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Showing papers in "Advances in Difference Equations in 2019"


Journal ArticleDOI
TL;DR: The numerical results for fractional SIRS-SI malaria model reveal that the recommended approach is very accurate and effective and has great impact on the dynamical system of human and mosquito populations.
Abstract: The present paper deals with a new fractional SIRS-SI model describing the transmission of malaria disease. The SIRS-SI malaria model is modified by using the Caputo–Fabrizio fractional operator for the inclusion of memory. We also suggest the utilization of vaccines, antimalarial medicines, and spraying for the treatment and control of the malaria disease. The theory of fixed point is utilized to examine the existence of the solution of a fractional SIRS-SI model describing spreading of malaria. The uniqueness of the solution of SIRS-SI model for malaria is also analyzed. It is shown that the treatments have great impact on the dynamical system of human and mosquito populations. The numerical simulation of fractional SIRS-SI malaria model is performed with the aid of HATM and Maple packages to show the effect of different parameters of the treatment of malaria disease. The numerical results for fractional SIRS-SI malaria model reveal that the recommended approach is very accurate and effective.

140 citations


Journal ArticleDOI
TL;DR: In this paper, a proper orthogonal decomposition (POD) is proposed to reduce the order of the classical Crank-Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs).
Abstract: In this paper, by means of a proper orthogonal decomposition (POD) we mainly reduce the order of the classical Crank–Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs). Toward this end, we will first review the CCNFD model for FOPTSGEs and the theoretical results (such as existence, stabilization, and convergence) of the CCNFD solutions. Then we establish an optimized Crank–Nicolson finite difference extrapolating (OCNFDE) model, including very few unknowns but holding the fully second-order accuracy for FOPTSGEs via POD. Next, by a matrix analysis we will discuss the existence, stabilization, and convergence of the OCNFDE solutions. Finally, we will use a numerical example to validate the validity of theoretical conclusions. Moreover, we show that the OCNFDE model is very valid for settling FOPTSGEs.

122 citations


Journal ArticleDOI
TL;DR: In this article, a new generalized exponential rational function method is employed to extract new solitary wave solutions for the Zakharov-Kuznetsov equation (ZKE), which exhibits the behavior of weakly nonlinear ion-acoustic waves in incorporated hot isothermal electrons and cold ions in the presence of a uniform magnetic field.
Abstract: In this paper, a new generalized exponential rational function method is employed to extract new solitary wave solutions for the Zakharov–Kuznetsov equation (ZKE). The ZKE exhibits the behavior of weakly nonlinear ion-acoustic waves in incorporated hot isothermal electrons and cold ions in the presence of a uniform magnetic field. Furthermore, the stability for the governing equations is investigated via the aspect of linear stability analysis. Numerical simulations are made to shed light on the characteristics of the obtained solutions.

102 citations


Journal ArticleDOI
TL;DR: In this article, the conditions for existence and uniqueness of solutions of fractional initial value problems are established using fixed point theorem and contraction principle, respectively, and the results reveal that the proposed method is most suitable in terms of computational cost efficiency, and accuracy which can be applied to find solutions of nonlinear fractional reaction-diffusion equations.
Abstract: This manuscript deals with fractional differential equations including Caputo–Fabrizio differential operator. The conditions for existence and uniqueness of solutions of fractional initial value problems is established using fixed point theorem and contraction principle, respectively. As an application, the iterative Laplace transform method (ILTM) is used to get an approximate solutions for nonlinear fractional reaction–diffusion equations, namely the Fitzhugh–Nagumo equation and the Fisher equation in the Caputo–Fabrizio sense. The obtained approximate solutions are compared with other available solutions from existing methods by using graphical representations and numerical computations. The results reveal that the proposed method is most suitable in terms of computational cost efficiency, and accuracy which can be applied to find solutions of nonlinear fractional reaction–diffusion equations.

