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 Advances in Difference Equations   [SJR: 0.647]   [H-I: 25]   [2 followers]  Follow       Open Access journal    ISSN (Print) 1687-1839 - ISSN (Online) 1687-1847    Published by SpringerOpen  [226 journals]
• Global existence of solutions for a fractional Caputo nonlocal thermistor
problem

• Abstract: We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then additional existence and continuation theorems are obtained, ensuring global existence of solutions.
PubDate: 2017-11-15

• Attracting and quasi-invariant sets of neutral stochastic
integro-differential equations with impulses driven by fractional Brownian
motion

• Abstract: The paper is devoted to investigating a class of neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion. By establishing two new impulsive integral inequalities which improve the inequalities established by Li (Neurocomputing 177:620-627, 2016) and Long et al. (Stat. Probab. Lett. 82(9):1699-1709, 2012), attracting and quasi-invariant sets of the system are obtained. Moreover, exponential stability of the mild solution is established with sufficient conditions.
PubDate: 2017-11-10

• Numerical solution of Volterra partial integro-differential equations
based on sinc-collocation method

• Abstract: We provide the numerical solution of a Volterra integro-differential equation of parabolic type with memory term subject to initial boundary value conditions. Finite difference method in combination with product trapezoidal integration rule is used to discretize the equation in time and sinc-collocation method is employed in space. A weakly singular kernel has been viewed as an important case in this study. The convergence analysis has been discussed in detail, which shows that the approach exponentially converges to the solution. Furthermore, numerical examples and illustrations are presented to prove the validity of the suggested method.
PubDate: 2017-11-10

• Optimality conditions for fractional variational problems with
Caputo-Fabrizio fractional derivatives

• Abstract: In this paper, we study the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrange function depending on a Caputo-Fabrizio fractional derivative. The new kernel of Caputo-Fabrizio fractional derivative has no singularity, which is critical to interpreting the memory aftermath of the system. This property was not precisely illustrated in the previous definitions. Two special cases of fractional variational problems are considered to demonstrate the application of the optimality conditions.
PubDate: 2017-11-09

• A nonstandard finite difference scheme for a multi-group epidemic model
with time delay

• Abstract: In this paper, we derive a discretized multi-group epidemic model with time delay by using a nonstandard finite difference (NSFD) scheme. A crucial observation regarding the advantage of the NSFD scheme is that the positivity and boundedness of solutions of the continuous model are preserved. Furthermore, we show that the discrete model has the same equilibria, and the conditions for their stability are identical in case of both the discrete and the corresponding continuous models. Specifically, if $$\mathfrak{R}_{0}\leq1$$ , then the disease-free equilibrium $$P_{0}$$ is globally asymptotically stable; if $$\mathfrak{R}_{0}>1$$ , then the infection equilibrium $$P_{*}$$ is globally asymptotically stable. The results imply that the discretization scheme can efficiently preserve the global dynamics of the original continuous model.
PubDate: 2017-11-09

• An alternating direction Galerkin method for a time-fractional partial
differential equation with damping in two space dimensions

• Abstract: In this paper, we propose an efficient alternating direction implicit (ADI) Galerkin method for solving the time-fractional partial differential equation with damping, where the fractional derivative is in the sense of Caputo with order in $$(1,2)$$ . The presented numerical scheme is based on the L2- $$1_{\sigma}$$ method in time and the Galerkin finite element method in space. The unconditional stability and convergence of the numerical scheme are both carefully proved. Numerical results are displayed for supporting the theoretical analysis.
PubDate: 2017-11-09

• Singular integral equations of convolution type with Hilbert kernel and a
discrete jump problem

• Abstract: One class of singular integral equations of convolution type with Hilbert kernel is studied in the space $$L^{2}[-\pi, \pi]$$ in the article. Such equations can be changed into either a system of discrete equations or a discrete jump problem depending on some parameter via the discrete Laurent transform. We can thus solve the equations with an explicit representation of solutions under certain conditions.
PubDate: 2017-11-09

• Multi-quasi-synchronization of coupled fractional-order neural networks
with delays via pinning impulsive control

• Abstract: We investigate the collective dynamics of multi-quasi-synchronization of coupled fractional-order neural networks with delays. Using the pinning impulsive strategy, we design a novel controller to pin the coupled networks to realize the multi-quasi-synchronization. Based on the comparison principle and mathematical analysis, we derive some novel criteria of the multi-quasi-synchronization. Moreover, we discuss the effects of coupling strength and pinning control matrix. Finally, some simulation examples show the effectiveness of the presented results.
PubDate: 2017-11-09

• Transmission dynamics of a Huanglongbing model with cross protection

• Abstract: Huanglongbing (HLB) is one of the most common widespread vector-borne transmission diseases through psyllid, which is called a kind of cancer of plant disease. In recent years, biologists have focused on the role of cross protection strategy to control the spread of HLB. In this paper, according to transmission mechanism of HLB, a deterministic model with cross protection is formulated. A threshold value $$R_{0}$$ is established to measure whether or not the disease is uniformly persistent. The existence of a backward bifurcation presents a further sub-threshold condition below $$R_{0}$$ for the spread of the disease. We also discuss the effects of cross protection and removing infected trees in spreading the disease. Numerical simulations suggest that cross protection is a promotion control measure, and replanting trees is bad for HLB control.
PubDate: 2017-11-06

