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

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

Deadline for manuscript submissions: closed (31 August 2018) | Viewed by 33384

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,

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 study 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, and, 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 (9 papers)

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Research

10 pages, 255 KiB  
Article
A Reliable Method for Solving Fractional Sturm–Liouville Problems
by M. M. Khashshan, Muhammed I. Syam and Ahlam Al Mokhmari
Mathematics 2018, 6(10), 176; https://doi.org/10.3390/math6100176 - 26 Sep 2018
Cited by 1 | Viewed by 2113
Abstract
In this paper, a reliable method for solving fractional Sturm–Liouville problem based on the operational matrix method is presented. Some of our numerical examples are presented. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
14 pages, 1002 KiB  
Article
The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense
by Panumart Sawangtong, Kamonchat Trachoo, Wannika Sawangtong and Benchawan Wiwattanapataphee
Mathematics 2018, 6(8), 129; https://doi.org/10.3390/math6080129 - 25 Jul 2018
Cited by 22 | Viewed by 6630
Abstract
It is well known that the Black-Scholes model is used to establish the behavior of the option pricing in the financial market. In this paper, we propose the modified version of Black-Scholes model with two assets based on the Liouville-Caputo fractional derivative. The [...] Read more.
It is well known that the Black-Scholes model is used to establish the behavior of the option pricing in the financial market. In this paper, we propose the modified version of Black-Scholes model with two assets based on the Liouville-Caputo fractional derivative. The analytical solution of the proposed model is investigated by the Laplace transform homotopy perturbation method. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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11 pages, 2458 KiB  
Article
An Implicit Hybrid Method for Solving Fractional Bagley-Torvik Boundary Value Problem
by Muhammed I. Syam, Azza Alsuwaidi, Asia Alneyadi, Safa Al Refai and Sondos Al Khaldi
Mathematics 2018, 6(7), 109; https://doi.org/10.3390/math6070109 - 25 Jun 2018
Cited by 6 | Viewed by 3274
Abstract
In this article, a modified implicit hybrid method for solving the fractional Bagley-Torvik boundary (BTB) value problem is investigated. This approach is of a higher order. We study the convergence, zero stability, consistency, and region of absolute stability of the modified implicit hybrid [...] Read more.
In this article, a modified implicit hybrid method for solving the fractional Bagley-Torvik boundary (BTB) value problem is investigated. This approach is of a higher order. We study the convergence, zero stability, consistency, and region of absolute stability of the modified implicit hybrid method. Three of our numerical examples are presented. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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19 pages, 264 KiB  
Article
Global Behavior of Certain Nonautonomous Linearizable Three Term Difference Equations
by E. J. Janowski and M. R. S. Kulenović
Mathematics 2018, 6(5), 79; https://doi.org/10.3390/math6050079 - 9 May 2018
Viewed by 2736
Abstract
We investigate the nonautonomous difference equation with real initial conditions and coefficients g i , i = 0 , 1 which are in general functions of n and/or the state variables x n , x n − 1 , … , and satisfy [...] Read more.
We investigate the nonautonomous difference equation with real initial conditions and coefficients g i , i = 0 , 1 which are in general functions of n and/or the state variables x n , x n − 1 , … , and satisfy g 0 + g 1 = 1 . We also obtain some global results about the behavior of solutions of the nonautonomous non-homogeneous difference equation where g i , i = 0 , 1 , 2 are functions of n and/or the state variables x n , x n − 1 , … , with g 0 + g 1 = 1 . Our results are based on the explicit formulas for solutions. We illustrate our results by numerous examples. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
22 pages, 332 KiB  
Article
A Numerical Method for Solving a Class of Nonlinear Second Order Fractional Volterra Integro-Differntial Type of Singularly Perturbed Problems
by Muhammed I. Syam and Mohammed Abu Omar
Mathematics 2018, 6(4), 48; https://doi.org/10.3390/math6040048 - 27 Mar 2018
Cited by 3 | Viewed by 3763
Abstract
