Special Issue "Differential/Difference Equations: Mathematical Modeling, Oscillation and Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 December 2020).

Special Issue Editors

Dr. Ioannis Dassios
E-Mail Website
Guest Editor
School of Electrical and Electronic Engineering, University College Dublin, D04 Dublin, Ireland
Interests: differential/difference equations; dynamical systems; modelling and stability analysis of electric power systems; mathematics of networks; fractional calculus; mathematical modelling (power systems, materials science, energy, macroeconomics, social media, etc.); optimization for the analysis of large-scale data sets; fluid mechanics; discrete calculus; Bayes control; e-learning
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Dr. Omar Bazighifan
E-Mail Website
Guest Editor
1. Department of Mathematics, Faculty of Science, Hadhramout University, 50512 Hadhramout, Yemen
2. Department of Mathematics, Faculty of Education, Seiyun University, 50512 Seiyun, Yemen
Interests: qualitative theory; ordinary differential equations; functional differential equations; dynamical systems; mathematical modeling
Special Issues and Collections in MDPI journals
Dr. Osama Moaaz
E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
Interests: differental equations; numerical analysis; analysis; applied mathematics; nonlinear dynamics; mathematical modelling; mathematical analysis; stability
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics.

This Special Issue will accept high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology.

Dr. Ioannis Dassios
Dr. Omar Bazighifan
Dr. Osama Moaaz
Guest Editors

Manuscript Submission Information

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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. Mathematics is an international peer-reviewed open access semimonthly 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 1600 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

  • Oscillation theory
  • Differential/difference equations
  • Partial differential equations
  • Dynamical systems
  • Fractional calculus
  • Delays
  • Mathematical modeling and oscillations

Published Papers (18 papers)

