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 Advances in Difference Equations   [SJR: 0.647]   [H-I: 25]   [1 followers]  Follow       Open Access journal    ISSN (Print) 1687-1839 - ISSN (Online) 1687-1847    Published by SpringerOpen  [225 journals]
• Global analysis of a new nonlinear stochastic differential competition
system with impulsive effect

• Abstract: Abstract We propose a new stochastic competition chemostat system with saturated growth rate and impulsive toxicant input. The main purpose of this paper is to study the stochastic dynamics of a high-dimensional impulsive stochastic chemostat model and find the threshold between persistence and extinction for the impulsive stochastic chemostat system. First, we investigate the stability of the periodic solution of a deterministic impulsive chemostat model and obtain the threshold between persistence and extinction for the system. Second, by using qualitative analysis method of impulsive stochastic differential equations, we obtain conditions for the extinction and persistence in mean of two microorganisms in the stochastic chemostat model. The results show that a stochastic disturbance or the impulsive effect can cause the microorganisms to go to extinction. Finally, we provide some examples together with numerical simulations to illustrate the analytical results and explain the biological implications.
PubDate: 2017-09-22

• Fractional-order model for biocontrol of the lesser date moth in palm
trees and its discretization

• Abstract: Abstract In this paper, a fractional-order model of palm trees, the lesser date moth and the predator is presented. Existence conditions of the local asymptotic stability of the equilibrium points of the fractional system are analyzed. We prove that the positive equilibrium point is globally stable also. The numerical simulations come to illustrate the dynamical behaviors of the model such as bifurcation and chaos phenomenon, and the numerical simulations confirm the validity of our theoretical results.
PubDate: 2017-09-21

• Stability analysis of a discrete competitive system with nonlinear
interinhibition terms

• Abstract: Abstract We propose and study a discrete competitive system of the following form: \begin{aligned} &x_{1}(n+1)=x_{1}(n)\exp{\biggl[r_{1}-a_{1}x_{1}(n)- \frac {b_{1}x_{2}(n)}{1+c_{2}x_{2}(n)}\biggr]}, \\ &x_{2}(n+1)=x_{2}(n)\exp{\biggl[r_{2}-a_{2}x_{2}(n)- \frac {b_{2}x_{1}(n)}{1+c_{1}x_{1}(n)}\biggr]}. \end{aligned} We obtain some conditions for the local stability of the equilibria. Using the iterative method and the comparison principle of a difference equation, we also obtain a set of sufficient conditions that ensure the global stability of the interior equilibrium. Numeric simulations show the feasibility of the main results. Our results supplement and complement some known results.
PubDate: 2017-09-20

• Existence and blowup of solutions for the modified Klein-Gordon-Zakharov
equations for plasmas with a quantum correction

• Abstract: Abstract This paper studies the existence and blowup of solutions for the modified Klein-Gordon-Zakharov equations for plasmas with a quantum correction, which describe the interaction between high frequency Langmuir waves and low frequency ion-acoustic waves in a plasma considering the quantum effects. Firstly the existence and uniqueness of the local smooth solutions are obtained by the a priori estimates and the Galerkin method. Secondly, and what is more, by introducing some auxiliary functionals and invariant manifolds, the authors study and derive a sharp threshold for the global existence and blowup of solutions by applying potential well argument and the concavity method. Furthermore, two more specific conditions of how small the initial data are for the solutions to exist globally are concluded by the dilation transformation.
PubDate: 2017-09-19

• Oscillations of differential equations generated by several deviating
arguments

• Abstract: Abstract Sufficient conditions, involving limsup and liminf, for the oscillation of all solutions of differential equations with several not necessarily monotone deviating arguments and nonnegative coefficients are established. Corresponding differential equations of both delayed and advanced type are studied. We illustrate the results and the improvement over other known oscillation criteria by examples, numerically solved in MATLAB.
PubDate: 2017-09-19

• Numerical analysis for the Klein-Gordon equation with mass parameter

• Abstract: Abstract A numerical analysis of the well-known linear partial differential equation describing the relativistic wave is presented in this work. Three different operators of fractional differentiation with power law, exponential decay law and Mittag-Leffler law are employed to extend the Klein-Gordon equation with mass parameter to the concept of fractional differentiation. The three models are solved numerically. The stability and the convergence of the numerical schemes are investigated in detail.
PubDate: 2017-09-16

• Qualitative behaviors of the high-order Lorenz-Stenflo chaotic system
arising in mathematical physics describing the atmospheric
acoustic-gravity waves

