Special Issue "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).

Special Issue Editor

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

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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

Open AccessArticle
Highly Accurate Numerical Technique for Population Models via Rational Chebyshev Collocation Method
Mathematics 2019, 7(10), 913; https://doi.org/10.3390/math7100913 - 01 Oct 2019
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
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Open AccessArticle
Asymptotic Almost-Periodicity for a Class of Weyl-Like Fractional Difference Equations
Mathematics 2019, 7(7), 592; https://doi.org/10.3390/math7070592 - 01 Jul 2019
Cited by 2
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
Open AccessArticle
Some Qualitative Behavior of Solutions of General Class of Difference Equations
Mathematics 2019, 7(7), 585; https://doi.org/10.3390/math7070585 - 01 Jul 2019
Cited by 6
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
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Open AccessArticle
A New Algorithm for Fractional Riccati Type Differential Equations by Using Haar Wavelet
Mathematics 2019, 7(6), 545; https://doi.org/10.3390/math7060545 - 14 Jun 2019
Cited by 2
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
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Open AccessArticle
An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations
Mathematics 2019, 7(5), 426; https://doi.org/10.3390/math7050426 - 13 May 2019
Cited by 6
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
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Open AccessArticle
Regularization and Error Estimate for the Poisson Equation with Discrete Data
Mathematics 2019, 7(5), 422; https://doi.org/10.3390/math7050422 - 10 May 2019
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
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Open AccessArticle
Application of the Laplace Homotopy Perturbation Method to the Black–Scholes Model Based on a European Put Option with Two Assets
Mathematics 2019, 7(4), 310; https://doi.org/10.3390/math7040310 - 27 Mar 2019
Cited by 1
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
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Open AccessArticle
Boundary Value Problems for Hybrid Caputo Fractional Differential Equations
Mathematics 2019, 7(3), 282; https://doi.org/10.3390/math7030282 - 19 Mar 2019
Cited by 2
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
Open AccessArticle
An Efficient Numerical Technique for the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equation
Mathematics 2019, 7(3), 265; https://doi.org/10.3390/math7030265 - 14 Mar 2019
Cited by 28
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
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Open AccessArticle
Certain Hermite–Hadamard Inequalities for Logarithmically Convex Functions with Applications
Mathematics 2019, 7(2), 163; https://doi.org/10.3390/math7020163 - 11 Feb 2019
Cited by 5
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
Open AccessArticle
Global Asymptotical Stability Analysis for Fractional Neural Networks with Time-Varying Delays
Mathematics 2019, 7(2), 138; https://doi.org/10.3390/math7020138 - 01 Feb 2019
Cited by 1
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
Open AccessArticle
Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids
Mathematics 2019, 7(1), 41; https://doi.org/10.3390/math7010041 - 03 Jan 2019
Cited by 19
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
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Open AccessArticle
Existence Results of a Coupled System of Caputo Fractional Hahn Difference Equations with Nonlocal Fractional Hahn Integral Boundary Value Conditions
Mathematics 2019, 7(1), 15; https://doi.org/10.3390/math7010015 - 24 Dec 2018
Cited by 2
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
Open AccessArticle
On Discrete Fractional Solutions of Non-Fuchsian Differential Equations
Mathematics 2018, 6(12), 308; https://doi.org/10.3390/math6120308 - 07 Dec 2018
Cited by 2
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
Open AccessArticle
New Numerical Method for Solving Tenth Order Boundary Value Problems
Mathematics 2018, 6(11), 245; https://doi.org/10.3390/math6110245 - 08 Nov 2018
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
Open AccessArticle
Nonlocal q-Symmetric Integral Boundary Value Problem for Sequential q-Symmetric Integrodifference Equations
Mathematics 2018, 6(11), 218; https://doi.org/10.3390/math6110218 - 25 Oct 2018
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
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