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 Journal of Patient-Reported OutcomesNumber of Followers: 0     Open Access journal ISSN (Online) 2509-8020 Published by SpringerOpen  [262 journals]
• The fractional form of a new three-dimensional generalized Hénon map

• Abstract: Abstract In this paper, we propose a fractional form of a new three-dimensional generalized Hénon map and study the existence of chaos and its control. Using bifurcation diagrams, phase portraits and Lyapunov exponents, we show that the general behavior of the proposed fractional map depends on the fractional order. We also present two control schemes for the proposed map, one that adaptively stabilizes the fractional map, and another to achieve the synchronization of the proposed fractional map.
PubDate: 2019-03-22

• A new result on stability analysis for discrete system with interval
time-varying delays

• Abstract: Abstract This paper is concerned with the stability criteria for a discrete-time linear system with interval time-varying delays. By using the time delay division we construct an augmented Lyapunov–Krasovskii functional for two delay subintervals. Moreover, we use a new summation inequality to estimate the derivatives of LKFs more accurately and derive less conservative criteria. Finally, we present two numerical examples to demonstrate that the obtained results are less conservative.
PubDate: 2019-03-22

• The classification of the single traveling wave solutions to ( 1 + 1 )
$(1+1)$ dimensional Gardner equation with variable coefficients

• Abstract: Abstract In this paper the classification of single traveling wave solutions of $$(1+1)$$ dimensional Gardner equation with variable coefficients is obtained by applying the complete discrimination system to the polynomial and trial equation methods. In particular, the corresponding solutions for the concrete parameters are constructed to show that each solution in the classification can be realized. Moreover, numerical simulations shown in the paper could help us better understand the nature of each solution.
PubDate: 2019-03-22

• State bounding estimation for a linear continuous-time singular system
with time-varying delay

• Abstract: Abstract This paper investigates the problem of a state bounding estimation for a linear continuous-time singular system with time-varying delay. By employing the maximal Lyapunov–Krasovskii functional and applying the new free-matrix-based integral inequality, some proper conditions are derived in terms of LMIs and a bounding estimation lemma and set are obtained for the studied singular system.
PubDate: 2019-03-22

• Anti-periodic boundary value problems with Riesz–Caputo derivative

• Abstract: Abstract This paper is concerned with a class of anti-periodic boundary value problems for fractional differential equations with the Riesz–Caputo derivative, which can reflected both the past and the future nonlocal memory effects. By means of new fractional Gronwall inequalities and some fixed point theorems, we obtain some existence results of solutions under the Lipschitz condition, the sublinear growth condition, the nonlinear growth condition and the comparison condition. Three examples are given to illustrate the results.
PubDate: 2019-03-21

• On the oscillation of fourth-order delay differential equations

• Abstract: Abstract In the paper, fourth-order delay differential equations of the form $$\bigl(r_{3} \bigl(r_{2} \bigl(r_{1}y' \bigr)' \bigr)' \bigr)'(t) + q(t) y \bigl( \tau (t) \bigr) = 0$$ under the assumption $$\int _{t_{0}}^{\infty }\frac{\mathrm {d}t}{r_{i}(t)} < \infty , \quad i = 1,2,3,$$ are investigated. Our newly proposed approach allows us to greatly reduce a number of conditions ensuring that all solutions of the studied equation oscillate. An example is also presented to test the strength and applicability of the results obtained.
PubDate: 2019-03-20

• A note on entire functions sharing a finite set with their difference
operators

• Abstract: Abstract In this note, we will show that an entire function is equal to its difference operator if it has a growth property and shares a set, where the set consists of two entire functions of smaller orders. This result generalizes a result of Li (Comput. Methods Funct. Theory 12:307–328, 2012 and partially answers Liu’s (J. Math. Anal. Appl. 359:384–393, 2009) question.
PubDate: 2019-03-19