• Abstract: Abstract The boundedness of chaotic systems plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors, the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, and chaos synchronization. However, as far as the authors know, there are only a few papers dealing with bounds of high-order chaotic systems due to their complex algebraic structure. To sort this out, in this paper, we study the bounds of a high-order Lorenz-Stenflo system arising in mathematical physics. Based on Lyapunov stability theory, we show that there exists a globally exponential attractive set for this system. The innovation of the paper is that we not only prove that this system is globally bounded for all the parameters, but also give a family of mathematical expressions of global exponential attractive sets of this system with respect to its parameters. We also study some other dynamical characteristics of this chaotic system such as invariant sets and chaotic behaviors. To justify the theoretical analysis, we carry out detailed numerical simulations.
PubDate: 2017-09-15

• Mathematical modeling of infectious disease transmission in macroalgae

• Abstract: Abstract Understanding the infectious diseases outbreak of algae can provide significant knowledge for disease control intervention and/or prevention. We consider here a disease caused by highly pathogenic organisms that can result in the death of algae. Even though a great deal of understanding about diseases of algae has been reached, studies concerning effects of the outbreak at the population level are still rare. For this reason, we computationally model an outbreak in the algae reservoir or container systems consisting of several patches or clusters of algae being infected with a contagious infectious disease. We computationally investigate the systems as well as make some predictions via the deterministic SEIR epidemic model. We consider the factors that could affect the spread of the disease including the number of patches, the size of initial infected population, the distance between patches or spatial range, and the basic reproduction number ( $$R_{0}$$ ). The results provide some information that may be beneficial to algae disease control, intervention or prevention.
PubDate: 2017-09-15

• Discrete Schrödinger equations in the nonperiodic and superlinear
cases: homoclinic solutions

• Abstract: Abstract Using variational methods, we study the existence and multiplicity of homoclinic solutions for a class of discrete Schrödinger equations in infinite m-dimensional lattices with nonlinearities being superlinear at infinity. Our results generalize some existing results in the literature by using some weaker conditions.
PubDate: 2017-09-15

• A general class of noninstantaneous impulsive fractional differential
inclusions in Banach spaces

• Abstract: Abstract In this paper we introduce the concept of a PC-mild solution to a general new class of noninstantaneous impulsive fractional differential inclusions involving the generalized Caputo derivative with the lower bound at zero in infinite dimensional Banach spaces. Using the formula of a PC-mild solution, we give two classes of sufficient conditions to guarantee the existence of PC-mild solutions via fixed point theorems for multivalued functions. Also we characterize the compactness of the solution set. We introduce the concept of generalized Ulam-Hyers stability and present a generalized Ulam-Hyers stability result using multivalued weakly Picard operator theory. Examples are given to illustrate the theoretical results.
PubDate: 2017-09-15

• An exponential B-spline collocation method for the fractional
sub-diffusion equation

• Abstract: Abstract In this article, we propose an exponential B-spline approach to obtain approximate solutions for the fractional sub-diffusion equation of Caputo type. The presented method is established via a uniform nodal collocation strategy by using an exponential B-spline based interpolation in conjunction with an effective finite difference scheme in time. The unique solvability is rigorously proved. The unconditional stability is well illustrated via a procedure closely resembling the classic von Neumann technique. A series of numerical examples are carried out, and by contrast to other algorithms available in the open literature, numerical results confirm the validity and superiority of our method.
PubDate: 2017-09-13

• Unconditional stable numerical techniques for a water-quality model in a
non-uniform flow stream

• Abstract: Abstract Two mathematical models are used to simulate water quality in a non-uniform flow stream. The first model is the hydrodynamic model that provides the velocity field and the water elevation. The second model is an advection-diffusion-reaction model that provides the pollutant concentration field. Both models are formulated as one-dimensional equations. The traditional Crank-Nicolson method is also used in the hydrodynamic model. At each step, the flow velocity fields calculated from the first model are the inputs into the second model. A new fourth-order scheme and a Saulyev scheme are simultaneously employed in the second model. This paper proposes a remarkably simple alteration to the fourth-order method so as to make it more accurate without any significant loss of computational efficiency. The results obtained indicate that the proposed new fourth-order scheme, coupled to the Saulyev method, does improve the prediction accuracy compared to that of the traditional methods.
PubDate: 2017-09-13

• Integral spectral Tchebyshev approach for solving space Riemann-Liouville

• Abstract: Abstract The principal aim of this paper is to analyze and implement two numerical algorithms for solving two kinds of space fractional linear advection-dispersion problems. The proposed numerical solutions are spectral and they are built on assuming the approximate solutions to be certain double shifted Tchebyshev basis. The two typical collocation and Petrov-Galerkin spectral methods are applied to obtain the desired numerical solutions. The special feature of the two proposed methods is that their applications enable one to reduce, through integration, the fractional problem under investigation into linear systems of algebraic equations, which can be efficiently solved via any suitable solver. The convergence and error analysis of the double shifted Tchebyshev basis are carefully investigated, aiming to illustrate the correctness and feasibility of the proposed double expansion. Finally, the efficiency, applicability, and high accuracy of the suggested algorithms are demonstrated by presenting some numerical examples accompanied with comparisons with some other existing techniques discussed in the literature.
PubDate: 2017-09-13