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Research

Article
Hyperbolic Center of Mass for a System of Particles in a Two-Dimensional Space with Constant Negative Curvature: An Application to the Curved 2-Body Problem
Mathematics 2021, 9(5), 531; https://doi.org/10.3390/math9050531 - 03 Mar 2021
Viewed by 339
Abstract
In this article, a simple expression for the center of mass of a system of material points in a two-dimensional surface of Gaussian constant negative curvature is given. By using the basic techniques of geometry, we obtained an expression in intrinsic coordinates, and [...] Read more.
In this article, a simple expression for the center of mass of a system of material points in a two-dimensional surface of Gaussian constant negative curvature is given. By using the basic techniques of geometry, we obtained an expression in intrinsic coordinates, and we showed how this extends the definition for the Euclidean case. The argument is constructive and serves to define the center of mass of a system of particles on the one-dimensional hyperbolic sphere LR1. Full article
Article
Some Important Criteria for Oscillation of Non-Linear Differential Equations with Middle Term
Mathematics 2021, 9(4), 346; https://doi.org/10.3390/math9040346 - 09 Feb 2021
Cited by 4 | Viewed by 396
Abstract
In this work, we present new oscillation conditions for the oscillation of the higher-order differential equations with the middle term. We obtain some oscillation criteria by a comparison method with first-order equations. The obtained results extend and simplify known conditions in the literature. [...] Read more.
In this work, we present new oscillation conditions for the oscillation of the higher-order differential equations with the middle term. We obtain some oscillation criteria by a comparison method with first-order equations. The obtained results extend and simplify known conditions in the literature. Furthermore, examining the validity of the proposed criteria is demonstrated via particular examples. Full article
Article
On the Approximate Solution of Partial Integro-Differential Equations Using the Pseudospectral Method Based on Chebyshev Cardinal Functions
Mathematics 2021, 9(3), 286; https://doi.org/10.3390/math9030286 - 01 Feb 2021
Viewed by 356
Abstract
In this paper, we apply the pseudospectral method based on the Chebyshev cardinal function to solve the parabolic partial integro-differential equations (PIDEs). Since these equations play a key role in mathematics, physics, and engineering, finding an appropriate solution is important. We use an [...] Read more.
In this paper, we apply the pseudospectral method based on the Chebyshev cardinal function to solve the parabolic partial integro-differential equations (PIDEs). Since these equations play a key role in mathematics, physics, and engineering, finding an appropriate solution is important. We use an efficient method to solve PIDEs, especially for the integral part. Unlike when using Chebyshev functions, when using Chebyshev cardinal functions it is no longer necessary to integrate to find expansion coefficients of a given function. This reduces the computation. The convergence analysis is investigated and some numerical examples guarantee our theoretical results. We compare the presented method with others. The results confirm the efficiency and accuracy of the method. Full article
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Article
On the Stability of la Cierva’s Autogiro
Mathematics 2020, 8(11), 2032; https://doi.org/10.3390/math8112032 - 15 Nov 2020
Viewed by 462
Abstract
In this paper, we rediscover in detail a series of unknown attempts that some Spanish mathematicians carried out in the 1930s to address a challenge posed by Mr. la Cierva in 1934, which consisted of mathematically justifying the stability of la Cierva’s autogiro, [...] Read more.
In this paper, we rediscover in detail a series of unknown attempts that some Spanish mathematicians carried out in the 1930s to address a challenge posed by Mr. la Cierva in 1934, which consisted of mathematically justifying the stability of la Cierva’s autogiro, the first practical use of the direct-lift rotary wing and one of the first helicopter type aircraft. Full article
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Article
Efficient Computation of Highly Oscillatory Fourier Transforms with Nearly Singular Amplitudes over Rectangle Domains
Mathematics 2020, 8(11), 1930; https://doi.org/10.3390/math8111930 - 02 Nov 2020
Viewed by 445
Abstract
In this paper, we consider fast and high-order algorithms for calculation of highly oscillatory and nearly singular integrals. Based on operators with regard to Chebyshev polynomials, we propose a class of spectral efficient Levin quadrature for oscillatory integrals over rectangle domains, and give [...] Read more.
In this paper, we consider fast and high-order algorithms for calculation of highly oscillatory and nearly singular integrals. Based on operators with regard to Chebyshev polynomials, we propose a class of spectral efficient Levin quadrature for oscillatory integrals over rectangle domains, and give detailed convergence analysis. Furthermore, with the help of adaptive mesh refinement, we are able to develop an efficient algorithm to compute highly oscillatory and nearly singular integrals. In contrast to existing methods, approximations derived from the new approach do not suffer from high oscillatory and singularity. Finally, several numerical experiments are included to illustrate the performance of given quadrature rules. Full article
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Article
Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order
