Authors:Shuhuang Xiang Pages: 1 - 7 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Shuhuang Xiang In this paper, some new asymptotic formulas on the decay of the coefficients of the Chebyshev expansions are presented and refined estimates on the asymptotics for functions of limited regularities are derived. Together with these, the optimal convergence rates of the polynomial interpolations in Chebyshev points are deduced.

Authors:Jaiok Roh Pages: 8 - 12 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Jaiok Roh In this paper, we want to see the properties of the smooth solutions u of the incompressible flows on an exterior domain Ω of R 2 . Specially, when the vorticity ω = ∇ × u has a bounded support, with suitable conditions we will show that there exists a constant C ( p , q ) such that ∥ u ∥ L p ( Ω ) ≤ C ∥ u ∥ L q ( Ω ) for 1 < p ≤ q ≤ ∞ .

Authors:Chao Li; Meimei Zhao Pages: 13 - 18 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Chao Li, Meimei Zhao In this paper, the improved fractional sub-equation method is applied to obtain analytical solutions of the (2+1)-dimensional space–time fractional Bogoyavlenskii’s breaking soliton equation involving Jumarie’s modified Riemann–Liouville derivative in mathematical physics. As a result, explicit analytical solutions are successfully obtained, which contain more generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions. The graphical representations have been used for demonstrating the nature of the obtained results.

Authors:Sun-Ho Choi; Intae Ryoo; Bum Il Hong Pages: 19 - 25 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Sun-Ho Choi, Intae Ryoo, Bum Il Hong We present a sufficient condition for synchronization of power grid model when the moment of inertia is zero. Unlike the Kuramoto model, the power grid model contains nonlinear terms of the first- and second-order derivatives. We assume that the natural frequencies are all the same and larger than coupling strength K > 0 . These conditions allow us to use a Lyapunov functional and a priori estimate method. From the Lyapunov functional, we can obtain the exponential decay result for the phase difference max i , j θ i ( t ) − θ j ( t ) , where θ i is the i th phase angle such that the ensemble approaches complete position synchronization.

Authors:Giuseppe Caristi; Shapour Heidarkhani; Amjad Salari; Stepan A. Tersian Pages: 26 - 33 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Giuseppe Caristi, Shapour Heidarkhani, Amjad Salari, Stepan A. Tersian In this paper, we study the existence of solutions for a class of degenerate nonlocal problems involving sub-linear nonlinearities, while the nonlinear part of the problem admits some hypotheses on the behavior at the origin or perturbation property. We obtain some new criteria for existence of two and infinitely many solutions of the problem using critical point theory. Some recent results are extended and improved. Some examples are presented to demonstrate the application of our main results.

Authors:Dongyang Shi; Pengcong Mu; Huaijun Yang Pages: 34 - 41 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Dongyang Shi, Pengcong Mu, Huaijun Yang In this paper, the superconvergence analysis of a two-grid method (TGM) is established for the semilinear parabolic equations. Based on the combination of the interpolation and Ritz projection technique, an important ingredient in the method, the superclose estimates in the H 1 -norm are deduced for the backward Euler fully-discrete TGM scheme. Moreover, through the interpolated postprocessing approach, the corresponding global superconvergence result is derived. Finally, some numerical results are provided to confirm the theoretical analysis, and also show that the computing cost of the proposed TGM is only half of the conventional Galerkin finite element methods (FEMs).

Authors:Maria V. Demina Pages: 42 - 48 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Maria V. Demina A novel algebraic method for finding invariant algebraic curves for a polynomial vector field in C 2 is introduced. The structure of irreducible invariant algebraic curves for Liénard dynamical systems x t = y , y t = − f ( x ) y − g ( x ) with deg g = deg f + 1 is obtained. It is shown that there exist Liénard systems that possess more complicated invariant algebraic curves than it was supposed before. As an example, all irreducible invariant algebraic curves for the Liénard differential system with deg f = 2 and deg g = 3 are obtained. All these results seem to be new.

Authors:Guangze Gu; Wei Zhang; Fukun Zhao Pages: 49 - 55 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Guangze Gu, Wei Zhang, Fukun Zhao In this paper, we obtain infinitely many small positive solutions of the following nonlocal problem − L K u = f ( x , u ) in Ω , u = 0 in R N ∖ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω , and L K is an integrodifferential operator of fractional Laplacian type. The character of this work is that we do not require any growth condition on f for u large.