89 citations


Journal ArticleDOI
TL;DR: In this paper, a new class of hybrid type fractional differential equations and inclusions via some non-local three-point boundary value conditions are investigated and some examples to illustrate their results are provided.
Abstract: We investigate some new class of hybrid type fractional differential equations and inclusions via some nonlocal three-point boundary value conditions. Also, we provide some examples to illustrate our results.

84 citations


Journal ArticleDOI
TL;DR: In this article, the generalization of natural convection flow of hybrid nanofluid in two infinite vertical parallel plates is considered and the Caputo-Fabrizio fractional derivative and the Laplace transform technique are used to develop exact analytical solutions for velocity and temperature profiles.
Abstract: This article deals with the generalization of natural convection flow of $Cu - Al_{2}O_{3} - H_{2}O$ hybrid nanofluid in two infinite vertical parallel plates. To demonstrate the flow phenomena in two parallel plates of hybrid nanofluids, the Brinkman type fluid model together with the energy equation is considered. The Caputo–Fabrizio fractional derivative and the Laplace transform technique are used to developed exact analytical solutions for velocity and temperature profiles. The general solutions for velocity and temperature profiles are brought into light through numerical computation and graphical representation. The obtained results show that the velocity and temperature profiles show dual behaviors for $0 < \alpha < 1$ and $0 < \beta < 1$ where α and β are the fractional parameters. It is noticed that, for a shorter time, the velocity and temperature distributions decrease with increasing values of the fractional parameters, whereas the trend reverses for a longer time. Moreover, it is found that the velocity and temperature profiles oppositely behave for the volume fraction of hybrid nanofluids.

75 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of solutions of a class of hybrid Caputo-Hadamard fractional differential inclusions with Dirichlet boundary conditions was studied and the Arzela-Ascoli theorem and some suitable theorems of fixed point theory were derived.
Abstract: In this manuscript, we talk over the existence of solutions of a class of hybrid Caputo–Hadamard fractional differential inclusions with Dirichlet boundary conditions. Our results are based on the Arzela–Ascoli theorem and some suitable theorems of fixed point theory. As well, to illustrate our results, we confront the exceptional case of the fractional differential inclusions with examples.

69 citations


Journal ArticleDOI
TL;DR: In this article, a modified analytical technique based on auxiliary parameters and residual power series method (RPSM) for Newell-Whitehead-Segel (NWS) equations of arbitrary order is presented.
Abstract: The main aim of this paper is to present a comparative study of modified analytical technique based on auxiliary parameters and residual power series method (RPSM) for Newell–Whitehead–Segel (NWS) equations of arbitrary order. The NWS equation is well defined and a famous nonlinear physical model, which is characterized by the presence of the strip patterns in two-dimensional systems and application in many areas such as mechanics, chemistry, and bioengineering. In this paper, we implement a modified analytical method based on auxiliary parameters and residual power series techniques to obtain quick and accurate solutions of the time-fractional NWS equations. Comparison of the obtained solutions with the present solutions reveal that both powerful analytical techniques are productive, fruitful, and adequate in solving any kind of nonlinear partial differential equations arising in several physical phenomena. We addressed $L_{2}$ and $L_{\infty }$ norms in both cases. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present aforesaid methods and noted excellent agreement. In this study, we use the fractional operators in Caputo sense.

69 citations


Journal ArticleDOI
TL;DR: In this article, the authors present analytical-approximate solution to the time-fractional nonlinear coupled Jaulent-Miodek system of equations which comes with an energy-dependent Schrodinger potential by means of a residual power series method (RSPM) and a q-homotopy analysis method (q-HAM).
Abstract: In this paper, we present analytical-approximate solution to the time-fractional nonlinear coupled Jaulent–Miodek system of equations which comes with an energy-dependent Schrodinger potential by means of a residual power series method (RSPM) and a q-homotopy analysis method (q-HAM). These methods produce convergent series solutions with easily computable components. Using a specific example, a comparison analysis is done between these methods and the exact solution. The numerical results show that present methods are competitive, powerful, reliable, and easy to implement for strongly nonlinear fractional differential equations.