• Existence of positive mild solutions for a class of fractional evolution
equations on the half line

• Abstract: The equivalent integral equation of a new form for a class of fractional evolution equations is obtained by the method of Laplace transform, which is different from those given in the existing literature. By the monotone iterative method without the assumption of lower and upper solutions, we present some new results on the existence of positive mild solutions for the abstract fractional evolution equations on the half-line.
PubDate: 2017-09-12

• Generalized finite difference/spectral Galerkin approximations for the
time-fractional telegraph equation

• Abstract: Abstract We discuss the numerical solution of the time-fractional telegraph equation. The main purpose of this work is to construct and analyze stable and high-order scheme for solving the time-fractional telegraph equation efficiently. The proposed method is based on a generalized finite difference scheme in time and Legendre spectral Galerkin method in space. Stability and convergence of the method are established rigorously. We prove that the temporal discretization scheme is unconditionally stable and the numerical solution converges to the exact one with order $$\mathcal {O}(\tau^{2-\alpha}+N^{1-\omega})$$ , where $$\tau, N$$ , and ω are the time step size, polynomial degree, and regularity of the exact solution, respectively. Numerical experiments are carried out to verify the theoretical claims.
PubDate: 2017-09-12

• The roles of maturation delay and vaccination on the spread of Dengue
virus and optimal control

• Abstract: Abstract A mathematical model of Dengue virus transmission between mosquitoes and humans, incorporating a control strategy of imperfect vaccination and vector maturation delay, is proposed in this paper. By using some analytical skills, we obtain the threshold conditions for the global attractiveness of two disease-free equilibria and prove the existence of a positive equilibrium for this model. Further, we investigate the sensitivity analysis of threshold conditions. Additionally, using the Pontryagin maximum principle, we obtain the optimal control strategy for the disease. Finally, numerical simulations are delivered to verify the correctness of the theoretical results, the feasibility of a vaccination control strategy, and the influences of the controlling parameters on the control and elimination of this disease. Theoretical results and numerical simulations show that the vaccination rate and effectiveness of vaccines are two key factors for the control of Dengue spread, and the manufacture of the Dengue vaccine is also architecturally significant.
PubDate: 2017-09-12

• Local uniqueness of positive solutions for a coupled system of fractional
differential equations with integral boundary conditions

• Abstract: Abstract In this paper, we study a coupled system of fractional boundary value problems subject to integral boundary conditions. By applying a recent fixed point theorem in ordered Banach spaces, we investigate the local existence and uniqueness of positive solutions for the coupled system. We show that the unique positive solution can be found in a product set, and that it can be approximated by constructing iterative sequences for any given initial point of the product set. As an application, an interesting example is presented to illustrate our main result.
PubDate: 2017-09-12

• An effective collocation technique to solve the singular Fredholm integral
equations with Cauchy kernel

• Abstract: Abstract In this paper, an effective numerical method to solve the Cauchy type singular Fredholm integral equations (CSFIEs) of the first kind is proposed. The collocation technique based on Bernstein polynomials is used for approximation the solution of various cases of CSFIEs. By transforming the problem into systems of linear algebraic equations, we see that this approach is computationally simple and attractive. Then the approximate solution of the problem in truncated series form is obtained by using the matrix form of this method. Convergence and error analyses of the presented method are mentioned. Finally, numerical experiments show the validity, accuracy, and efficiency of the proposed method.
PubDate: 2017-09-12

• Bounded and periodic solutions to the linear first-order difference
equation on the integer domain

• Abstract: Abstract The existence of bounded solutions to the linear first-order difference equation on the set of all integers is studied. Some sufficient conditions for the existence of solutions converging to zero when $$n\to -\infty$$ , as well as when $$n\to+\infty$$ , are also given. For the case when the coefficients of the equation are periodic, the long-term behavior of non-periodic solutions is studied.
PubDate: 2017-09-12

• The modified two-dimensional Toda lattice with self-consistent sources

• Abstract: Abstract In this paper, we derive the Grammian determinant solutions to the modified two-dimensional Toda lattice, and then we construct the modified two-dimensional Toda lattice with self-consistent sources via the source generation procedure. We show the integrability of the modified two-dimensional Toda lattice with self-consistent sources by presenting its Casoratian and Grammian structure of the N-soliton solution. It is also demonstrated that the commutativity between the source generation procedure and Bäcklund transformation is valid for the two-dimensional Toda lattice.
PubDate: 2017-09-11

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