In this paper, we study a class of fractional nonlinear second order Volterra integro-differential type of singularly perturbed problems with fractional order. We divide the problem into two subproblems. The first subproblems is the reduced problem when ϵ = 0 . The second [...] Read more.
In this paper, we study a class of fractional nonlinear second order Volterra integro-differential type of singularly perturbed problems with fractional order. We divide the problem into two subproblems. The first subproblems is the reduced problem when ϵ = 0 . The second subproblems is fractional Volterra integro-differential problem. We use the finite difference method to solve the first problem and the reproducing kernel method to solve the second problem. In addition, we use the pade’ approximation. The results show that the proposed analytical method can achieve excellent results in predicting the solutions of such problems. Theoretical results are presented. Numerical results are presented to show the efficiency of the proposed method. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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12 pages, 259 KiB  
Article
Existence of Solution, Filippov’s Theorem and Compactness of the Set of Solutions for a Third-Order Differential Inclusion with Three- Point Boundary Conditions
by Ali Rezaiguia and Smail Kelaiaia
Mathematics 2018, 6(3), 40; https://doi.org/10.3390/math6030040 - 8 Mar 2018
Cited by 2 | Viewed by 2958
Abstract
In this paper, we study a third-order differential inclusion with three-point boundary conditions. We prove the existence of a solution under convexity conditions on the multi-valued right-hand side; the proof is based on a nonlinear alternative of Leray-Schauder type. We also study the [...] Read more.
In this paper, we study a third-order differential inclusion with three-point boundary conditions. We prove the existence of a solution under convexity conditions on the multi-valued right-hand side; the proof is based on a nonlinear alternative of Leray-Schauder type. We also study the compactness of the set of solutions and establish some Filippov’s- type results for this problem. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
13 pages, 4078 KiB  
Article
Global Dynamics of Certain Mix Monotone Difference Equation
by Senada Kalabušić, Mehmed Nurkanović and Zehra Nurkanović
Mathematics 2018, 6(1), 10; https://doi.org/10.3390/math6010010 - 12 Jan 2018
Cited by 9 | Viewed by 3057
Abstract
We investigate global dynamics of the following second order rational difference equation [...] Read more.
We investigate global dynamics of the following second order rational difference equation x n + 1 = x n x n − 1 + α x n + β x n − 1 a x n x n − 1 + b x n − 1 , where the parameters α , β , a , b are positive real numbers and initial conditions x − 1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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255 KiB  
Article
A Numerical Solution of Fractional Lienard’s Equation by Using the Residual Power Series Method
by Muhammed I. Syam
Mathematics 2018, 6(1), 1; https://doi.org/10.3390/math6010001 - 22 Dec 2017
Cited by 8 | Viewed by 3909
Abstract
In this paper, we investigate a numerical solution of Lienard’s equation. The residual power series (RPS) method is implemented to find an approximate solution to this problem. The proposed method is a combination of the fractional Taylor series and the residual functions. Numerical [...] Read more.
In this paper, we investigate a numerical solution of Lienard’s equation. The residual power series (RPS) method is implemented to find an approximate solution to this problem. The proposed method is a combination of the fractional Taylor series and the residual functions. Numerical and theoretical results are presented. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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265 KiB  
Article
Solving the Lane–Emden Equation within a Reproducing Kernel Method and Group Preserving Scheme
by Mir Sajjad Hashemi, Ali Akgül, Mustafa Inc, Idrees Sedeeq Mustafa and Dumitru Baleanu
Mathematics 2017, 5(4), 77; https://doi.org/10.3390/math5040077 - 12 Dec 2017
Cited by 16 | Viewed by 3227
Abstract
We apply the reproducing kernel method and group preserving scheme for investigating the Lane–Emden equation. The reproducing kernel method is implemented by the useful reproducing kernel functions and the numerical approximations are given. These approximations demonstrate the preciseness of the investigated techniques. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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