65 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of solution of boundary value problems with integral type boundary conditions in the frame of some Caputo type fractional operators has been shown to be a fixed point theorem that extends and unifies existing results in the literature.
Abstract: In this paper, we consider a fixed point theorem that extends and unifies several existing results in the literature. We apply the proven fixed point results on the existence of solution of ordinary boundary value problems and fractional boundary value problems with integral type boundary conditions in the frame of some Caputo type fractional operators.

64 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of solutions for a three-step crisis fractional integro-differential equation under some boundary conditions is investigated, where the boundary conditions are defined by the authors.
Abstract: One of the interesting fractional integro-differential equations is the three step crisis equation which has been reviewed recently. In this paper, we investigate the existence of solutions for a three step crisis fractional integro-differential equation under some boundary conditions.

Journal ArticleDOI
TL;DR: In this paper, a fractional residual power series method (FRPS) was applied to solving a class of fractional stiff systems. And the results showed that the proposed method is analytically effective and convenient, and it can be implemented in a large number of engineering problems.
Abstract: A powerful analytical approach, namely the fractional residual power series method (FRPS), is applied successfully in this work to solving a class of fractional stiff systems. The methodology of the FRPS method gets a Maclaurin expansion of the solution in rapidly convergent form and apparent sequences based on the Caputo sense without any restriction hypothesis. This approach is tested on a fractional stiff system with nonlinearity ranging. Meanwhile, stability and convergence study are presented in the domain of interest. Illustrative examples justify that the proposed method is analytically effective and convenient, and it can be implemented in a large number of engineering problems. A numerical comparison for the experimental data with another well-known method, the reproducing kernel method, is given. The graphical consequences illuminate the simplicity and reliability of the FRPS method in the determination of the RPS solutions consistently.

Journal ArticleDOI
TL;DR: In this paper, new conditions are obtained for the oscillation of solutions of the even-order delay equation, where the Riccati transformation is used to ensure the stability of the solution.
Abstract: In this work, new conditions are obtained for the oscillation of solutions of the even-order equation $$ \bigl( r ( \zeta ) z^{ ( n-1 ) } ( \zeta ) \bigr) ^{\prime }+ \int _{a}^{b}q ( \zeta ,s ) f \bigl( x \bigl( g ( \zeta ,s ) \bigr) \bigr) \,\mathrm{d}s=0, \quad \zeta \geq \zeta _{0}, $$ where $n\geq 2$ is an even integer and $z ( \zeta ) =x ^{\alpha } ( \zeta ) +p ( \zeta ) x ( \sigma ( \zeta ) ) $ . By using the theory of comparison with first-order delay equations and the technique of Riccati transformation, we get two various conditions to ensure oscillation of solutions of this equation. Moreover, the importance of the obtained conditions is illustrated via some examples.

Journal ArticleDOI
TL;DR: In this paper, the authors developed and analyzed a Caputo-Fabrizio fractional derivative model for the HIV/AIDS epidemic which includes an antiretroviral treatment compartment.
Abstract: In recent years, many new definitions of fractional derivatives have been proposed and used to develop mathematical models for a wide variety of real-world systems containing memory, history, or nonlocal effects. The main purpose of the present paper is to develop and analyze a Caputo–Fabrizio fractional derivative model for the HIV/AIDS epidemic which includes an antiretroviral treatment compartment. The existence and uniqueness of the system of solutions of the model are established using a fixed-point theorem and an iterative method. The model is shown to have a disease-free and an endemic equilibrium point. Conditions are derived for the existence of the endemic equilibrium point and for the local asymptotic stability of the disease-free equilibrium point. The results confirm that the disease-free equilibrium point becomes increasingly stable as the fractional order is reduced. Numerical simulations are carried out using a three-step Adams–Bashforth predictor method for a range of fractional orders to illustrate the effects of varying the fractional order and to support the theoretical results.

