Special Issue "Advances in Differential and Difference Equations with Applications 2020"

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

Deadline for manuscript submissions: closed (20 May 2020).

Printed Edition Available!
A printed edition of this Special Issue is available here.

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
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Special Issue Information

Dear Colleagues,

This issue is a continuation of the previous successful Special Issue “Advances in Differential and Difference Equations with Applications 2019”.

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

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Keywords

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

Published Papers (19 papers)

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Research

Article
On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback
Mathematics 2020, 8(6), 992; https://doi.org/10.3390/math8060992 - 17 Jun 2020
Cited by 1 | Viewed by 665
Abstract
We consider a linear continuous-time control system with time-invariant linear bounded operator coefficients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary [...] Read more.
We consider a linear continuous-time control system with time-invariant linear bounded operator coefficients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment for the upper Bohl exponent by state feedback control. We prove that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented. Full article
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Article
About Some Possible Implementations of the Fractional Calculus
Mathematics 2020, 8(6), 893; https://doi.org/10.3390/math8060893 - 02 Jun 2020
Cited by 2 | Viewed by 830
Abstract
We present a partial panoramic view of possible contexts and applications of the fractional calculus. In this context, we show some different applications of fractional calculus to different models in ordinary differential equation (ODE) and partial differential equation (PDE) formulations ranging from the [...] Read more.
We present a partial panoramic view of possible contexts and applications of the fractional calculus. In this context, we show some different applications of fractional calculus to different models in ordinary differential equation (ODE) and partial differential equation (PDE) formulations ranging from the basic equations of mechanics to diffusion and Dirac equations. Full article
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Article
Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefficients by Linear Stationary Feedback
Mathematics 2020, 8(5), 853; https://doi.org/10.3390/math8050853 - 24 May 2020
Cited by 2 | Viewed by 869
Abstract
We consider a control system defined by a linear time-varying differential equation of n-th order with uncertain bounded coefficients. The problem of exponential stabilization of the system with an arbitrary given decay rate by linear static state or output feedback with constant [...] Read more.
We consider a control system defined by a linear time-varying differential equation of n-th order with uncertain bounded coefficients. The problem of exponential stabilization of the system with an arbitrary given decay rate by linear static state or output feedback with constant gain coefficients is studied. We prove that every system is exponentially stabilizable with any pregiven decay rate by linear time-invariant static state feedback. The proof is based on the Levin’s theorem on sufficient conditions for absolute non-oscillatory stability of solutions to a linear differential equation. We obtain sufficient conditions of exponential stabilization with any pregiven decay rate for a linear differential equation with uncertain bounded coefficients by linear time-invariant static output feedback. Illustrative examples are considered. Full article
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Article
The Sign of the Green Function of an n-th Order Linear Boundary Value Problem
Mathematics 2020, 8(5), 673; https://doi.org/10.3390/math8050673 - 29 Apr 2020
Cited by 1 | Viewed by 773
Abstract
This paper provides results on the sign of the Green function (and its partial derivatives) of an n-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The dependence of the absolute value of the Green function [...] Read more.
This paper provides results on the sign of the Green function (and its partial derivatives) of an n-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The dependence of the absolute value of the Green function and some of its partial derivatives with respect to the extremes where the boundary conditions are set is also assessed. Full article
Article
Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation
Mathematics 2020, 8(4), 588; https://doi.org/10.3390/math8040588 - 15 Apr 2020
