symmetry 2023, 15(10), 1917; https://doi.org/10.3390/sym15101917 - 14 Oct 2023
Cited by 1 | Viewed by 875
Abstract
In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on [...] Read more.
In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on a more general type of relaxation functions, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
27 pages, 6300 KiB  
Article
Methodology for Solving Engineering Problems of Burgers–Huxley Coupled with Symmetric Boundary Conditions by Means of the Network Simulation Method
by Juan Francisco Sánchez-Pérez, Fulgencio Marín-García, Enrique Castro, Gonzalo García-Ros, Manuel Conesa and Joaquín Solano-Ramírez
Symmetry 2023, 15(9), 1740; https://doi.org/10.3390/sym15091740 - 11 Sep 2023
Cited by 4 | Viewed by 938
Abstract
The Burgers–Huxley equation is a partial differential equation which is based on the Burgers equation, involving diffusion, accumulation, drag, and species generation or sink phenomena. This equation is commonly used in fluid mechanics, air pollutant emissions, chloride diffusion in concrete, non-linear acoustics, and [...] Read more.
The Burgers–Huxley equation is a partial differential equation which is based on the Burgers equation, involving diffusion, accumulation, drag, and species generation or sink phenomena. This equation is commonly used in fluid mechanics, air pollutant emissions, chloride diffusion in concrete, non-linear acoustics, and other areas. A general methodology is proposed in this work to solve the mentioned equation or coupled systems formed by it using the network simulation method. Additionally, the implementation of the most common possible boundary conditions in different engineering problems is indicated, including the Neumann condition that enables symmetry to be applied to the problem, reducing computation times. The method consists mainly of establishing an analogy between the variables of the differential equations and the electrical voltage at a central node. The methodology is also explained in detail, facilitating its implementation to similar engineering problems, since the equivalence, for example, between the different types of spatial and time derivatives and its correspondence with the electrical device is detailed. As an example, several cases of both the equation and a coupled system are solved by varying the boundary conditions on one side and applying symmetry on the other. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
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15 pages, 368 KiB  
Article
A Pair of Optimized Nyström Methods with Symmetric Hybrid Points for the Numerical Solution of Second-Order Singular Boundary Value Problems
by Higinio Ramos, Mufutau Ajani Rufai and Bruno Carpentieri
Symmetry 2023, 15(9), 1720; https://doi.org/10.3390/sym15091720 - 7 Sep 2023
Cited by 2 | Viewed by 1013
Abstract
This paper introduces an efficient approach for solving Lane–Emden–Fowler problems. Our method utilizes two Nyström schemes to perform the integration. To overcome the singularity at the left end of the interval, we combine an optimized scheme of Nyström type with a set of [...] Read more.
This paper introduces an efficient approach for solving Lane–Emden–Fowler problems. Our method utilizes two Nyström schemes to perform the integration. To overcome the singularity at the left end of the interval, we combine an optimized scheme of Nyström type with a set of Nyström formulas that are used at the fist subinterval. The optimized technique is obtained after imposing the vanishing of some of the local truncation errors, which results in a set of symmetric hybrid points. By solving an algebraic system of equations, our proposed approach generates simultaneous approximations at all grid points, resulting in a highly effective technique that outperforms several existing numerical methods in the literature. To assess the efficiency and accuracy of our approach, we perform some numerical tests on diverse real-world problems, including singular boundary value problems (SBVPs) from chemical kinetics. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
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12 pages, 320 KiB  
Article
Several Characterizations of Δh-Doped Special Polynomials Associated with Appell Sequences
by Rabab Alyusof and Shahid Ahmmad Wani
Symmetry 2023, 15(7), 1315; https://doi.org/10.3390/sym15071315 - 27 Jun 2023
Cited by 3 | Viewed by 798
Abstract
The study presented in this paper follows the line of research created by the fact that by employing the monomiality principle, new outcomes are produced. This article deals with the inducement of Δh tangent-based Appell polynomials and derivation of certain of its [...] Read more.
The study presented in this paper follows the line of research created by the fact that by employing the monomiality principle, new outcomes are produced. This article deals with the inducement of Δh tangent-based Appell polynomials and derivation of certain of its characterizations such as explicit form, determinant form, monomiality principle, etc. These polynomials are designed to exhibit certain symmetries themselves or to capture and describe symmetrical patterns in mathematical structures. Further, certain members of Δh Appell polynomials such as Δh Bernoulli, Euler, and Genocchi polynomials are taken, and their corresponding results are obtained. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
9 pages, 268 KiB  
Article
An Existence Result of Positive Solutions for the Bending Elastic Beam Equations
by Yongxiang Li and Dan Wang
Symmetry 2023, 15(2), 405; https://doi.org/10.3390/sym15020405 - 3 Feb 2023
Cited by 3 | Viewed by 975
Abstract
This paper is concerned with the existence of positive solutions to the fourth-order boundary value problem u(4)(x)=f(x,u(x),u″(x)) on the interval [0, [...] Read more.
This paper is concerned with the existence of positive solutions to the fourth-order boundary value problem u(4)(x)=f(x,u(x),u″(x)) on the interval [0, 1] with the boundary condition u(0)=u(1)=u″(0)=u″(1)=0, which models a statically bending elastic beam whose two ends are simply supported. Without assuming that the nonlinearity f(x, u, v) is nonnegative, an existence result of positive solutions is obtained under the inequality conditions that |(u, v)| is small or large enough. The discussion is based on the method of lower and upper solutions. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
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19 pages, 624 KiB  
Article
Analytical Solutions for Fractional Differential Equations Using a General Conformable Multiple Laplace Transform Decomposition Method
by Honggang Jia
Symmetry 2023, 15(2), 389; https://doi.org/10.3390/sym15020389 - 1 Feb 2023
Cited by 2 | Viewed by 2434
Abstract
In this paper, a new analytical technique is proposed for solving fractional partial differential equations. This method is referred to as the general conformal multiple Laplace transform decomposition method. It is a combination of the multiple Laplace transform method and the Adomian decomposition [...] Read more.
