Showing papers in "Advances in Difference Equations in 2017"

On a new class of fractional operators

Abstract: This manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. We introduce new fractional integration and differentiation operators. We define spaces and present some theorems related to these operators.

A new fractional model for giving up smoking dynamics

Abstract: The key purpose of the present work is to examine a fractional giving up smoking model pertaining to a new fractional derivative with non-singular kernel. The numerical simulations are conducted with the aid of an iterative technique. The existence of the solution is discussed by employing the fixed point postulate, and the uniqueness of the solution is also proved. The effect of various parameters is shown graphically. The numerical results for the smoking model associated with the new fractional derivative are compared with numerical results for a smoking model pertaining to the standard derivative and Caputo fractional derivative.

Existence and uniqueness of weak solutions for a class of fractional superdiffusion equations

Abstract: In this paper, we consider the existence and uniqueness of weak solutions for a class of fractional superdiffusion equations with initial-boundary conditions. For a multidimensional fractional drift superdiffusion equation, we just consider the simplest case with divergence-free drift velocity $u \in L^{2}(\Omega)$ only depending on the spatial variable x. Finally, exploiting the Schauder fixed point theorem combined with the Arzela-Ascoli compactness theorem, we obtain the existence and uniqueness of weak solutions in the standard Banach space $C([0,T]; H_{0}^{1}(\Omega))$ for a class of fractional superdiffusion equations.

A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative

Abstract: We present a new method to investigate some fractional integro-differential equations involving the Caputo-Fabrizio derivation and we prove the existence of approximate solutions for these problems. We provide three examples to illustrate our main results. By checking those, one gets the possibility of using some discontinuous mappings as coefficients in the fractional integro-differential equations.

Fractional operators with exponential kernels and a Lyapunov type inequality

Abstract: In this article, we extend fractional calculus with nonsingular exponential kernels, initiated recently by Caputo and Fabrizio, to higher order. The extension is given to both left and right fractional derivatives and integrals. We prove existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial value problems by using Banach contraction theorem. Then we prove Lyapunov type inequality for the Riemann type fractional boundary value problems within the exponential kernels. Illustrative examples are analyzed and an application about Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

Monotonicity results for fractional difference operators with discrete exponential kernels

Abstract: We prove that if the Caputo-Fabrizio nabla fractional difference operator $({}^{\mathrm{CFR}}_{a-1} abla^{\alpha}y)(t)$ of order $0<\alpha\leq1$ and starting at $a-1$ is positive for $t=a,a+1,\ldots$ , then $y(t)$ is α-increasing. Conversely, if $y(t)$ is increasing and $y(a)\geq0$ , then $({}^{\mathrm{CFR}}_{a-1} abla^{\alpha}y)(t)\geq0$ . A monotonicity result for the Caputo-type fractional difference operator is proved as well. As an application, we prove a fractional difference version of the mean-value theorem and make a comparison to the classical discrete fractional case.

Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel

Abstract: This work presents the homotopy perturbation transform method for nonlinear fractional partial differential equations of the Caputo-Fabrizio fractional operator. Perturbative expansion polynomials are considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding the exact solutions of the fractional equations proposed, for the special case when the limit of the integral order of the time derivative is considered.

On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations

Abstract: We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order.

A generalized Lyapunov-type inequality in the frame of conformable derivatives

Abstract: We prove a generalized Lyapunov-type inequality for a conformable boundary value problem (BVP) of order $\alpha \in (1,2]$ . Indeed, it is shown that if the boundary value problem $$ \bigl(\textbf{T}_{\alpha }^{c} x\bigr) (t)+r(t)x(t)=0,\quad t \in (c,d), x(c)=x(d)=0 $$ has a nontrivial solution, where r is a real-valued continuous function on $[c,d]$ , then 1 $$ \int_{c}^{d} \bigl\vert r(t) \bigr\vert \,dt> \frac{\alpha^{\alpha }}{(\alpha -1)^{\alpha -1}(d-c)^{ \alpha -1}}. $$ Moreover, a Lyapunov type inequality of the form 2 $$ \int_{c}^{d} \bigl\vert r(t) \bigr\vert \,dt> \frac{3\alpha -1}{(d-c)^{2\alpha -1}} \biggl( \frac{3 \alpha -1}{2\alpha -1} \biggr) ^{\frac{2\alpha -1}{\alpha }},\quad \frac{1}{2}< \alpha \leq 1, $$ is obtained for a sequential conformable BVP. Some examples are given and an application to conformable Sturm-Liouville eigenvalue problem is analyzed.

