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Advances in Differential and Difference Equations with Applications 2019
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Advances in Differential and Difference Equations with Applications 2019

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 May 2019) | Viewed by 43368

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, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

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

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 appear 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.

Prof. Dr. Dumitru Baleanu
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. 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 2600 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 (16 papers)

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Research

15 pages, 1796 KiB  
Article
Highly Accurate Numerical Technique for Population Models via Rational Chebyshev Collocation Method
by Mohamed Abdel-Latif Ramadan, Dumitru Baleanu and Mahmoud Abdel-Ghany Nassar
Mathematics 2019, 7(10), 913; https://0-doi-org.brum.beds.ac.uk/10.3390/math7100913 - 01 Oct 2019
Cited by 5 | Viewed by 1849
Abstract
The present work introduces the application of rational Chebyshev collocation technique for approximating bio-mathematical problems of continuous population models for single and interacting species (C.P.M.). We study systematically the logistic growth model in a population, prey-predator model: Lotka-Volterra system (L.V.M.), the simple two-species [...] Read more.
The present work introduces the application of rational Chebyshev collocation technique for approximating bio-mathematical problems of continuous population models for single and interacting species (C.P.M.). We study systematically the logistic growth model in a population, prey-predator model: Lotka-Volterra system (L.V.M.), the simple two-species Lotka-Volterra competition model (L.V.C.M.) and the prey-predator model with limit cycle periodic behavior (P.P.M.). For testing the accuracy, the numerical results for our method and others existing methods as well as the exact solution are compared. The obtained numerical results indicate the ability, the reliability and the accuracy of the present method. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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24 pages, 308 KiB  
Article
Asymptotic Almost-Periodicity for a Class of Weyl-Like Fractional Difference Equations
by Junfei Cao, Amar Debbouche and Yong Zhou
Mathematics 2019, 7(7), 592; https://0-doi-org.brum.beds.ac.uk/10.3390/math7070592 - 01 Jul 2019
Cited by 2 | Viewed by 2082
Abstract
This work deal with asymptotic almost-periodicity of mild solutions for a class of difference equations with a Weyl-like fractional difference in Banach space. Based on a combination of a decomposition technique and the Krasnoselskii’s fixed point theorem, we establish some new existence theorems [...] Read more.
This work deal with asymptotic almost-periodicity of mild solutions for a class of difference equations with a Weyl-like fractional difference in Banach space. Based on a combination of a decomposition technique and the Krasnoselskii’s fixed point theorem, we establish some new existence theorems of mild solutions with asymptotic almost-periodicity. Our results extend some related conclusions, since (locally) Lipschitz assumption on the nonlinear perturbation is not needed and with Lipschitz assumption becoming a special case. An example is presented to validate the application of our results. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
12 pages, 831 KiB  
Article
Some Qualitative Behavior of Solutions of General Class of Difference Equations
by Osama Moaaz, Dimplekumar Chalishajar and Omar Bazighifan
Mathematics 2019, 7(7), 585; https://0-doi-org.brum.beds.ac.uk/10.3390/math7070585 - 01 Jul 2019
Cited by 36 | Viewed by 2717
Abstract
In this work, we consider the general class of difference equations (covered many equations that have been studied by other authors or that have never been studied before), as a means of establishing general theorems, for the asymptotic behavior of its solutions. Namely, [...] Read more.
In this work, we consider the general class of difference equations (covered many equations that have been studied by other authors or that have never been studied before), as a means of establishing general theorems, for the asymptotic behavior of its solutions. Namely, we state new necessary and sufficient conditions for local asymptotic stability of these equations. In addition, we study the periodic solution with period two and three. Our results essentially extend and improve the earlier ones. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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12 pages, 288 KiB  
Article
A New Algorithm for Fractional Riccati Type Differential Equations by Using Haar Wavelet
by M. Motawi Khashan, Rohul Amin and Muhammed I. Syam