Authors:Ricardo Almeida Pages: 56 - 62 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Ricardo Almeida In this paper we focus on models consisting of fractional differential equations to describe the dynamics of certain epidemics. The population is divided into susceptible, exposed, infectious, and recovered (SEIR), with treatment policies. We present an analytical study and show that the model has two equilibrium points (disease free equilibrium and endemic equilibrium). Local asymptotic stability is proven for both cases. Numerical simulations are presented to illustrate the conclusions.

Authors:De-Yin Liu; Wen-Rong Sun Pages: 63 - 69 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): De-Yin Liu, Wen-Rong Sun We derive the nonlocal sixth-order nonlinear Schrödinger (NLS) equation which admits rational solutions. Through the Lax pair, we prove the integrability of the nonlocal NLS equation. Furthermore, with the Darboux transformation, analytic rational solutions up to the second order are explicitly given.

Authors:Peng Chen; Xiaochun Liu Pages: 70 - 75 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Peng Chen, Xiaochun Liu In this paper we prove the existence of ground states of linearly coupled systems of Choquard type. Asymptotic behaviour of ground states is also studied.

Authors:Daiyong Wu; Hongyong Zhao; Yuzhen Bai Pages: 76 - 83 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Daiyong Wu, Hongyong Zhao, Yuzhen Bai In this paper, we investigate a degenerate reaction–diffusion/reaction–aggregation–diffusion model with non-smooth reaction term. The strong and weak properties of travelling wave front solutions for this model are analyzed. Under degenerate reaction–diffusion/reaction–aggregation–diffusion term and non-smooth reaction term, we obtain how strong and weak properties change.

Authors:Xiancheng Gao; Hongjun Gao Pages: 84 - 89 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Xiancheng Gao, Hongjun Gao In this paper, the semigroup S ( t ) generated by unbounded linear operator A is not Hölder-continuous at zero. By assuming the regularity of initial condition, the mild solution u ( t ) ∈ C β ( [ 0 , T ] ; V ) is obtained. Then the local exponential stability of evolution equations driven by Hölder-continuous paths with Hölder exponent H ∈ ( 1 ∕ 2 , 1 ) is established. This result can be directly applied to the evolution equations with fractional Brownian motion with Hurst parameter H ∈ ( 1 ∕ 2 , 1 ) .

Authors:Tieshan He; Chaolong Zhang; Dongqing Wu; Kaihao Liang Pages: 90 - 95 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Tieshan He, Chaolong Zhang, Dongqing Wu, Kaihao Liang We consider the Brézis–Nirenbergproblem: { − Δ u = u 2 ⋆ − 2 u + λ u in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N , N ≥ 3 , 2 ⋆ = 2 N N − 2 is the critical Sobolev exponent and λ > 0 . Our main result asserts that if N ≥ 4 then there exists a pair of sign-changing solutions of the problem for every λ ∈ ( 0 , λ 1 ( Ω ) ) , λ 1 ( Ω ) being the first eigenvalue of − Δ in Ω with Dirichlet boundary conditions, while if N = 3 then a pair of sign-changing solutions exists for λ slightly smaller than λ 1 ( Ω ) . Our approach uses variational methods together with flow invariance arguments.

Authors:Leonid Shaikhet Pages: 103 - 110 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Leonid Shaikhet A nonlinear difference equation with delay and logarithmic nonlinearity is considered. Some properties of asymptotic behavior of the solution of this equation under stochastic perturbations are discussed. In particular, stability and instability of the positive and zero equilibria are studied. All obtained results are illustrated by numerical simulations of solutions of the considered equation.

Authors:Bashir Ahmad; Madeaha Alghanmi; Sotiris K. Ntouyas; Ahmed Alsaedi Pages: 111 - 117 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Bashir Ahmad, Madeaha Alghanmi, Sotiris K. Ntouyas, Ahmed Alsaedi In this paper, we obtain the sufficient conditions for the uniqueness of solutions for a boundary value problem of fractional differential equations involving generalized fractional derivative supplemented with Stieltjes and generalized fractional integral boundary conditions.

Authors:A.O. Ignatyev Pages: 124 - 129 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): A.O. Ignatyev A. M. Lyapunov proved the inequality that makes it possible to estimate the distance between two consecutive zeros a and b of solutions of a linear differential equation of the second order x ̈ ( t ) + q ( t ) x ( t ) = 0 where q ( t ) is a continuous function for t ∈ [ a , b ] . In the present note, a similar problem is solved for a differential equation of the form d d t x ̇ 1 − x ̇ 2 + p ( t ) x ̇ + q ( t ) x = 0 . The obtained inequality is applied to the estimate of the period of a periodic solution of relativistic differential Van der Pol equation.

