Showing papers in "Advances in Difference Equations in 2016"

Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels

Abstract: In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration by parts formulas are proved. Then a discrete variational problem is considered with an illustrative example. Finally, some more tools for these derivatives and their discrete versions have been obtained.

Extinction for a discrete competition system with the effect of toxic substances

Abstract: A nonautonomous discrete competitive system with nonlinear inter-inhibition terms and one toxin producing species is studied in this paper. Sufficient conditions which guarantee the extinction of one of the components are obtained and the global attractivity of the other one is proved. Our results supplement some existing ones. Numerical simulations show the feasibility of our results.

Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular

Abstract: In this work, we present an analysis based on a combination of the Laplace transform and homotopy methods in order to provide a new analytical approximated solutions of the fractional partial differential equations (FPDEs) in the Liouville-Caputo and Caputo-Fabrizio sense. So, a general scheme to find the approximated solutions of the FPDE is formulated. The effectiveness of this method is demonstrated by comparing exact solutions of the fractional equations proposed with the solutions here obtained.

Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation

Abstract: Recently, Caputo and Fabrizio proposed a new derivative with fractional order without singular kernel. The derivative has several interesting properties that are useful for modeling in many branches of sciences. For instance, the derivative is able to describe substance heterogeneities and configurations with different scales. In order to accommodate researchers dealing with numerical analysis, we propose a numerical approximation in time and space. We modified the advection dispersion equation by replacing the time derivative with the new fractional derivative. We solve numerically the modified equation using the proposed numerical approximation. The stability and convergence analysis of the numerical scheme were presented together with some simulations.

Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel

Abstract: This paper presents the procedure to obtain analytical solutions of Lienard type model of a fluid transmission line represented by the Caputo-Fabrizio fractional operator. For such a model, we derive a new approximated analytical solution by using the Laplace homotopy analysis method. Both the efficiency and the accuracy of the method are verified by comparing the obtained solutions with the exact analytical solution. Good agreement between them is confirmed.

Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions

Abstract: In this article we discuss some of the qualitative properties of fractional difference operators. We especially focus on the connections between the fractional difference operator and the monotonicity and convexity of functions. In the integer-order setting, these connections are elementary and well known. However, in the fractional-order setting the connections are very complicated and muddled. We survey some of the known results and suggest avenues for future research. In addition, we discuss the asymptotic behavior of solutions to fractional difference equations and how the nonlocal structure of the fractional difference can be used to deduce these asymptotic properties.

Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces

Abstract: In this article, we prove that the ω-periodic discrete evolution family $\Gamma:= \{\rho(n,k): n, k \in\mathbb{Z}_{+}, n\geq k\}$ of bounded linear operators is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions. More precisely, we prove that if for each real number γ and each sequence $(\xi(n))$ taken from some Banach space, the approximate solution of the nonautonomous ω-periodic discrete system $\theta _{n+1} = \Lambda_{n}\theta_{n}$ , $n\in\mathbb{Z}_{+}$ is represented by $\phi _{n+1}=\Lambda_{n}\phi_{n}+e^{i\gamma(n+1)}\xi(n+1)$ , $n\in\mathbb{Z}_{+}$ ; $\phi_{0}=\theta_{0}$ , then the Hyers-Ulam stability of the nonautonomous ω-periodic discrete system $\theta_{n+1} = \Lambda_{n}\theta_{n}$ , $n\in\mathbb{Z}_{+}$ is equivalent to its uniform exponential stability.

ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect

Abstract: Two-dimensional Rossby solitary waves propagating in a line have attracted much attention in the past decade, whereas there is few research on three-dimensional Rossby solitary waves. But as is well known, three-dimensional Rossby solitary waves are more suitable for real ocean and atmosphere conditions. In this paper, using multiscale and perturbation expansion method, a new Zakharov-Kuznetsov (ZK)-Burgers equation is derived to describe three-dimensional Rossby solitary waves that propagate in a plane. By analyzing the equation we obtain the conservation laws of three-dimensional Rossby solitary waves. Based on the sine-cosine method, we give the classical solitary wave solutions of the ZK equation; on the other hand, by the Hirota method we also obtain the rational solutions, which are similar to the solutions of the Benjamin-Ono (BO) equation, the solutions of which can describe the algebraic solitary waves. The rational solutions of the ZK equations are worth of attention. Finally, with the help of the classical solitary wave solutions, similar to the fiber soliton communication, we discuss the dissipation and chirp effect of three-dimensional Rossby solitary waves.