Mathematics 2019, 7(6), 545; https://0-doi-org.brum.beds.ac.uk/10.3390/math7060545 - 14 Jun 2019
Cited by 19 | Viewed by 3005
Abstract
In this paper, a new collocation method based on Haar wavelet is developed for numerical solution of Riccati type differential equations with non-integer order. The fractional derivatives are considered in the Caputo sense. The method is applied to one test problem. The maximum [...] Read more.
In this paper, a new collocation method based on Haar wavelet is developed for numerical solution of Riccati type differential equations with non-integer order. The fractional derivatives are considered in the Caputo sense. The method is applied to one test problem. The maximum absolute estimated error functions are calculated, and the performance of the process is demonstrated by calculating the maximum absolute estimated error functions for a distinct number of nodal points. The results show that the method is applicable and efficient. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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19 pages, 1608 KiB  
Article
An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations
by Hassan Khan, Rasool Shah, Poom Kumam, Dumitru Baleanu and Muhammad Arif
Mathematics 2019, 7(5), 426; https://0-doi-org.brum.beds.ac.uk/10.3390/math7050426 - 13 May 2019
Cited by 32 | Viewed by 3446
Abstract
In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-order telegraph equations by using the proposed method. Some numerical examples are [...] Read more.
In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-order telegraph equations by using the proposed method. Some numerical examples are presented to understand the procedure of the Laplace-Adomian decomposition method. As the Laplace-Adomian decomposition procedure has shown the least volume of calculations and high rate of convergence compared to other analytical techniques, the Laplace-Adomian decomposition method is considered to be one of the best analytical techniques for solving fractional-order, non-linear partial differential equations—particularly the fractional-order telegraph equation. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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20 pages, 1259 KiB  
Article
Regularization and Error Estimate for the Poisson Equation with Discrete Data
by Nguyen Anh Triet, Nguyen Duc Phuong, Van Thinh Nguyen and Can Nguyen-Huu
Mathematics 2019, 7(5), 422; https://0-doi-org.brum.beds.ac.uk/10.3390/math7050422 - 10 May 2019
Viewed by 2476
Abstract
In this work, we focus on the Cauchy problem for the Poisson equation in the two dimensional domain, where the initial data is disturbed by random noise. In general, the problem is severely ill-posed in the sense of Hadamard, i.e., the solution does [...] Read more.
In this work, we focus on the Cauchy problem for the Poisson equation in the two dimensional domain, where the initial data is disturbed by random noise. In general, the problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. To regularize the instable solution of the problem, we have applied a nonparametric regression associated with the truncation method. Eventually, a numerical example has been carried out, the result shows that our regularization method is converged; and the error has been enhanced once the number of observation points is increased. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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11 pages, 498 KiB  
Article
Application of the Laplace Homotopy Perturbation Method to the Black–Scholes Model Based on a European Put Option with Two Assets
by Din Prathumwan and Kamonchat Trachoo
Mathematics 2019, 7(4), 310; https://0-doi-org.brum.beds.ac.uk/10.3390/math7040310 - 27 Mar 2019
Cited by 5 | Viewed by 3416
Abstract
In this paper, the Laplace homotopy perturbation method (LHPM) is applied to obtain the approximate solution of Black–Scholes partial differential equations for a European put option with two assets. Different from all other approximation methods, LHPM provides a simple way to get the [...] Read more.
In this paper, the Laplace homotopy perturbation method (LHPM) is applied to obtain the approximate solution of Black–Scholes partial differential equations for a European put option with two assets. Different from all other approximation methods, LHPM provides a simple way to get the explicit solution which is represented in the form of a Mellin–Ross function. The numerical examples represent that the solution from the proposed method is easy and effective. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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11 pages, 275 KiB  
Article
Boundary Value Problems for Hybrid Caputo Fractional Differential Equations
by Zidane Baitiche, Kaddour Guerbati, Mouffak Benchohra and Yong Zhou
Mathematics 2019, 7(3), 282; https://0-doi-org.brum.beds.ac.uk/10.3390/math7030282 - 19 Mar 2019
Cited by 23 | Viewed by 2836
Abstract