Journal ArticleDOI
TL;DR: In this article, the existence, uniqueness, and Ulam-Hyers-Mittag-Leffler stability of solutions to a class of ψ-Hilfer fractional-order delay differential equations were proved.
Abstract: In this paper, we present results on the existence, uniqueness, and Ulam–Hyers–Mittag-Leffler stability of solutions to a class of ψ-Hilfer fractional-order delay differential equations. We use the Picard operator method and a generalized Gronwall inequality involved in a ψ-Riemann–Liouville fractional integral. Finally, we give two examples to illustrate our main theorems.

Journal ArticleDOI
TL;DR: In this article, the Grunwald-Letnikov fractional derivative of the Riemann ζ function is computed in a simplified form that reduces the computational cost, and a quasisymmetric form of the aforementioned functional equation is derived.
Abstract: This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grunwald–Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a quasisymmetric form of the aforementioned functional equation is derived (symmetric up to one complex multiplicative constant). The second part of the paper examines the link with the distribution of prime numbers. The Dirichlet η function suggests the introduction of a complex strip as a fractional counterpart of the critical strip. Analytic properties are shown, particularly that a Dirichlet series can be linked with this strip and expressed as a sum of the fractional derivatives of ζ. Finally, Theorem 4.3 links the fractional derivative of ζ with the distribution of prime numbers in the left half-plane.

Journal ArticleDOI
TL;DR: In this paper, the generalized nonlocal proportional fractional integrals and derivatives were used to derive the reverse Minkowski inequalities and some other fractional integral inequalities by utilizing generalized PRF integrals.
Abstract: Recent research has gained more attention on conformable integrals and derivatives to derive the various type of inequalities. One of the recent advancements in the field of fractional calculus is the generalized nonlocal proportional fractional integrals and derivatives lately introduced by Jarad et al. (Eur. Phys. J. Special Topics 226:3457–3471, 2017) comprising the exponential functions in the kernels. The principal aim of this paper is to establish reverse Minkowski inequalities and some other fractional integral inequalities by utilizing generalized proportional fractional integrals. Also, two new theorems connected with this inequality as well as other inequalities associated with the generalized proportional fractional integrals are established.

Journal ArticleDOI
TL;DR: In this paper, the authors extended the application of the reproducing-kernel method and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α.
Abstract: The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, $0<\alpha\leq1$ , concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.

Journal ArticleDOI
TL;DR: In this article, the authors considered a general class of nonlinear singular fractional DEs with p-Laplacian for the existence and uniqueness (EU) of a positive solution and the Hyers-Ulam stability.
Abstract: In this article, we consider a study of a general class of nonlinear singular fractional DEs with p-Laplacian for the existence and uniqueness (EU) of a positive solution and the Hyers–Ulam (HU) stability. To proceed, we use classical fixed point theorem and properties of a p-Laplacian operator. The fractional DE is converted into an integral alternative form with the help of the Green’s function. The Green’s function is analyzed as regards its nature and then, with the help of a fixed point approach, the existence of a positive solution and uniqueness are studied. After the EU of a positive solution, the HU-stability and an application are considered. The suggested singular fractional DE with $\phi _{p}$ is more general than the one considered in (Khan et al. in Eur. Phys. J. Plus 133:26, 2018)

Journal ArticleDOI
TL;DR: In this article, the generalized proportional Hadamard fractional integrals were introduced and several inequalities for convex functions were established in the framework of the defined class of fractional integral functions.
Abstract: In the article, we introduce the generalized proportional Hadamard fractional integrals and establish several inequalities for convex functions in the framework of the defined class of fractional integrals. The given results are generalizations of some known results.