Global Mittag-Leffler projective synchronization for fractional-order neural networks: an LMI-based approach

Abstract: This paper is concerned with the global projective synchronization issue for fractional neural networks in the Mittag-Leffler stability sense. Firstly, a fractional-order differential inequality in the existing literature for the Caputo fractional derivative, with $0<\alpha< 1$ , is improved, which plays a central role in the synchronization analysis. Secondly, hybrid control strategies are designed via combing open loop control and adaptive control, and unknown control parameters are determined by the adaptive fractional updated laws to achieve global projective synchronization. In addition, applying the fractional Lyapunov approach and Mittag-Leffler function, the projective synchronization conditions are addressed in terms of linear matrix inequalities (LMIs) to ensure the synchronization. Finally, two examples are given to demonstrate the validity of the proposed method.

Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets

Abstract: In this paper, a collocation method based on Laguerre wavelets is proposed for the numerical solutions of linear and nonlinear singular boundary value problems. Laguerre wavelet expansions together with operational matrix of integration are used to convert the problems into systems of algebraic equations which can be efficiently solved by suitable solvers. Illustrative examples are given to demonstrate the validity and applicability of this technique, and the results have been compared with the exact solutions.

Positive Solutions for Hadamard Fractional Differential Equations on Infinite Domain

Abstract: This paper investigates the existence of nonnegative multiple solutions for nonlinear fractional differential equations of Hadamard type, with nonlocal fractional integral boundary conditions on an unbounded domain by means of Leggett-Williams and Guo-Krasnoselskii’s fixed point theorems. Two examples are discussed for illustration of the main work.

New results on higher-order Daehee and Bernoulli numbers and polynomials

Abstract: We derive a new matrix representation for higher-order Daehee numbers and polynomials, higher-order λ-Daehee numbers and polynomials, and twisted λ-Daehee numbers and polynomials of order k This helps us to obtain simple and short proofs of many previous results on higher-order Daehee numbers and polynomials Moreover, we obtain recurrence relations, explicit formulas, and some new results for these numbers and polynomials Furthermore, we investigate the relation between these numbers and polynomials and Stirling, Norlund, and Bernoulli numbers of higher-order Some numerical results and program are introduced using Mathcad for generating higher-order Daehee numbers and polynomials The results of this article generalize the results derived very recently by El-Desouky and Mustafa (Appl Math Sci 9(73):3593-3610, 2015)

A generalized virus dynamics model with cell-to-cell transmission and cure rate

Abstract: Viruses can be spread and transmitted through two fundamental modes, one by virus-to-cell infection and the other by direct cell-to-cell transmission. In this paper, we propose a new generalized virus dynamics model, which incorporates both modes and takes into account the cure of infected cells. We first show mathematically and biologically the well-posedness of our model. Further, an explicit formula for the basic reproduction number $R_{0}$ of the model is determined. By analyzing the characteristic equations we establish the local stability of the disease-free equilibrium and the chronic infection equilibrium in terms of $R_{0}$ . The global behavior of the model is investigated by constructing an appropriate Lyapunov functional for disease-free equilibrium and by applying geometrical approach to chronic infection equilibrium. Moreover, mathematical virus models and results presented in many previous studies are generalized and improved.

Monotone iterative technique for a nonlinear fractional q-difference equation of Caputo type

Abstract: By establishing a comparison theorem and applying the monotone iterative technique combined with the method of lower and upper solutions, we investigate the existence of extremal solutions of the initial value problem for fractional q-difference equation involving Caputo derivative. An example is presented to illustrate the main result.

Global dynamics of an age-structured in-host viral infection model with humoral immunity

Abstract: An age-structured in-host viral infection model with humoral immunity, consisting of partial differential and ordinary differential equations, is investigated. By calculation we get the basic reproduction number $\Re_{0}$ and the immune-activated reproduction number $\Re_{1}$ . By analyzing the characteristic equations, the local stability of an infection-free steady state, an immune-inactivated infected steady state and an immune-activated infected steady state of the model is established. By using suitable Lyapunov functionals and LaSalle’s invariance principle, it is proved that if $\Re_{0}<1$ , the infection-free steady state is globally asymptotically stable; if $\Re _{1}<1<\Re_{0}$ , the immune-inactivated infected steady state is globally asymptotically stable; and if $\Re_{1}>1$ , the immune-activated infected steady state is globally asymptotically stable. Numerical simulations are carried out to illustrate the theoretical results.