In this paper, we discuss the existence of solutions for a hybrid boundary value problem of Caputo fractional differential equations. The main tool used in our study is associated with the technique of measures of noncompactness. As an application, we give an example [...] Read more.
In this paper, we discuss the existence of solutions for a hybrid boundary value problem of Caputo fractional differential equations. The main tool used in our study is associated with the technique of measures of noncompactness. As an application, we give an example to illustrate our results. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
18 pages, 2526 KiB  
Article
An Efficient Numerical Technique for the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equation
by Pundikala Veeresha, Doddabhadrappla Gowda Prakasha and Dumitru Baleanu
Mathematics 2019, 7(3), 265; https://0-doi-org.brum.beds.ac.uk/10.3390/math7030265 - 14 Mar 2019
Cited by 50 | Viewed by 3164
Abstract
The q -homotopy analysis transform method ( q -HATM) is employed to find the solution for the fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation in the present frame work. To ensure the applicability and efficiency of the proposed algorithm, we consider three distinct initial conditions with [...] Read more.
The q -homotopy analysis transform method ( q -HATM) is employed to find the solution for the fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation in the present frame work. To ensure the applicability and efficiency of the proposed algorithm, we consider three distinct initial conditions with two of them having Jacobi elliptic functions. The numerical simulations have been conducted to verify that the proposed scheme is reliable and accurate. Moreover, the uniqueness and convergence analysis for the projected problem is also presented. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyze the complex problems arising in science and technology. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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12 pages, 283 KiB  
Article
Certain Hermite–Hadamard Inequalities for Logarithmically Convex Functions with Applications
by Shilpi Jain, Khaled Mehrez, Dumitru Baleanu and Praveen Agarwal
Mathematics 2019, 7(2), 163; https://0-doi-org.brum.beds.ac.uk/10.3390/math7020163 - 11 Feb 2019
Cited by 37 | Viewed by 2926
Abstract
In this paper, we discuss various estimates to the right-hand (resp. left-hand) side of the Hermite–Hadamard inequality for functions whose absolute values of the second (resp. first) derivatives to positive real powers are log-convex. As an application, we derive certain inequalities involving the [...] Read more.
In this paper, we discuss various estimates to the right-hand (resp. left-hand) side of the Hermite–Hadamard inequality for functions whose absolute values of the second (resp. first) derivatives to positive real powers are log-convex. As an application, we derive certain inequalities involving the q-digamma and q-polygamma functions, respectively. As a consequence, new inequalities for the q-analogue of the harmonic numbers in terms of the q-polygamma functions are derived. Moreover, several inequalities for special means are also considered. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
8 pages, 252 KiB  
Article
Global Asymptotical Stability Analysis for Fractional Neural Networks with Time-Varying Delays
by Zhixin Zhang, Yufeng Zhang, Jia-Bao Liu and Jiang Wei
Mathematics 2019, 7(2), 138; https://0-doi-org.brum.beds.ac.uk/10.3390/math7020138 - 01 Feb 2019
Cited by 2 | Viewed by 2145
Abstract
In this paper, the global asymptotical stability of Riemann-Liouville fractional-order neural networks with time-varying delays is studied. By combining the Lyapunov functional function and LMI approach, some sufficient criteria that guarantee the global asymptotical stability of such fractional-order neural networks with both discrete [...] Read more.
In this paper, the global asymptotical stability of Riemann-Liouville fractional-order neural networks with time-varying delays is studied. By combining the Lyapunov functional function and LMI approach, some sufficient criteria that guarantee the global asymptotical stability of such fractional-order neural networks with both discrete time-varying delay and distributed time-varying delay are derived. The stability criteria is suitable for application and easy to be verified by software. Lastly, some numerical examples are presented to check the validity of the obtained results. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
13 pages, 275 KiB  
Article
Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids
by Lei Fu, Yaodeng Chen and Hongwei Yang
Mathematics 2019, 7(1), 41; https://0-doi-org.brum.beds.ac.uk/10.3390/math7010041 - 03 Jan 2019
Cited by 34 | Viewed by 3526
Abstract
In this paper, the theoretical model of Rossby waves in two-layer fluids is studied. A single quasi-geostrophic vortex equation is used to derive various models of Rossby waves in a one-layer fluid in previous research. In order to explore the propagation and interaction [...] Read more.