Authors:Fajie Wang; Qingsong Hua; Chein-Shan Liu Pages: 130 - 136 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Fajie Wang, Qingsong Hua, Chein-Shan Liu This paper develops a novel approach for detecting unknown boundaries in the two-dimensional anisotropic heat conduction equations based on the boundary function method, in which a partial homogenization function satisfied the over-specified Cauchy data on an arc is derived to effectively solve the inverse geometry problem. After the homogenized technique, the governing equation is transformed into the one in a reduced domain, whose numerical solution is expanded by a sequence of boundary functions, automatically satisfying the homogeneous boundary conditions on the arc. The nonlinear equation will be formed and then solved by the Newton iterative method. Two numerical examples are provided to demonstrate the ability and accuracy of the proposed scheme.

Authors:Wei Shi; Kai Liu Pages: 137 - 142 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Wei Shi, Kai Liu This article presents a new analytical formula for the Cauchy problem of the wave equation with variable coefficients, which is a much simpler solution than that given by the Poisson formula. The derivation is based on the variation-of-constants formula and the theory of pseudodifferential operator. The formula is applied to an example to illustrate the feasibility.

Authors:Yue Wu; Ji-Huan He Pages: 143 - 147 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Yue Wu, Ji-Huan He A general derivation of Euler–Lagrange equation of Samuelson’s variational principle in economics is elucidated without Lagrange multipliers, and the optimal solutions and prices can be determined easily.

Authors:Anouar Bahrouni; Hichem Ounaies; Vicenţiu D. Rădulescu Pages: 148 - 154 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Anouar Bahrouni, Hichem Ounaies, Vicenţiu D. Rădulescu We establish the existence of entire compactly supported solutions for a class of Schrödinger equations with competing terms and indefinite potentials. The analysis developed in this paper corresponds to the case of small perturbations of the reaction term.

Authors:Guowei Dai; Hua Luo Pages: 155 - 159 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Guowei Dai, Hua Luo We study the global structure of admissible solutions for the following k -Hessian equation S k D 2 u = λ k f ( − u ) . By bifurcation and topological methods, we determine the interval of λ for the existence of admissible solution for this problem.

Authors:Xiaoli Chen; Yana Di; Jinqiao Duan; Dongfang Li Pages: 160 - 167 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Xiaoli Chen, Yana Di, Jinqiao Duan, Dongfang Li This paper is concerned with the construction and analysis of linearized numerical methods for solving the two-dimensional nonlinear time fractional Schrödinger equations. By adding different correction terms, two linearized compact alternating direction implicit (ADI) methods are proposed. Convergence of the proposed methods is obtained. Numerical results are presented to verify the accuracy and efficiency of the proposed schemes.

Authors:Jasmina Djordjevic; Cristiana J. Silva; Delfim F.M. Torres Pages: 168 - 175 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Jasmina Djordjevic, Cristiana J. Silva, Delfim F.M. Torres We propose a stochastic SICA epidemic model for HIV transmission, described by stochastic ordinary differential equations, and discuss its perturbation by environmental white noise. Existence and uniqueness of the global positive solution to the stochastic HIV system is proven, and conditions under which extinction and persistence in mean hold, are given. The theoretical results are illustrated via numerical simulations.

Authors:Yun-guang Lu Pages: 176 - 180 Abstract: Publication date: October 2018 Source:Applied Mathematics Letters, Volume 84 Author(s): Yun-guang Lu In this paper, we remove the restriction A ′ ( x ) ≥ 0 in the paper ‘Lu (2011)’, the restriction z 0 ( x ) ≤ 0 or w 0 ( x ) ≤ 0 in the paper ‘Klingenberg and Lu (1997)’, and obtain the global existence of entropy solutions to the isothermal gas dynamics system in a divergent nozzle with friction.

Authors:Yuhua Long; Jiali Chen Pages: 7 - 14 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Yuhua Long, Jiali Chen By using the invariant set of descending flow and variational method, we establish the existence of multiple solutions to a class of second-order discrete Neumann boundary value problems. The solutions include sign-changing solutions, positive solutions, and negative solutions. An example is given to illustrate our results.