Effect of awareness programs and travel-blocking operations in the control of HIV/AIDS outbreaks: a multi-domains SIR model

Abstract: The main goal of this article is to exhibit the importance of awareness programs and travel-blocking operations, in the prevention of HIV/AIDS outbreaks, based on a multi-domains SIR epidemic model. The model devised describes the spatial-temporal spread of HIV/AIDS, in p neighboring domains, taking into account the epidemiological diversity of their populations. For a first case of the study, we focus on discussing the benefits of awareness campaigns that aim to raise public consciousness among susceptible people, about the danger of HIV/AIDS. Thus, for the reason that intra-domains interventions are not always sufficient to prevent people belonging to a domain (neighborhood, town or city for example), from a rapid expansion of HIV/AIDS, we propose the travel-blocking strategy, as a second approach presenting inter-domains interventions that health-policy makers could follow, by restricting the number of suspicious people that can participate in spreading HIV/AIDS via travel, coming from domains at high-risk of infection to enter domains with lower risk. The optimal control theory, based on Pontryagin’s maximum principle, is applied twice in this paper, for the characterizations of the awareness and travel-blocking controls. The numerical results associated to the multi-points boundary value problems are obtained based on the Forward-Backward Sweep Method combined with progressive-regressive Runge-Kutta fourth-order schemes.

Two-dimensional product-type system of difference equations solvable in closed form

Abstract: A solvable two-dimensional product-type system of difference equations of interest is presented. Closed form formulas for its general solution are given.

Positive solution for singular fractional differential equations involving derivatives

Abstract: By using the fixed point theorem for the mixed monotone operator, the existence of unique positive solutions for singular nonlocal boundary value problems of fractional differential equations is established. An example is provided to illustrate the main results.

Qualitative behavior of a smoking model

Abstract: We study the qualitative behavior of a smoking model in which the population is divided into five classes, that is, non-smokers, smokers, people who temporarily quit smoking, people who permanently quit smoking, and people who are associated with illness due to smoking. The global asymptotic stability of the unique positive equilibrium point is presented. More precisely, a graph-theoretic method is used to prove the global stability of the unique positive equilibrium point.

Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

Abstract: In this article, we study the existence of iterative positive solutions for a class of singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions, where the nonlinear term may be singular both for time and space variables. By using the properties of the Green function and the fixed point theorem of mixed monotone operators in cones we obtain some results on the existence and uniqueness of positive solutions. We also construct successively some sequences for approximating the unique solution. Our results include the multipoint boundary problems and integral boundary problems as special cases, and we also extend and improve many known results including singular and non-singular cases.

Application of measures of noncompactness to the infinite system of second-order differential equations in ℓ p $\ell_{p}$ spaces

Abstract: In this article, we use the technique based upon measures of noncompactness in conjunction with a Darbo-type fixed point theorem with a view to studying the existence of solutions of infinite systems of second-order differential equations in the Banach sequence space $\ell_{p}$ . An illustrative example is also given in support of our existence result.

A new construction of a fractional derivative mask for image edge analysis based on Riemann-Liouville fractional derivative

Abstract: We present a new way of constructing a fractional-based convolution mask with an application to image edge analysis. The mask was constructed based on the Riemann-Liouville fractional derivative which is a special form of the Srivastava-Owa operator. This operator is generally known to be robust in solving a range of differential equations due to its inherent property of memory effect. However, its application in constructing a convolution mask can be devastating if not carefully constructed. In this paper, we show another effective way of constructing a fractional-based convolution mask that is able to find edges in detail quite significantly. The resulting mask can trap both local discontinuities in intensity and its derivatives as well as locating Dirac edges. The experiments conducted on the mask were done using some selected well known synthetic and Medical images with realistic geometry. Using visual perception and performing both mean square error and peak signal-to-noise ratios analysis, the method demonstrated significant advantages over other known methods.