In this paper, a new analytical technique is proposed for solving fractional partial differential equations. This method is referred to as the general conformal multiple Laplace transform decomposition method. It is a combination of the multiple Laplace transform method and the Adomian decomposition method. The main theoretical results of using this method are presented. In addition, illustrative examples are provided to demonstrate the validity and symmetry of the presented method. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
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12 pages, 343 KiB  
Article
Construction of General Implicit-Block Method with Three-Points for Solving Seventh-Order Ordinary Differential Equations
by Mohammed Yousif Turki, Mohammed Mahmood Salih and Mohammed S. Mechee
Symmetry 2022, 14(8), 1605; https://doi.org/10.3390/sym14081605 - 4 Aug 2022
Viewed by 1390
Abstract
In order to solve general seventh-order ordinary differential equations (ODEs), this study will develop an implicit block method with three points of the form [...] Read more.
In order to solve general seventh-order ordinary differential equations (ODEs), this study will develop an implicit block method with three points of the form y(7)(ξ)=f(ξ,y(ξ),y′(ξ),y″(ξ),y‴(ξ),y(4)(ξ),y(5)(ξ),y(6)(ξ)) directly. The general implicit block method with Hermite interpolation in three points (GIBM3P) has been derived to solve general seventh-order initial value problems (IVPs) using the basic functions of Hermite interpolating polynomials. A block multi-step method is constructed to be suitable with the numerical approximation at three points. However, the construction of the new method has been presented while the numerical results of the implementations are used to prove the efficiency and the accuracy of the proposed method which compared with the RK and RKM numerical methods together to analytical method. We established the characteristics of the proposed method, including order and zero-stability. Applications of various IVP problems are also discussed, and the outcomes are very encouraging for the suggested approach. The proposed GIBM3P method yields more accurate numerical solutions to the test problems than the existing RK method, which are in good agreement with analytical and RKM method solutions. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
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20 pages, 456 KiB  
Article
Analytically Pricing Formula for Contingent Claim with Polynomial Payoff under ECIR Process
by Fukiat Nualsri and Khamron Mekchay
Symmetry 2022, 14(5), 933; https://doi.org/10.3390/sym14050933 - 4 May 2022
Cited by 4 | Viewed by 1839
Abstract
Contingent claims, such as bonds, swaps, and options, are financial derivatives whose payoffs depend on uncertain future real values of underlying assets which emphasize various real-world applications. In general, valuations for contingent claims can be derived from the conditional expectations of underlying assets. [...] Read more.
Contingent claims, such as bonds, swaps, and options, are financial derivatives whose payoffs depend on uncertain future real values of underlying assets which emphasize various real-world applications. In general, valuations for contingent claims can be derived from the conditional expectations of underlying assets. For a simple process, the moments are usually directly obtained from its transition probability density function (PDF). However, if the transition PDF is unavailable in simple form, the derivations of the moments and the contingent claim prices will not be accessible in closed forms. This paper provides a closed-form formula for pricing contingent claims with nonlinear payoff under a no-arbitrage principle when underlying assets follow the extended Cox–Ingersoll–Ross (ECIR) process with the symmetry properties of the Brownian motion. The formula proposed here is a consequence of successfully solving an explicit solution for a system of recurrence partial differential equations in which its solution subtly depends on the conditional moments. For the particular CIR process, we obtain simple closed-form formulas by solving the Riccati differential equation. Furthermore, we carry out a complete investigation of the convergent case for those formulas. In case such as that of the unsolvable Riccati differential equation, ECIR case, a numerical method for numerically evaluating the mentioned analytical formulas and numerical validations for the formulas are examined. The validity and efficiency of the formulas are numerically demonstrated by comparison with results from Monte Carlo simulations for various modeling parameters. Finally, the proposed formula is applied to the value contingent claims such as coupon bonds, interest rate swaps, and arrears swaps. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
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21 pages, 351 KiB  
Article
On a Nonlocal Coupled System of Hilfer Generalized Proportional Fractional Differential Equations
by Ayub Samadi, Sotiris K. Ntouyas and Jessada Tariboon
Symmetry 2022, 14(4), 738; https://doi.org/10.3390/sym14040738 - 4 Apr 2022
Cited by 10 | Viewed by 1602
Abstract
This paper studies the existence and uniqueness of solutions for a coupled system of Hilfer-type generalized proportional fractional differential equations supplemented with nonlocal asymmetric multipoint boundary conditions. We consider both the scalar and the Banach space case. We apply standard fixed-point theorems to [...] Read more.
This paper studies the existence and uniqueness of solutions for a coupled system of Hilfer-type generalized proportional fractional differential equations supplemented with nonlocal asymmetric multipoint boundary conditions. We consider both the scalar and the Banach space case. We apply standard fixed-point theorems to derive the desired results. In the scalar case, we apply Banach’s fixed-point theorem, the Leray–Schauder alternative, and Krasnosel’skiĭ’s fixed-point theorem. The Banach space case is based on Mönch’s fixed-point theorem and the technique of the measure of noncompactness. Examples illustrating the main results are presented. Symmetric distance between itself and its derivative can be investigated by replacing the proportional number equal to one half. Full article
(This article belongs to the Special Issue Differential Equations and Applied Mathematics)
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