• Efficient finite element numerical solution of the variable coefficient
fractional subdiffusion equation

• Abstract: Abstract Based on the weighted and shifted Grünwald formula, a fully discrete finite element scheme is derived for the variable coefficient time-fractional subdiffusion equation. Firstly, the unconditional stable and convergent of the fully discrete scheme in $$L^{1}(H^{1})$$ -norm is proved. Secondly, through a new estimate approach, the superclose properties are obtained. The global superconvergence order $$\mathcal{O}(\tau ^{2}+h^{m+1})$$ is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.
PubDate: 2019-03-19

• A pest control model with birth pulse and residual and delay effects of
pesticides

• Abstract: Abstract Pesticides often cause residual and delayed effects on pests. Considering these effects, we use a pollution emission model to simulate the process of spraying pesticides. Many pests reproduce only at a fixed time in a year. So a pest control model with birth pulse and spraying pesticides is proposed. Using the limit system of the developed model, we analyze the dynamics of the system. The stability of the trivial equilibrium and the positive equilibrium of the model is analyzed, and the threshold conditions of pest eradication and permanence of the system are given. We obtain the optimal frequency of spraying pesticides by numerical simulations. The important parameters related to the pest eradication or permanence of the system are given by analyzing the sensitivity of the parameters. Finally, biological explanations are provided.
PubDate: 2019-03-19

• Solution of fractional Volterra–Fredholm integro-differential equations
under mixed boundary conditions by using the HOBW method

• Abstract: 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.
PubDate: 2019-03-19

• Stability of a stochastic discrete mutualism system

• Abstract: Abstract Under the assumption that stochastic white noise perturbations are directly proportional to the deviation of the state from the equilibrium states of a continuous mutualistic model, we use the Euler–Maruyama discretization method to obtain a two-species stochastic discrete mutualism model. For this stochastic model, we establish conditions on the asymptotic mean square stability of the positive equilibrium state and the almost sure asymptotic stability of the three boundary equilibrium states. The theoretical results are supported with numerical simulations.
PubDate: 2019-03-18

• Legendre spectral element method for solving sine-Gordon equation

• Abstract: Abstract In this paper, we study the Legendre spectral element method for solving the sine-Gordon equation in one dimension. Firstly, we discretize the equation by Legendre spectral element in space and then discretize the time by the second-order leap-frog method. We study the stability and convergence of the method and show the convergence of our method. Finally, we show the results with numerical examples.
PubDate: 2019-03-18

• Multiple positive solutions for nonlinear mixed fractional differential
equation with p -Laplacian operator

• Abstract: Abstract In this article, multiple positive solutions are considered for nonlinear mixed fractional differential equations with a p-Laplacian operator. Using the Avery–Peterson fixed point theorem, we conclude to the existence of positive solutions for the fractional boundary value problem. An example is also presented to illustrate the effectiveness of the main result.
PubDate: 2019-03-18

• Representation by several orthogonal polynomials for sums of finite
products of Chebyshev polynomials of the first, third and fourth kinds

• Abstract: Abstract The classical linearization problem concerns with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. As a generalization of this, we consider here sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the ones previously studied. We represent each of them as linear combinations of Hermite, extended Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. Here, the coefficients involve some terminating hypergeometric functions $${}_{2}F_{1}$$ , $${}_{2}F_{2}$$ , and $${}_{1}F_{1}$$ . These representations are obtained by explicit computations.
PubDate: 2019-03-18

• Wave propagation in a infectious disease model with non-local diffusion

• Abstract: In this paper, we propose a nonlocal diffusion infectious disease model with nonlinear incidences and distributed delay to model the transmission of the epidemic. By a fixed point theorem and a limiting argument, we establish the existence of traveling wave solutions for the model. Meanwhile, we obtain the non-existence of traveling wave solutions for the model via two-sided Laplace transform. It is found that the threshold dynamics of traveling wave solutions are entirely determined by the basic reproduction number of the corresponding spatially-homogenous delayed differential system and the minimum wave speed. A typical example is given for supporting our abstract results. Moreover, the effect of the diffusive rate of the infected individuals on the minimum wave speed is discussed.
PubDate: 2019-03-15