In this paper, the theoretical model of Rossby waves in two-layer fluids is studied. A single quasi-geostrophic vortex equation is used to derive various models of Rossby waves in a one-layer fluid in previous research. In order to explore the propagation and interaction of Rossby waves in two-layer fluids, from the classical quasi-geodesic vortex equations, by employing the multi-scale analysis and turbulence method, we derived a new (2+1)-dimensional coupled equations set, namely the generalized Zakharov-Kuznetsov(gZK) equations set. The gZK equations set is an extension of a single ZK equation; they can describe two kinds of weakly nonlinear waves interaction by multiple coupling terms. Then, for the first time, based on the semi-inverse method and the variational method, a new fractional-order model which is the time-space fractional coupled gZK equations set is derived successfully, which is greatly different from the single fractional equation. Finally, group solutions of the time-space fractional coupled gZK equations set are obtained with the help of the improved ( G ′ / G ) -expansion method. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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21 pages, 339 KiB  
Article
Existence Results of a Coupled System of Caputo Fractional Hahn Difference Equations with Nonlocal Fractional Hahn Integral Boundary Value Conditions
by Thongchai Dumrongpokaphan, Nichaphat Patanarapeelert and Thanin Sitthiwirattham
Mathematics 2019, 7(1), 15; https://0-doi-org.brum.beds.ac.uk/10.3390/math7010015 - 24 Dec 2018
Cited by 5 | Viewed by 1998
Abstract
In this article, we propose a coupled system of Caputo fractional Hahn difference equations with nonlocal fractional Hahn integral boundary conditions. The existence and uniqueness result of solution for the problem is studied by using the Banach’s fixed point theorem. Furthermore, the existence [...] Read more.
In this article, we propose a coupled system of Caputo fractional Hahn difference equations with nonlocal fractional Hahn integral boundary conditions. The existence and uniqueness result of solution for the problem is studied by using the Banach’s fixed point theorem. Furthermore, the existence of at least one solution is presented by using the Schauder fixed point theorem. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
9 pages, 271 KiB  
Article
On Discrete Fractional Solutions of Non-Fuchsian Differential Equations
by Resat Yilmazer, Mustafa Inc and Mustafa Bayram
Mathematics 2018, 6(12), 308; https://0-doi-org.brum.beds.ac.uk/10.3390/math6120308 - 07 Dec 2018
Cited by 2 | Viewed by 2006
Abstract
In this article, we obtain new fractional solutions of the general class of non-Fuchsian differential equations by using discrete fractional nabla operator ∇ η ( 0 < η < 1 ) . This operator is applied to homogeneous and nonhomogeneous linear ordinary differential [...] Read more.
In this article, we obtain new fractional solutions of the general class of non-Fuchsian differential equations by using discrete fractional nabla operator ∇ η ( 0 < η < 1 ) . This operator is applied to homogeneous and nonhomogeneous linear ordinary differential equations. Thus, we obtain new solutions in fractional forms by a newly developed method. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
9 pages, 241 KiB  
Article
New Numerical Method for Solving Tenth Order Boundary Value Problems
by Ali Akgül, Esra Karatas Akgül, Dumitru Baleanu and Mustafa Inc
Mathematics 2018, 6(11), 245; https://0-doi-org.brum.beds.ac.uk/10.3390/math6110245 - 08 Nov 2018
Cited by 6 | Viewed by 2709
Abstract
In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate [...] Read more.
In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate solutions. We obtain very efficient results by this method. We show our numerical results by tables. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
9 pages, 235 KiB  
Article
Nonlocal q-Symmetric Integral Boundary Value Problem for Sequential q-Symmetric Integrodifference Equations
by Rujira Ouncharoen, Nichaphat Patanarapeelert and Thanin Sitthiwirattham
Mathematics 2018, 6(11), 218; https://0-doi-org.brum.beds.ac.uk/10.3390/math6110218 - 25 Oct 2018
Cited by 9 | Viewed by 1798
Abstract
In this paper, we prove the sufficient conditions for the existence results of a solution of a nonlocal q-symmetric integral boundary value problem for a sequential q-symmetric integrodifference equation by using the Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. [...] Read more.
In this paper, we prove the sufficient conditions for the existence results of a solution of a nonlocal q-symmetric integral boundary value problem for a sequential q-symmetric integrodifference equation by using the Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Some examples are also presented to illustrate our results. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)
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