Cited by 4 | Viewed by 610
Abstract
In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, [...] Read more.
In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, by power transformation, and determine the probabilistic characteristics of the process, i.e., its analytic expression, the transition probability density function and the trend functions. We then study the statistical inference in this process. The parameters present in the model are studied by using the maximum likelihood estimation method, based on discrete sampling, thus obtaining the expression of the likelihood estimators and their ergodic properties. We then obtain the power process of the stochastic lognormal diffusion as the limit of the Gompertz process being studied and go on to obtain all the probabilistic characteristics and the statistical inference. Finally, the proposed model is applied to simulated data. Full article
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Article
Asymptotic Behavior of Solutions of the Third Order Nonlinear Mixed Type Neutral Differential Equations
Mathematics 2020, 8(4), 485; https://doi.org/10.3390/math8040485 - 01 Apr 2020
Cited by 6 | Viewed by 676
Abstract
The objective of our paper is to study asymptotic properties of the class of third order neutral differential equations with advanced and delayed arguments. Our results supplement and improve some known results obtained in the literature. An illustrative example is provided. Full article
Article
Differential Equations Associated with Two Variable Degenerate Hermite Polynomials
Mathematics 2020, 8(2), 228; https://doi.org/10.3390/math8020228 - 10 Feb 2020
Cited by 3 | Viewed by 719
Abstract
In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of [...] Read more.
In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the two variable degenerate Hermite equations. Full article
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Article
Influence of Single- and Multi-Wall Carbon Nanotubes on Magnetohydrodynamic Stagnation Point Nanofluid Flow over Variable Thicker Surface with Concave and Convex Effects
Mathematics 2020, 8(1), 104; https://doi.org/10.3390/math8010104 - 08 Jan 2020
Cited by 37 | Viewed by 1292
Abstract
This paper reports a theoretical study on the magnetohydrodynamic flow and heat exchange of carbon nanotubes (CNTs)-based nanoliquid over a variable thicker surface. Two types of carbon nanotubes (CNTs) are accounted for saturation in base fluid. Particularly, the single-walled and multi-walled carbon nanotubes, [...] Read more.
This paper reports a theoretical study on the magnetohydrodynamic flow and heat exchange of carbon nanotubes (CNTs)-based nanoliquid over a variable thicker surface. Two types of carbon nanotubes (CNTs) are accounted for saturation in base fluid. Particularly, the single-walled and multi-walled carbon nanotubes, best known as SWCNTs and MWCNTs, are used. Kerosene oil is taken as the base fluid for the suspension of nanoparticles. The model involves the impact of the thermal radiation and induced magnetic field. However, a tiny Reynolds number is assumed to ignore the magnetic induction. The system of nonlinear equations is obtained by reasonably adjusted transformations. The analytic solution is obtained by utilizing a notable procedure called optimal homotopy analysis technique (O-HAM). The impact of prominent parameters, such as the magnetic field parameter, Brownian diffusion, Thermophoresis, and others, on the dimensionless velocity field and thermal distribution is reported graphically. A comprehensive discussion is given after each graph that summarizes the influence of the respective parameters on the flow profiles. The behavior of the friction coefficient and the rate of heat transfer (Nusselt number) at the surface (y = 0) are given at the end of the text in tabular form. Some existing solutions of the specific cases have been checked as the special case of the solution acquired here. The results indicate that MWCNTs cause enhancement in the velocity field compared with SWCNTs when there is an increment in nanoparticle volume fraction. Furthermore, the temperature profile rises with an increment in radiation estimator for both SWCNT and MWCNT and, finally, the heat transfer rate lessens for increments in the magnetic parameter for both types of nanotubes. Full article
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Article
Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration
Mathematics 2020, 8(1), 96; https://doi.org/10.3390/math8010096 - 07 Jan 2020
Cited by 3 | Viewed by 1533
Abstract
In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor [...] Read more.
In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable. Full article
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Article
Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations
Mathematics 2019, 7(12), 1158; https://doi.org/10.3390/math7121158 - 01 Dec 2019
Cited by 2 | Viewed by 875
Abstract
In this paper, we compare a multi-step method and a multi-stage method for stiff initial value problems. Traditionally, the multi-step method has been preferred than the multi-stage for a stiff problem, to avoid an enormous amount of computational costs required to solve a [...] Read more.
In this paper, we compare a multi-step method and a multi-stage method for stiff initial value problems. Traditionally, the multi-step method has been preferred than the multi-stage for a stiff problem, to avoid an enormous amount of computational costs required to solve a massive linear system provided by the linearization of a highly stiff system. We investigate the possibility of usage of multi-stage methods for stiff systems by discussing the difference between the two methods in several numerical experiments. Moreover, the advantages of multi-stage methods are heuristically presented even for nonlinear stiff systems through several numerical tests. Full article
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Article
Second Order Semilinear Volterra-Type Integro-Differential Equations with Non-Instantaneous Impulses
Mathematics 2019, 7(12), 1134; https://doi.org/10.3390/math7121134 - 20 Nov 2019
Cited by 1 | Viewed by 1028
Abstract
We consider a non-instantaneous system represented by a second order nonlinear differential equation in a Banach space E. We use the family of linear bounded operators introduced by Kozak, Darbo fixed point method and Kuratowski measure of noncompactness. A new set of [...] Read more.
We consider a non-instantaneous system represented by a second order nonlinear differential equation in a Banach space E. We use the family of linear bounded operators introduced by Kozak, Darbo fixed point method and Kuratowski measure of noncompactness. A new set of sufficient conditions is formulated which guarantees the existence of the solution of the non-instantaneous system. An example is also discussed to illustrate the efficiency of the obtained results. Full article
Article
A New Scheme Using Cubic B-Spline to Solve Non-Linear Differential Equations Arising in Visco-Elastic Flows and Hydrodynamic Stability Problems
Mathematics 2019, 7(11), 1078; https://doi.org/10.3390/math7111078 - 08 Nov 2019
Cited by 8 | Viewed by 1607
Abstract
This study deals with the numerical solution of the non-linear differential equations (DEs) arising in the study of hydrodynamics and hydro-magnetic stability problems using a new cubic B-spline scheme (CBS). The main idea is that we have modified the boundary value problems (BVPs) [...] Read more.
This study deals with the numerical solution of the non-linear differential equations (DEs) arising in the study of hydrodynamics and hydro-magnetic stability problems using a new cubic B-spline scheme (CBS). The main idea is that we have modified the boundary value problems (BVPs) to produce a new system of linear equations. The algorithm developed here is not only for the approximation solutions of the 10th order BVPs but also estimate from 1st derivative to 10th derivative of the exact solution as well. Some examples are illustrated to show the feasibility and competence of the proposed scheme. Full article
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Article
Identification of Source Term for the Time-Fractional Diffusion-Wave Equation by Fractional Tikhonov Method
Mathematics 2019, 7(10), 934; https://doi.org/10.3390/math7100934 - 10 Oct 2019
Cited by 3 | Viewed by 837
Abstract
In this article, we consider an inverse problem to determine an unknown source term in a space-time-fractional diffusion equation. The inverse problems are often ill-posed. By an example, we show that this problem is NOT well-posed in the Hadamard sense, i.e., this problem [...] Read more.
In this article, we consider an inverse problem to determine an unknown source term in a space-time-fractional diffusion equation. The inverse problems are often ill-posed. By an example, we show that this problem is NOT well-posed in the Hadamard sense, i.e., this problem does not satisfy the last condition-the solution’s behavior changes continuously with the input data. It leads to having a regularization model for this problem. We use the Tikhonov method to solve the problem. In the theoretical results, we also propose a priori and a posteriori parameter choice rules and analyze them. Full article
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Article
Approximate Solutions of Time Fractional Diffusion Wave Models
Mathematics 2019, 7(10), 923; https://doi.org/10.3390/math7100923 - 03 Oct 2019
Cited by 4 | Viewed by 1228
Abstract
In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. [...] Read more.