• On numerical techniques for solving the fractional logistic differential
equation

• Abstract: Abstract This paper studied the existence and uniqueness of the solution of the fractional logistic differential equation using Hadamard derivative and integral. Previous work has shown that there is not an exact solution to this fractional model. Hence several numerical approaches, such as generalized Euler’s method (GEM), power series expansion (PSE) method, and Caputo–Fabrizio (CF) method, were used to compute the solution. The classical solution obtained from the first order non-linear differential equation was also considered to enable the comparison of error levels.
PubDate: 2019-03-14

• Periodic solutions for p -Laplacian neutral differential equation with
multiple delay and variable coefficients

• Abstract: Abstract In this paper, we first discuss some properties of the neutral operator with multiple delays and variable coefficients $$(Ax)(t):=x(t)-\sum_{i=1}^{n}c_{i}(t)x(t-\delta _{i})$$ . Afterwards, by using an extension of Mawhin’s continuation theorem, a second order p-Laplacian neutral differential equation $$\Biggl(\phi _{p} \Biggl(x(t)-\sum_{i=1}^{n}c_{i}(t)x(t- \delta _{i}) \Biggr)' \Biggr)'=\tilde{f} \bigl(t,x(t),x'(t)\bigr)$$ is studied. Some new results on the existence of a periodic solution are obtained. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from those known in the literature.
PubDate: 2019-03-13

• On the existence and uniqueness of ( N , λ ) $(N,\lambda )$ -periodic
solutions to a class of Volterra difference equations

• Abstract: Abstract In this paper we introduce the class of $$(N,\lambda )$$ -periodic vector-valued sequences and show several notable properties of this new class. This class includes periodic, anti-periodic, Bloch and unbounded sequences. Furthermore, we show the existence and uniqueness of $$(N, \lambda )$$ -periodic solutions to the following class of Volterra difference equations with infinite delay: $$u(n+1)=\alpha \sum_{j=-\infty }^{n}a(n-j)u(j)+f \bigl(n,u(n) \bigr), \quad n \in \mathbb{Z}, \alpha \in \mathbb{C},$$ where the kernel a and the nonlinear term f satisfy suitable conditions.
PubDate: 2019-03-13

• Bifurcation analysis for the Kaldor–Kalecki model with two delays

• Abstract: Abstract In this paper, a Kaldor–Kalecki model of business cycle with two discrete time delays is considered. Firstly, by analyzing the corresponding characteristic equations, the local stability of the positive equilibrium is discussed. Choosing delay (or the adjustment coefficient in the goods market α) as bifurcation parameter, the existence of Hopf bifurcation is investigated in detail. Secondly, by combining the normal form method with the center manifold theorem, we are able to determine the direction of the bifurcation and the stability of the bifurcated periodic solutions. Finally, some numerical simulations are carried out to illustrate the theoretical results.
PubDate: 2019-03-13

• G-stability one-leg hybrid methods for solving DAEs

• Abstract: Abstract This paper introduces the solution of differential algebraic equations using two hybrid classes and their twin one-leg with improved stability properties. Physical systems of interest in control theory are sometimes described by systems of differential algebraic equations (DAEs) and ordinary differential equations (ODEs) which are zero index DAEs. The study of the first hybrid class includes the order of convergence, A(α)-stability, stability regions, and G-stability for its one-leg twin in two cases: for step ( $$k=1$$ ) and steps ( $$k=2$$ ). For the second class, G-stability of its one-leg twin is studied in two cases: for steps ( $$k=2$$ ) and steps ( $$k=3$$ ). Test problems are introduced with different step size at different end points.
PubDate: 2019-03-12

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