Optimal harvesting control and dynamics of two-species stochastic model with delays

Abstract: Taking the stochastic effects on growth rate and harvesting effort into account, we propose a stochastic delay model of species in two habitats. The main aim of this paper is to investigate optimal harvesting and dynamics of the stochastic delay model. By using the stochastic analysis theory and differential inequality technology, we firstly obtain sufficient conditions for persistence in the mean and extinction. Furthermore, the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY) are gained by using Hessian matrix, the ergodic method, and optimal harvesting theory of differential equations. To illustrate the performance of the theoretical results, we present a series of numerical simulations of these cases with respect to different noise disturbance coefficients.

Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel

Abstract: In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.

On solutions of fractional Riccati differential equations

Abstract: We apply an iterative reproducing kernel Hilbert space method to get the solutions of fractional Riccati differential equations. The analysis implemented in this work forms a crucial step in the process of development of fractional calculus. The fractional derivative is described in the Caputo sense. Outcomes are demonstrated graphically and in tabulated forms to see the power of the method. Numerical experiments are illustrated to prove the ability of the method. Numerical results are compared with some existing methods.

Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian

Abstract: In this article, we study a class of fractional coupled systems with Riemann-Stieltjes integral boundary conditions and generalized p-Laplacian which involves two different parameters. Based on the Guo-Krasnosel’skii fixed point theorem, some new results on the existence and nonexistence of positive solutions for the fractional system are received, the impact of the two different parameters on the existence and nonexistence of positive solutions is also investigated. An example is then given to illuminate the application of the main results.

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.

Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type

Abstract: This article deals with some existence and Ulam-Hyers-Rassias stability results for a class of functional differential equations involving the Hilfer-Hadamard fractional derivative. An application is made of a Schauder fixed point theorem for the existence of solutions. Next we prove that our problem is generalized Ulam-Hyers-Rassias stable.

Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition

Abstract: In this paper, we investigate four different types of Ulam stability, i.e., Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for a class of nonlinear implicit fractional differential equations with non-instantaneous integral impulses and nonlinear integral boundary condition. We also establish certain conditions for the existence and uniqueness of solutions for such a class of fractional differential equations using Caputo fractional derivative. The arguments are based on generalized Diaz-Margolis’s fixed point theorem. We provide two examples, which shows the validity of our main results.

On the asymptotic behavior of fourth-order functional differential equations

Abstract: The aim of this work is to study asymptotic properties of a class of fourth-order delay differential equations. Our results in this paper not only generalize some previous results, but also improve the earlier ones. Examples are considered to elucidate the main results.

Adaptive tracking control for a class of uncertain nonlinear systems with infinite number of actuator failures using neural networks

Abstract: We consider adaptive compensation for infinite number of actuator failures in the tracking control of uncertain nonlinear systems. We construct an adaptive controller by combining the common Lyapunov function approach and the structural characteristic of neural networks. The proposed control strategy is feasible under the presupposition that the systems have a nonstrict-feedback structure. We prove that the states of the closed-loop system are bounded and the tracking error converges to a small neighborhood of the origin under the designed controllers, even though there are an infinite number of actuator failures. At last, the validity of the proposed control scheme is demonstrated by two examples.

Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis

Abstract: In this paper, a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis is proposed and analysed. We explain the effects of stochastic disturbance on disease transmission. To this end, firstly, we investigated the dynamic properties of the system neglecting stochastic disturbance and obtained the threshold and the conditions for the extinction and the permanence of two kinds of epidemic diseases by considering the stability of the equilibria of the deterministic system. Secondly, we paid prime attention on the threshold dynamics of the stochastic system and established the conditions for the extinction and the permanence of two kinds of epidemic diseases. We found that there exists a significant difference between the threshold of the deterministic system and that of the stochastic system. Moreover, it has been established that the persistent of infectious disease analysed by use of deterministic system becomes extinct under the same conditions due to the stochastic disturbance. This implies that a stochastic disturbance has significant impact on the spread of infectious diseases and the larger stochastic disturbance leads to control the epidemic diseases. In order to illustrate the dynamic difference between the deterministic system and the stochastic system, there have been given a series of numerical simulations by using different noise disturbance coefficients.

Positive solutions for a class of fractional 3-point boundary value problems at resonance

Abstract: In this paper, we study the nonlocal fractional differential equation: $$\left \{ \textstyle\begin{array}{@{}l} D^{\alpha}_{0+}u(t)+f(t,u(t))=0 ,\quad 0< t< 1,\\ u(0)=0,\qquad u(1)=\eta u(\xi), \end{array}\displaystyle \right . $$ where $1 < \alpha< 2$ , $0 < \xi< 1$ , $\eta\xi^{\alpha-1}= 1$ , $D^{\alpha}_{0+}$ is the standard Riemann-Liouville derivative, $f:[0,1]\times[0,+\infty)\rightarrow\mathbb{R}$ is continuous. The existence and uniqueness of positive solutions are obtained by means of the fixed point index theory and iterative technique.

Estimation of parameters in a structured SIR model

Abstract: In this paper, an age-structured epidemiological process is considered. The disease model is based on a SIR model with unknown parameters. We addressed two important issues to analyzing the model and its parameters. One issue is concerned with the theoretical existence of unique solution, the identifiability problem. The second issue is how to estimate the parameters in the model. We propose an iterative algorithm to study the identifiability of the system and a method to estimate the parameters which are identifiable. A least squares approach based on a finite set of observations helps us to estimate the initial values of the parameters. Finally, we test the proposed algorithms.

Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method

Abstract: In this paper, a wavelet numerical method for solving nonlinear Volterra integro-differential equations of fractional order is presented. The method is based upon Euler wavelet approximations. The Euler wavelet is first presented and an operational matrix of fractional-order integration is derived. By using the operational matrix, the nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which is solved through known numerical algorithms. Also, various types of solutions, with smooth, non-smooth, and even singular behavior have been considered. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input

Abstract: A mathematical model describing continuous microbial culture and harvest in a chemostat, incorporating a control strategy and defined by impulsive differential equations, is presented and investigated. Theoretical results indicate that the model has a microbe-extinction periodic solution, which is globally attractive if the threshold $R_{1}$ is less than unity, and the model is permanent if the threshold $R_{2}$ is greater than unity. Further, we consider the control strategy under time delay and periodical impulsive effect. Analysis shows that continuous microbial culture and harvest process can be implemented by adjusting time delay, impulsive period or input amount of flocculant. Finally, we give an example with numerical simulations to illustrate the control strategy.

A new operational matrix of Caputo fractional derivatives of Fermat polynomials: an application for solving the Bagley-Torvik equation

Abstract: Herein, an innovative operational matrix of fractional-order derivatives (sensu Caputo) of Fermat polynomials is presented. This matrix is used for solving the fractional Bagley-Torvik equation with the aid of tau spectral method. The basic approach of this algorithm depends on converting the fractional differential equation with its initial (boundary) conditions into a system of algebraic equations in the unknown expansion coefficients. The convergence and error analysis of the suggested expansion are carefully discussed in detail based on introducing some new inequalities, including the modified Bessel function of the first kind. The developed algorithm is tested via exhibiting some numerical examples with comparisons. The obtained numerical results ensure that the proposed approximate solutions are accurate and comparable to the analytical ones.

Enhanced robust finite-time passivity for Markovian jumping discrete-time BAM neural networks with leakage delay.