Mathematics 2020, 8(11), 1897; https://doi.org/10.3390/math8111897 - 31 Oct 2020
Cited by 1 | Viewed by 423
Abstract
In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented [...] Read more.
In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding. Full article
Article
Multi-Wavelets Galerkin Method for Solving the System of Volterra Integral Equations
Mathematics 2020, 8(8), 1369; https://doi.org/10.3390/math8081369 - 15 Aug 2020
Viewed by 734
Abstract
In this work, an efficient algorithm is proposed for solving the system of Volterra integral equations based on wavelet Galerkin method. This problem is reduced to a set of algebraic equations using the operational matrix of integration and wavelet transform matrix. For linear [...] Read more.
In this work, an efficient algorithm is proposed for solving the system of Volterra integral equations based on wavelet Galerkin method. This problem is reduced to a set of algebraic equations using the operational matrix of integration and wavelet transform matrix. For linear type, the computational effort decreases by thresholding. The convergence analysis of the proposed scheme has been investigated and it is shown that its convergence is of order O(2Jr), where J is the refinement level and r is the multiplicity of multi-wavelets. Several numerical tests are provided to illustrate the ability and efficiency of the method. Full article
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Article
Oscillatory Behavior of a Type of Generalized Proportional Fractional Differential Equations with Forcing and Damping Terms
Mathematics 2020, 8(6), 1037; https://doi.org/10.3390/math8061037 - 25 Jun 2020
Cited by 1 | Viewed by 625
Abstract
In this paper, we study the oscillatory behavior of solutions for a type of generalized proportional fractional differential equations with forcing and damping terms. Several oscillation criteria are established for the proposed equations in terms of Riemann-Liouville and Caputo settings. The results of [...] Read more.
In this paper, we study the oscillatory behavior of solutions for a type of generalized proportional fractional differential equations with forcing and damping terms. Several oscillation criteria are established for the proposed equations in terms of Riemann-Liouville and Caputo settings. The results of this paper generalize some existing theorems in the literature. Indeed, it is shown that for particular choices of parameters, the obtained conditions in this paper reduce our theorems to some known results. Numerical examples are constructed to demonstrate the effectiveness of the our main theorems. Furthermore, we present and illustrate an example which does not satisfy the assumptions of our theorem and whose solution demonstrates nonoscillatory behavior. Full article
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Article
The Analytical Analysis of Time-Fractional Fornberg–Whitham Equations
Mathematics 2020, 8(6), 987; https://doi.org/10.3390/math8060987 - 16 Jun 2020
Cited by 5 | Viewed by 648
Abstract
This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of [...] Read more.
This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems. Full article
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Article
Establishing New Criteria for Oscillation of Odd-Order Nonlinear Differential Equations
Mathematics 2020, 8(6), 937; https://doi.org/10.3390/math8060937 - 08 Jun 2020
Cited by 4 | Viewed by 441
Abstract
By establishing new conditions for the non-existence of so-called Kneser solutions, we can generate sufficient conditions to ensure that all solutions of odd-order equations are oscillatory. Our results improve and expand the previous results in the literature. Full article
Article
Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation
Mathematics 2020, 8(6), 908; https://doi.org/10.3390/math8060908 - 03 Jun 2020
Cited by 24 | Viewed by 815
Abstract
In this paper, a new solution process of ( 1 / G ) -expansion and ( G / G , 1 / G ) -expansion methods has been proposed for the analytic solution of the Zhiber-Shabat (Z-S) equation. Rather than the classical ( G / G , 1 / G ) -expansion method, a solution function in different formats has been produced with the help of the proposed process. New complex rational, hyperbolic, rational and trigonometric types solutions of the Z-S equation have been constructed. By giving arbitrary values to the constants in the obtained solutions, it can help to add physical meaning to the traveling wave solutions, whereas traveling wave has an important place in applied sciences and illuminates many physical phenomena. 3D, 2D and contour graphs are displayed to show the stationary wave or the state of the wave at any moment with the values given to these constants. Conditions that guarantee the existence of traveling wave solutions are given. Comparison of ( G / G , 1 / G ) -expansion method and ( 1 / G ) -expansion method, which are important instruments in the analytical solution, has been made. In addition, the advantages and disadvantages of these two methods have been discussed. These methods are reliable and efficient methods to obtain analytic solutions of nonlinear evolution equations (NLEEs). Full article
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Article
Improved Oscillation Criteria for 2nd-Order Neutral Differential Equations with Distributed Deviating Arguments
Mathematics 2020, 8(5), 849; https://doi.org/10.3390/math8050849 - 23 May 2020
Cited by 4 | Viewed by 611
Abstract
In this study, we establish new sufficient conditions for oscillation of solutions of second-order neutral differential equations with distributed deviating arguments. By employing a refinement of the Riccati transformations and comparison principles, we obtain new oscillation criteria that complement and improve some results [...] Read more.
