Special Issue "Advances in Differential and Difference Equations with Applications 2021"

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

Deadline for manuscript submissions: 31 May 2022.

Special Issue Editor

Prof. Dr. Dumitru Baleanu
grade E-Mail Website
Guest Editor
1. Institute of Space Sciences, P.O. BOX MG-23, RO-077125 Magurele-Bucharest, Romania;
2. Department of Mathematics, Cankaya University, Ankara 06530, Turkey
Interests: fractional dynamics; fractional differential equations; discrete mathematics; fractals; image processing; bio-informatics; mathematical biology; soliton theory; Lie symmetry; dynamic systems on time scales; computational complexity; the wavelet method
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

It is very well known that differential and difference equations are extreme representations of complex dynamical systems.

During the last few years, the theory of fractional differentiation has been successfully applied to the study of anomalous social and physical behaviors, where scaling power law of fractional order appears universal as an empirical description of such complex phenomena. Recently, the difference counterpart of fractional calculus has started to be intensively used for a better characterization of some real-world phenomena. Systems of delay differential equations have started to occupy a central place of importance in various areas of science, particularly in biological areas.

This Special Issue deals with the theory and application of differential and difference equations, especially in science and engineering, and will accept high-quality papers having original research results.

The purpose of this Special Issue is to bring mathematicians together with physicists, engineers, as well as other scientists, for whom differential and difference equations are valuable research tools.

This issue is a continuation of the previous successful Special Issue “Advances in Differential and Difference Equations with Applications 2020”.

Prof. Dr. Dumitru Baleanu
Guest Editor

Manuscript Submission Information

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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

  • Differential equations
  • Fractional differential equations
  • Difference equations
  • Discrete fractional equations
  • Delay differential equations
  • Mathematical Physics

Published Papers (8 papers)

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Research

Article
Sturm–Liouville Differential Equations Involving Kurzweil–Henstock Integrable Functions
Mathematics 2021, 9(12), 1403; https://doi.org/10.3390/math9121403 - 17 Jun 2021
Viewed by 171
Abstract
In this paper, we give sufficient conditions for the existence and uniqueness of the solution of Sturm–Liouville equations subject to Dirichlet boundary value conditions and involving Kurzweil–Henstock integrable functions on unbounded intervals. We also present a finite element method scheme for Kurzweil–Henstock integrable [...] Read more.
In this paper, we give sufficient conditions for the existence and uniqueness of the solution of Sturm–Liouville equations subject to Dirichlet boundary value conditions and involving Kurzweil–Henstock integrable functions on unbounded intervals. We also present a finite element method scheme for Kurzweil–Henstock integrable functions. Full article
Article
Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity
Mathematics 2021, 9(12), 1398; https://doi.org/10.3390/math9121398 - 16 Jun 2021
Viewed by 155
Abstract
We consider the orbital stability of solitary waves to the double dispersion equation uttuxx+h1uxxxxh2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,aR,bR,b0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c20,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3. Full article
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Article
Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators
Mathematics 2021, 9(11), 1193; https://doi.org/10.3390/math9111193 - 25 May 2021
Viewed by 276
Abstract
The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by [...] Read more.
The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations. Full article
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Article
More Effective Results for Testing Oscillation of Non-Canonical Neutral Delay Differential Equations
Mathematics 2021, 9(10), 1114; https://doi.org/10.3390/math9101114 - 14 May 2021
Viewed by 228
Abstract
In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some [...] Read more.
In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results. Full article
Article
New Oscillation Theorems for Second-Order Differential Equations with Canonical and Non-Canonical Operator via Riccati Transformation
Mathematics 2021, 9(10), 1111; https://doi.org/10.3390/math9101111 - 14 May 2021
Cited by 3 | Viewed by 461
Abstract
In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form (a(ξ)((v(ξ)+b(ξ)v(ϑ(ξ)))))+c(ξ)G1(v(κ(ξ)))+d(ξ)G2(v(ς(ξ)))=0 under canonical and non-canonical operators, that is, ξ0dξa(ξ)= and ξ0dξa(ξ)<. We use the Riccati transformation to prove our main results. Furthermore, some examples are provided to show the effectiveness and feasibility of the main results. Full article
Article
Fractional System of Korteweg-De Vries Equations via Elzaki Transform
Mathematics 2021, 9(6), 673; https://doi.org/10.3390/math9060673 - 22 Mar 2021
Viewed by 467
Abstract
In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in [...] Read more.
In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in series form to analyze the analytical results of fractional-order coupled Korteweg-de Vries equations. To understand the analytical procedure of Iteration transform method, some numerical problems are presented for the analytical result of fractional-order coupled Korteweg-de Vries equations. It is also demonstrated that the current technique’s solutions are in good agreement with the exact results. The numerical solutions show that only a few terms are sufficient for obtaining an approximate result, which is efficient, accurate, and reliable. Full article
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Article
Existence and Stability Results on Hadamard Type Fractional Time-Delay Semilinear Differential Equations
Mathematics 2020, 8(8), 1242; https://doi.org/10.3390/math8081242 - 30 Jul 2020
Viewed by 523
Abstract
A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed. This combines the classic Mittag–Leffler type matrix function with a logarithm and delayed Mittag–Leffler type matrix function. With the help of this introduced delayed perturbation of the Mittag–Leffler type matrix [...] Read more.
A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed. This combines the classic Mittag–Leffler type matrix function with a logarithm and delayed Mittag–Leffler type matrix function. With the help of this introduced delayed perturbation of the Mittag–Leffler type matrix function with a logarithm, we provide an explicit form for solutions to non-homogeneous Hadamard-type fractional time-delay linear differential equations. We also examine the existence, uniqueness, and Ulam–Hyers stability of Hadamard-type fractional time-delay nonlinear equations. Full article
Article
More Effective Criteria for Oscillation of Second-Order Differential Equations with Neutral Arguments
Mathematics 2020, 8(6), 986; https://doi.org/10.3390/math8060986 - 16 Jun 2020
Cited by 8 | Viewed by 505
Abstract
The motivation for this paper is to create new criteria for oscillation of solutions of second-order nonlinear neutral differential equations. In more than one respect, our results improve several related ones in the literature. As proof of the effectiveness of the new criteria, [...] Read more.
The motivation for this paper is to create new criteria for oscillation of solutions of second-order nonlinear neutral differential equations. In more than one respect, our results improve several related ones in the literature. As proof of the effectiveness of the new criteria, we offer more than one practical example. Full article
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