Journal ArticleDOI
TL;DR: In this article, the synchronization problem of N-coupled fractional-order chaotic systems with ring connection via bidirectional coupling is discussed. But the authors focus on the problem of finding the appropriate controllers to transform the fractionalorder error dynamical system into a nonlinear system with antisymmetric structure.
Abstract: This paper discusses the synchronization problem of N-coupled fractional-order chaotic systems with ring connection via bidirectional coupling. On the basis of the direct design method, we design the appropriate controllers to transform the fractional-order error dynamical system into a nonlinear system with antisymmetric structure. By choosing appropriate fractional-order Lyapunov functions and employing the fractional-order Lyapunov-based stability theory, several sufficient conditions are obtained to ensure the asymptotical stabilization of the fractional-order error system at the origin. The proposed method is universal, simple, and theoretically rigorous. Finally, some numerical examples are presented to illustrate the validity of theoretical results.

Journal ArticleDOI
TL;DR: In this article, a coupled system of implicit impulsive boundary value problems (IBVPs) of fractional differential equations (FODEs) was studied and conditions for the existence and uniqueness of positive solutions were obtained.
Abstract: In this paper, we study a coupled system of implicit impulsive boundary value problems (IBVPs) of fractional differential equations (FODEs). We use the Schaefer fixed point and Banach contraction theorems to obtain conditions for the existence and uniqueness of positive solutions. We discuss Hyers–Ulam (HU) type stability of the concerned solutions and provide an example for illustration of the obtained results.

Journal ArticleDOI
TL;DR: In this article, a class of $4{th}$ -order neutral delay differential equations with continuously distributed delay is studied, and a new oscillation criterion using the Riccati transformation is established.
Abstract: In this paper, a class of $4{th}$ -order neutral delay differential equations with continuously distributed delay is studied. We establish a new oscillation criterion using the Riccati transformation. An example illustrating the results is also given.

Journal ArticleDOI
TL;DR: In this article, a new approximate technique is introduced to find a solution of FVFIDE with mixed boundary conditions, where the fractional derivatives are replaced by the Caputo operator, and the solution is demonstrated by the hybrid orthonormal Bernstein and block-pulse functions wavelet method.
Abstract: A new approximate technique is introduced to find a solution of FVFIDE with mixed boundary conditions. This paper started from the meaning of Caputo fractional differential operator. The fractional derivatives are replaced by the Caputo operator, and the solution is demonstrated by the hybrid orthonormal Bernstein and block-pulse functions wavelet method (HOBW). We demonstrate the convergence analysis for this technique to emphasize its reliability. The applicability of the HOBW is demonstrated using three examples. The approximate results of this technique are compared with the correct solutions, which shows that this technique has approval with the correct solutions to the problems.

Journal ArticleDOI
TL;DR: In this article, the authors considered a nonlocal boundary value problem of nonlinear implicit fractional Langevin equation with noninstantaneous impulses and studied the existence, uniqueness and generalized Ulam-Hyers-Rassias stability of the proposed model with the help of fixed point approach, over generalized complete metric space.
Abstract: In this paper, we consider a nonlocal boundary value problem of nonlinear implicit fractional Langevin equation with noninstantaneous impulses. We study the existence, uniqueness and generalized Ulam–Hyers–Rassias stability of the proposed model with the help of fixed point approach, over generalized complete metric space. We give an example which supports our main result.

Journal ArticleDOI
TL;DR: In this article, a generalized impulsive fractional order differential equation (DE) involving a nonlinear p-Laplacian operator was considered and the existence of a solution, uniqueness and the Hyers-Ulam stability was established.
Abstract: We deal with three important aspects of a generalized impulsive fractional order differential equation (DE) involving a nonlinear p-Laplacian operator: the existence of a solution, the uniqueness and the Hyers–Ulam stability. Our problem involves Caputo’s fractional derivative. For these goals, we establish an equivalent fractional integral form of the problem and use a topological degree approach for the existence and uniqueness of the solution (EUS). Next, we check the stability of the suggested problem and then demonstrate the results via an illustrative example. In the literature, we could not find articles on the Hyers–Ulam stability of the impulsive fractional order DEs with $\phi _{p}$ operator.