In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before. Full article
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Article
Numerical Study for Darcy–Forchheimer Flow of Nanofluid due to a Rotating Disk with Binary Chemical Reaction and Arrhenius Activation Energy
Mathematics 2019, 7(10), 921; https://doi.org/10.3390/math7100921 - 02 Oct 2019
Cited by 32 | Viewed by 1348
Abstract
The present article investigates Darcy–Forchheimer 3D nanoliquid flow because of a rotating disk with Arrhenius activation energy. Flow is created by rotating disk. Impacts of thermophoresis and Brownian dispersion are accounted for. Convective states of thermal and mass transport at surface of a [...] Read more.
The present article investigates Darcy–Forchheimer 3D nanoliquid flow because of a rotating disk with Arrhenius activation energy. Flow is created by rotating disk. Impacts of thermophoresis and Brownian dispersion are accounted for. Convective states of thermal and mass transport at surface of a rotating disk are imposed. The nonlinear systems have been deduced by transformation technique. Shooting method is employed to construct the numerical arrangement of subsequent problem. Plots are organized just to investigate how velocities, concentration, and temperature are influenced by distinct emerging flow variables. Surface drag coefficients and local Sherwood and Nusselt numbers are also plotted and discussed. Our results indicate that the temperature and concentration are enhanced for larger values of porosity parameter and Forchheimer number. Full article
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Article
On Fractional Symmetric Hahn Calculus
Mathematics 2019, 7(10), 873; https://doi.org/10.3390/math7100873 - 20 Sep 2019
Cited by 6 | Viewed by 692
Abstract
In this paper, we study fractional symmetric Hahn difference calculus. The new idea of the symmetric Hahn difference operator, the fractional symmetric Hahn integral, and the fractional symmetric Hahn operators of Riemann–Liouville and Caputo types are presented. In addition, we formulate some fundamental [...] Read more.
In this paper, we study fractional symmetric Hahn difference calculus. The new idea of the symmetric Hahn difference operator, the fractional symmetric Hahn integral, and the fractional symmetric Hahn operators of Riemann–Liouville and Caputo types are presented. In addition, we formulate some fundamental properties based on these fractional symmetric Hahn operators. Full article
Article
Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case
Mathematics 2019, 7(8), 735; https://doi.org/10.3390/math7080735 - 12 Aug 2019
Cited by 7 | Viewed by 919
Abstract
We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, [...] Read more.
We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter—Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input. Full article
Article
On the Solution of an Imprecisely Defined Nonlinear Time-Fractional Dynamical Model of Marriage
Mathematics 2019, 7(8), 689; https://doi.org/10.3390/math7080689 - 01 Aug 2019
Cited by 20 | Viewed by 1378
Abstract
The present paper investigates the numerical solution of an imprecisely defined nonlinear coupled time-fractional dynamical model of marriage (FDMM). Uncertainties are assumed to exist in the dynamical system parameters, as well as in the initial conditions that are formulated by triangular normalized fuzzy [...] Read more.
The present paper investigates the numerical solution of an imprecisely defined nonlinear coupled time-fractional dynamical model of marriage (FDMM). Uncertainties are assumed to exist in the dynamical system parameters, as well as in the initial conditions that are formulated by triangular normalized fuzzy sets. The corresponding fractional dynamical system has first been converted to an interval-based fuzzy nonlinear coupled system with the help of a single-parametric gamma-cut form. Further, the double-parametric form (DPF) of fuzzy numbers has been used to handle the uncertainty. The fractional reduced differential transform method (FRDTM) has been applied to this transformed DPF system for obtaining the approximate solution of the FDMM. Validation of this method was ensured by comparing it with other methods taking the gamma-cut as being equal to one. Full article
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Article
Structure of Non-Oscillatory Solutions for Second Order Dynamic Equations on Time Scales
Mathematics 2019, 7(8), 680; https://doi.org/10.3390/math7080680 - 30 Jul 2019
Viewed by 836
Abstract
In this paper, we make a detailed analysis of the structure of non-oscillatory solutions for second order superlinear and sublinear dynamic equations on time scales. The sufficient and necessary conditions for existence of non-oscillatory solutions are presented. Full article
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