Abstract: This paper is concerned with the problem of enhanced results on robust finite-time passivity for uncertain discrete-time Markovian jumping BAM delayed neural networks with leakage delay. By implementing a proper Lyapunov-Krasovskii functional candidate, the reciprocally convex combination method together with linear matrix inequality technique, several sufficient conditions are derived for varying the passivity of discrete-time BAM neural networks. An important feature presented in our paper is that we utilize the reciprocally convex combination lemma in the main section and the relevance of that lemma arises from the derivation of stability by using Jensen’s inequality. Further, the zero inequalities help to propose the sufficient conditions for finite-time boundedness and passivity for uncertainties. Finally, the enhancement of the feasible region of the proposed criteria is shown via numerical examples with simulation to illustrate the applicability and usefulness of the proposed method.

A finite difference scheme based on cubic trigonometric B-splines for a time fractional diffusion-wave equation

Abstract: In this paper, we propose an efficient numerical scheme for the approximate solution of a time fractional diffusion-wave equation with reaction term based on cubic trigonometric basis functions. The time fractional derivative is approximated by the usual finite difference formulation, and the derivative in space is discretized using cubic trigonometric B-spline functions. A stability analysis of the scheme is conducted to confirm that the scheme does not amplify errors. Computational experiments are also performed to further establish the accuracy and validity of the proposed scheme. The results obtained are compared with finite difference schemes based on the Hermite formula and radial basis functions. It is found that our numerical approach performs superior to the existing methods due to its simple implementation, straightforward interpolation and very low computational cost. A convergence analysis of the scheme is also discussed.

Analytical study for time and time-space fractional Burgers’ equation

Abstract: In this paper, the variational iteration method (VIM) is applied to solve the time and space-time fractional Burgers’ equation for various initial conditions. VIM solutions are computed for the fractional Burgers’ equation to show the behavior of VIM solutions as the fractional derivative parameter is changed. The results obtained by VIM are compared with exact solutions and also with expansions of the exact solutions. VIM solutions are found to be in excellent agreement with these exact solutions.

Stability and Hopf bifurcation analysis of fractional-order complex-valued neural networks with time delays

Abstract: This paper considers a class of fractional-order complex-valued Hopfield neural networks (CVHNNs) with time delay for analyzing the dynamic behaviors such as local asymptotic stability and Hopf bifurcation. In the case of a neural network with hub and ring structure, the stability of the equilibrium state is investigated by analyzing the eigenvalue of the corresponding characteristic matrix for the hub and ring structured fractional-order time delay models using a Laplace transformation for the Caputo-fractional derivatives. Some sufficient conditions are established to guarantee the uniqueness of the equilibrium point. In addition, conditions for the occurrence of a Hopf bifurcation are also presented. Finally, numerical examples are given to demonstrate the effectiveness of the derived results.

Effects of HIV infection on CD4+ T-cell population based on a fractional-order model

Abstract: In this paper, we study the HIV infection model based on fractional derivative with particular focus on the degree of T-cell depletion that can be caused by viral cytopathicity. The arbitrary order of the fractional derivatives gives an additional degree of freedom to fit more realistic levels of CD4+ cell depletion seen in many AIDS patients. We propose an implicit numerical scheme for the fractional-order HIV model using a finite difference approximation of the Caputo derivative. The fractional system has two equilibrium points, namely the uninfected equilibrium point and the infected equilibrium point. We investigate the stability of both equilibrium points. Further we examine the dynamical behavior of the system by finding a bifurcation point based on the viral death rate and the number of new virions produced by infected CD4+ T-cells to investigate the influence of the fractional derivative on the HIV dynamics. Finally numerical simulations are carried out to illustrate the analytical results.

Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse

Abstract: In this paper, a predator-prey model with Holling type-I functional response and multi state impulsive feedback control is established, where the intensity of pesticide spraying and the release amount of natural enemies are linearly dependent on the given threshold in the second impulse. Firstly, the existence of order-1 periodic solution of the system is investigated by successor functions and Bendixson theorem of impulsive differential equations, then the stability of periodic solutions is proved by the analogue of the Poincare criterion. Furthermore, in order to reduce the actual total cost and obtain the best economic benefit, the optimal economic threshold is obtained, which provides the optimal strategy for the practical application. Finally, numerical simulations for specific examples are carried out to illustrate the feasibility of the above conclusions.