• On fractional Hahn calculus

• Abstract: In this paper, the new concepts of Hahn difference operators are introduced. The properties of fractional Hahn calculus in the sense of a forward Hahn difference operator are introduced and developed.
PubDate: 2017-11-03

• Approximate solutions of a sum-type fractional integro-differential
equation by using Chebyshev and Legendre polynomials

• Abstract: We investigate the existence of solutions for a sum-type fractional integro-differential problem via the Caputo differentiation. By using the shifted Legendre and Chebyshev polynomials, we provide a numerical method for finding solutions for the problem. In this way, we give some examples to illustrate our results.
PubDate: 2017-11-02

• Stationary distribution and extinction of a stochastic SIRS epidemic model
with information intervention

• Abstract: In this paper, a new SIRS epidemic model which considers the influence of information intervention and environmental noise is studied. The study shows that information intervention and white noise have great effects on the disease. First, we show that there is global existence and positivity of the solution. Then, we prove that the stochastic basic production $$\mathscr{R}_{s}$$ is a threshold which determines the extinction or persistence of the disease. When the intensity of noise is large, we obtain $$\mathscr{R}_{s}<1$$ and the disease will die out. When the intensity of noise is small, then $$\mathscr{R}_{s}>1$$ and a sufficient condition for the existence of stationary distribution is obtained, which means the disease is prevalent. Finally, the main results are illustrated by numerical simulations.
PubDate: 2017-11-02

• Existence results for hybrid fractional neutral differential equations

• Abstract: We discuss the existence of solutions of initial value problems for a class of hybrid fractional neutral differential equations. To prove the main results, we use a hybrid fixed point theorem for the sum of three operators. We also derive the dependence of a solution on the initial data and present an example to illustrate the results.
PubDate: 2017-11-02

• Existence and stability of solutions to non-linear neutral stochastic
functional differential equations in the framework of G-Brownian motion

• Abstract: In the past decades, quantitative study of different disciplines such as system sciences, physics, ecological sciences, engineering, economics and biological sciences, have been driven by new modeling known as stochastic dynamical systems. This paper aims at studying these important dynamical systems in the framework of G-Brownian motion and G-expectation. It is demonstrated that, under the contractive condition, the weakened linear growth condition and the non-Lipschitz condition, a neutral stochastic functional differential equation in the G-frame has at most one solution. Hölder’s inequality, Gronwall’s inequality, the Burkholder-Davis-Gundy (in short BDG) inequalities, Bihari’s inequality and the Picard approximation scheme are used to establish the uniqueness-and-existence theorem. In addition, the stability in mean square is developed for the above mentioned stochastic dynamical systems in the G-frame.
PubDate: 2017-10-30

• H ∞ ${H} _{ {\infty}}$ analysis and switching control for uncertain
discrete switched time-delay systems by discrete Wirtinger inequality

• Abstract: In this paper, the $$H_{\infty}$$ performance analysis and switching control of uncertain discrete switched systems with time delay and linear fractional perturbations are considered via a switching signal design. Lyapunov-Krasovskii type functional and discrete Wirtinger inequality are used in our approach to improve the conservativeness of the past research results. Less LMI variables and shorter program running time are provided than our past proposed results. Finally, two numerical examples are given to show the improvement of the developed results.
PubDate: 2017-10-27

• Existence results for a class of generalized fractional boundary value
problems

• Abstract: In this paper, we study a class of generalized fractional order three-point boundary value problems that involve fractional derivative defined in terms of weight and scale functions. Using several fixed point theorems, the existence and uniqueness results are obtained.
PubDate: 2017-10-27

• Source degenerate identification problems with smoothing overdetermination

• Abstract: We consider degenerate identification problems with smoothing overdetermination in abstract spaces. We establish an identifiability result using a projection method and suitable hypotheses on the operators involved and develop an identification method by reformulating the problem into a nondegenerate problem. Then we use perturbation results for linear operators to solve the regular problem. The introduced identification method permits one to solve the problems under the minimum restrictions on the input data. Finally, we provide applications to degenerate differential equations that appear in mathematical physics to support the theoretical results.
PubDate: 2017-10-26

• Finite-time synchronization of delayed complex dynamical network via
pinning control

• Abstract: We investigate a finite-time synchronization problem of hybrid-coupled delayed dynamical network via pinning control. According to linear feedback principle and finite-time control theory, the finite-time synchronization can be achieved by pinning control with suitable continuous finite-time controller. Some sufficient conditions are given for finite-time synchronization of undirected and directed complex network by applying finite-time stability lemma. Numerical simulations are finally presented to demonstrate the effectiveness of the theoretical results.
PubDate: 2017-10-25

• Normalized Bernstein polynomials in solving space-time fractional
diffusion equation

• Abstract: In this paper, we solve a time-space fractional diffusion equation. Our methods are based on normalized Bernstein polynomials. For the space domain, we use a set of normalized Bernstein polynomials and for the time domain, which is a semi-infinite domain, we offer an algebraic map to make the rational normalized Bernstein functions. This study uses Galerkin and collocation methods. The integrals in the Galerkin method are established with Chebyshev interpolation. We implemented the proposed methods for some examples that are presented to demonstrate the theoretical results. To confirm the accuracy, error analysis is carried out.
PubDate: 2017-10-25

• On Chebyshev polynomials and their applications

• Abstract: The main purpose of this paper is, using some properties of the Chebyshev polynomials, to study the power sum problems for the sinx and cosx functions and to obtain some interesting computational formulas.
PubDate: 2017-10-24

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