Journal ArticleDOI
Osama Moaaz1
TL;DR: In this paper, sufficient conditions for the oscillation of neutral differential equation second order NDEs were provided. But the Riccati substitution technique was not employed to obtain the conditions.
Abstract: The aim of this work is to offer sufficient conditions for the oscillation of neutral differential equation second order $$ \bigl( r ( t ) \bigl[ \bigl( y ( t ) +p ( t ) y \bigl( \tau ( t ) \bigr) \bigr) ^{\prime } \bigr] ^{\gamma } \bigr) ^{\prime }+f \bigl( t,y \bigl( \sigma ( t ) \bigr) \bigr) =0, $$ where $\int ^{\infty }r^{-1/\gamma } ( s ) \,\mathrm{d}s= \infty $. Based on the comparison with first order delay equations and by employ the Riccati substitution technique, we improve and complement a number of well-known results. Some examples are provided to show the importance of these results.

Journal ArticleDOI
TL;DR: In this article, the Hermite-Hadamard and midpoint types inequalities were established via a parameter ρ ∈ [0, 1]$676 for a function F whose ρ is η-quasiconvex on ρ with ρ ≥ 0.
Abstract: We establish new quantum Hermite–Hadamard and midpoint types inequalities via a parameter $\mu \in [0,1]$ for a function F whose $|{}_{\alpha }D_{q}F|^{u}$ is η-quasiconvex on $[\alpha ,\beta ]$ with $u\geq 1$ . Results obtained in this paper generalize, sharpen, and extend some results in the literature. For example, see (Noor et al. in Appl. Math. Comput. 251:675–679, 2015; Alp et al. in J. King Saud Univ., Sci. 30:193–203, 2018) and (Kunt et al. in Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat. 112:969–992, 2018). By choosing different values of μ, loads of novel estimates can be deduced. We also present some illustrative examples to show how some consequences of our results may be applied to derive more quantum inequalities.

Journal ArticleDOI
TL;DR: In this article, three fractional chaotic maps based on the well known 3D Stefanski, Rossler, and Wang maps are proposed and the dynamics of the proposed fractional maps are investigated experimentally by means of phase portraits, bifurcation diagrams, and Lyapunov exponents.
Abstract: In this paper, we propose three fractional chaotic maps based on the well known 3D Stefanski, Rossler, and Wang maps. The dynamics of the proposed fractional maps are investigated experimentally by means of phase portraits, bifurcation diagrams, and Lyapunov exponents. In addition, three control laws are introduced for these fractional maps and the convergence of the controlled states towards zero is guaranteed by means of the stability theory of linear fractional discrete systems. Furthermore, a combined synchronization scheme is introduced whereby the fractional Rossler map is considered as a drive system with the response system being a combination of the remaining two maps. Numerical results are presented throughout the paper to illustrate the findings.

Journal ArticleDOI
TL;DR: In this paper, a stochastic SIS epidemic model with nonlinear incidence rate is proposed and analyzed, and the authors show that the condition for the epidemic disease to go to extinction is weaker than that of the deterministic system.
Abstract: In this paper, considering the impact of stochastic environment noise on infection rate, a stochastic SIS epidemic model with nonlinear incidence rate is proposed and analyzed. Firstly, for the corresponding deterministic system, the threshold which determines the extinction or permanence of the disease is obtained by analyzing the stability of the equilibria. Then, for the stochastic system, the global dynamics is investigated by using the theory of stochastic differential equations; especially the threshold dynamics is explored when the stochastic environment noise is small. The results show that the condition for the epidemic disease to go to extinction in the stochastic system is weaker than that of the deterministic system, which implies that stochastic noise has a significant impact on the spread of infectious diseases and the larger stochastic noise is conducive to controlling the epidemic diseases. To illustrate this phenomenon, we give some computer simulations with different intensities of the stochastic noise.