A new approach for one-dimensional sine-Gordon equation

Abstract: In this work, we use a reproducing kernel method for investigating the sine-Gordon equation with initial and boundary conditions. Numerical experiments are studied to show the efficiency of the technique. The acquired results are compared with the exact solutions and results obtained by different methods. These results indicate that the reproducing kernel method is very effective.

Extremal solutions for some periodic fractional differential equations

Abstract: By using the lower and upper solution method, the existence of an iterative solution for a class of fractional periodic boundary value problems, $$\begin{aligned}& D_{0+}^{\alpha}u(t)=f\bigl(t, u(t)\bigr),\quad t \in(0, h),\\& \lim_{t \to0^{+}}t^{1-\alpha}u(t) = h^{1-\alpha}u(h), \end{aligned}$$ is discussed, where $0< h<+\infty$ , $f\in C([0, h]\times R, R)$ , $D_{0+}^{\alpha}u (t) $ is the Riemann-Liouville fractional derivative, $0<\alpha< 1$ . Different from other well-known results, a new condition on the nonlinear term is given to guarantee the equivalence between the solution of the periodic boundary value problem and the fixed point of the corresponding operator. Moreover, the existence of extremal solutions for the problem is given.

Convergences of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes

Abstract: A stage-structured predator-prey model (stage structure for both predator and prey) with modified Leslie-Gower and Holling-II schemes is studied in this paper. Using the iterative technique method and the fluctuation lemma, sufficient conditions which guarantee the global stability of the positive equilibrium and boundary equilibrium are obtained. Our results indicate that for a stage-structured predator-prey community, both the stage structure and the death rate of the mature species are the important factors that lead to the permanence or extinction of the system.

Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments

Abstract: In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for boundary value problems of Hadamard-type fractional functional differential equations and inclusions involving both retarded and advanced arguments. We make use of the standard tools of fixed point theory to obtain the main results.

Global dynamics for discrete-time analog of viral infection model with nonlinear incidence and CTL immune response

Abstract: In this paper, a discrete-time analog of a viral infection model with nonlinear incidence and CTL immune response is established by using the Micken non-standard finite difference scheme. The two basic reproduction numbers $R_{0}$ and $R_{1}$ are defined. The basic properties on the positivity and boundedness of solutions and the existence of the virus-free, the no-immune, and the infected equilibria are established. By using the Lyapunov functions and linearization methods, the global stability of the equilibria for the model is established. That is, when $R_{0}\leq1$ then the virus-free equilibrium is globally asymptotically stable, and under the additional assumption $(A_{4})$ when $R_{0}>1$ and $R_{1}\leq1$ then the no-immune equilibrium is globally asymptotically stable and when $R_{0}>1$ and $R_{1}>1$ then the infected equilibrium is globally asymptotically stable. Furthermore, the numerical simulations show that even if assumption $(A_{4})$ does not hold, the no-immune equilibrium and the infected equilibrium also may be globally asymptotically stable.

A PD-type iterative learning control algorithm for singular discrete systems

Abstract: Based on a specific decomposition of discrete singular systems, in this paper, we study the problem of state tracking control by using PD-type algorithm of iterative learning control. The convergence conditions and theoretical analysis of the PD-type algorithm are presented in detail. An illustrative example supporting the theoretical results and the effectiveness of the PD-type iterative learning control algorithm for discrete singular systems is shown at the end of the paper.

A geometric approach for solving the density-dependent diffusion Nagumo equation

Abstract: In this paper, some solutions of the density-dependent diffusion Nagumo equation are obtained by using a new approach, the Lie symmetry group-preserving scheme (LSGPS). The effects of various model parameters on the solution are investigated graphically using LSGPS. Finally, a different reduction method of PDEs is applied to construct two new analytical solutions and a first integral of the Nagumo equation.

Stability analysis of a fractional-order epidemics model with multiple equilibriums

Abstract: In this paper, we extend the SIR model with vaccination into a fractional-order model by using a system of fractional ordinary differential equations in the sense of the Caputo derivative of order $\alpha\in(0,1]$ . By applying fractional calculus, we give a detailed analysis of the equilibrium points of the model. In particular, we analytically obtain a certain threshold value of the basic reproduction number $R_{0}$ and describe the existence conditions of multiple equilibrium points. Moreover, it is shown that the stability region of the equilibrium points increases by choosing an appropriate value of the fractional order α. Finally, the analytical results are confirmed by some numerical simulations for real data related to pertussis disease.