In this study, we establish new sufficient conditions for oscillation of solutions of second-order neutral differential equations with distributed deviating arguments. By employing a refinement of the Riccati transformations and comparison principles, we obtain new oscillation criteria that complement and improve some results reported in the literature. Examples are provided to illustrate the main results. Full article
Article
A Class of Quantum Briot–Bouquet Differential Equations with Complex Coefficients
Mathematics 2020, 8(5), 794; https://doi.org/10.3390/math8050794 - 14 May 2020
Cited by 1 | Viewed by 561
Abstract
Quantum inequalities (QI) are local restraints on the magnitude and range of formulas. Quantum inequalities have been established to have a different range of applications. In this paper, we aim to introduce a study of QI in a complex domain. The idea basically, [...] Read more.
Quantum inequalities (QI) are local restraints on the magnitude and range of formulas. Quantum inequalities have been established to have a different range of applications. In this paper, we aim to introduce a study of QI in a complex domain. The idea basically, comes from employing the notion of subordination. We shall formulate a new q-differential operator (generalized of Dunkl operator of the first type) and employ it to define the classes of QI. Moreover, we employ the q-Dunkl operator to extend the class of Briot–Bouquet differential equations. We investigate the upper solution and exam the oscillation solution under some analytic functions. Full article
Article
New Oscillation Criteria for Advanced Differential Equations of Fourth Order
Mathematics 2020, 8(5), 728; https://doi.org/10.3390/math8050728 - 06 May 2020
Cited by 21 | Viewed by 691
Abstract
The main objective of this paper is to establish new oscillation results of solutions to a class of fourth-order advanced differential equations with delayed arguments. The key idea of our approach is to use the Riccati transformation and the theory of comparison with [...] Read more.
The main objective of this paper is to establish new oscillation results of solutions to a class of fourth-order advanced differential equations with delayed arguments. The key idea of our approach is to use the Riccati transformation and the theory of comparison with first and second-order delay equations. Four examples are provided to illustrate the main results. Full article
Article
New Results for Kneser Solutions of Third-Order Nonlinear Neutral Differential Equations
Mathematics 2020, 8(5), 686; https://doi.org/10.3390/math8050686 - 01 May 2020
Cited by 5 | Viewed by 453
Abstract
In this paper, we consider a certain class of third-order nonlinear delay differential equations r w α v + q v x β ς v = 0 , for v v 0 , where w v = x v + p v x ϑ v . We obtain new criteria for oscillation of all solutions of this nonlinear equation. Our results complement and improve some previous results in the literature. An example is considered to illustrate our main results. Full article
Article
Improved Approach for Studying Oscillatory Properties of Fourth-Order Advanced Differential Equations with p-Laplacian Like Operator
Mathematics 2020, 8(5), 656; https://doi.org/10.3390/math8050656 - 26 Apr 2020
Cited by 18 | Viewed by 640
Abstract
This paper aims to study the oscillatory properties of fourth-order advanced differential equations with p-Laplacian like operator. By using the technique of Riccati transformation and the theory of comparison with first-order delay equations, we will establish some new oscillation criteria for this [...] Read more.
This paper aims to study the oscillatory properties of fourth-order advanced differential equations with p-Laplacian like operator. By using the technique of Riccati transformation and the theory of comparison with first-order delay equations, we will establish some new oscillation criteria for this equation. Some examples are considered to illustrate the main results. Full article
Article
Oscillation Theorems for Nonlinear Differential Equations of Fourth-Order
Mathematics 2020, 8(4), 520; https://doi.org/10.3390/math8040520 - 03 Apr 2020
Cited by 23 | Viewed by 712
Abstract
We study the oscillatory behavior of a class of fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation with middle term. Our theorems extend and complement a number of related results reported in the literature. One example is [...] Read more.
We study the oscillatory behavior of a class of fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation with middle term. Our theorems extend and complement a number of related results reported in the literature. One example is provided to illustrate the main results. Full article
Article
New Aspects for Non-Existence of Kneser Solutions of Neutral Differential Equations with Odd-Order
Mathematics 2020, 8(4), 494; https://doi.org/10.3390/math8040494 - 02 Apr 2020
Cited by 9 | Viewed by 488
Abstract
Some new oscillatory and asymptotic properties of solutions of neutral differential equations with odd-order are established. Through the new results, we give sufficient conditions for the oscillation of all solutions of the studied equations, and this is an improvement of the relevant results. [...] Read more.
Some new oscillatory and asymptotic properties of solutions of neutral differential equations with odd-order are established. Through the new results, we give sufficient conditions for the oscillation of all solutions of the studied equations, and this is an improvement of the relevant results. The efficiency of the obtained criteria is illustrated via example. Full article
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