Authors:Ying Lv; Shiqing Zhang Pages: 15 - 20 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Ying Lv, Shiqing Zhang Based on the works of Gordon (1977) and Zhang and Zhou (2001) on the variational minimizing properties for Keplerian orbits and Lagrangian solutions of Newtonian 2-body and 3-body problems, we use the constrained variational principle of Ambrosetti and Coti Zelati (1990) to compute the Lagrangian actions on Keplerian and Lagrangian elliptical solutions with fixed energies. We also find an interesting relation between the period and the energy for Lagrangian elliptical solutions with Newtonian potentials.

Authors:Zhong-Zhi Bai; Wen-Ting Wu Pages: 21 - 26 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Zhong-Zhi Bai, Wen-Ting Wu For solving large sparse systems of linear equations by iteration methods, we further generalize the greedy randomized Kaczmarz method by introducing a relaxation parameter in the involved probability criterion, obtaining a class of relaxed greedy randomized Kaczmarz methods. We prove the convergence of these methods when the linear system is consistent, and show that these methods can be more efficient than the greedy randomized Kaczmarz method if the relaxation parameter is chosen appropriately.

Authors:Qingshan Zhang Pages: 27 - 32 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Qingshan Zhang In this paper we consider the following competitive two-species chemotaxis system with two chemicals u t = Δ u − χ 1 ∇ ⋅ ( u ∇ v ) + μ 1 u ( 1 − u − a 1 w ) , x ∈ Ω , t > 0 , 0 = Δ v − v + w , x ∈ Ω , t > 0 , w t = Δ w − χ 2 ∇ ⋅ ( w ∇ z ) + μ 2 w ( 1 − a 2 u − w ) , x ∈ Ω , t > 0 , 0 = Δ z − z + u , x ∈ Ω , t > 0 in a smooth bounded domain Ω ⊂ R n with n ≥ 1 , where χ i ≥ 0 , a i ≥ 0 and μ i > 0 ( i = 1 , 2 ) . For the case a 1 > 1 > a 2 ≥ 0 , it will be proved that if χ 1 χ 2 < μ 1 μ 2 , χ 1 ≤ a 1 μ 1 and χ 2 < μ 2 , then the initial–boundary value problem with homogeneous Neumann boundary condition admits a unique global bounded solution and ( u , v , w , z ) → ( 0 , 1 , 1 , 0 ) uniformly on Ω ̄ as t → ∞... PubDate: 2018-04-15T12:28:10Z DOI: 10.1016/j.aml.2018.03.012 Issue No:Vol. 83 (2018)

Authors:Shundong Zhu; Junfeng Song Pages: 33 - 39 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Shundong Zhu, Junfeng Song Based on the residual symmetry theorem, the residual symmetry is obtained for the (2+1)-dimensional generalized Broer–Kaup (GBK) equations. The multiple residual symmetries are presented and localized by introducing prolonged systems, and then n th Bäcklund transformation for the GBK system in terms of determinant is derived. By means of the consistent tanh expansion (CTE) method, we get various interaction solutions which describe soliton interacting with other nonlinear waves including multiple resonant soliton wave, error function wave, periodic wave and rational function wave.

Authors:Said R. Grace; John R. Graef Pages: 40 - 45 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Said R. Grace, John R. Graef In this paper the authors examine the asymptotic behavior of solutions of a certain third order forced integro-differential equations with δ -Laplacian. Their main goal is to investigate whether nonoscillatory solutions behave at infinity like certain nontrivial nonlinear functions. They apply a technique involving Young’s, Hölder’s, and Gronwall’s inequalities.

Authors:Jiao Wei; Xianguo Geng Pages: 46 - 52 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Jiao Wei, Xianguo Geng A super Sasa–Satsuma hierarchy associated with a 3 × 3 matrix spectral problem is proposed with the aid of the zero-curvature equation and Lenard recursion equations. A typical member in the hierarchy is the super Sasa–Satsuma equation. The super bi-Hamiltonian structures of the super Sasa–Satsuma hierarchy are constructed by utilizing the super trace identity. The infinite conservation laws of the super Sasa–Satsuma equation are presented by resorting to the spectral parameter expansion.

Authors:P. Amster; A. Déboli; M.P. Kuna Pages: 53 - 58 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): P. Amster, A. Déboli, M.P. Kuna A coupled Gompertz-like system of delay differential equations is considered. We prove the existence of T -periodic solutions under resonance assuming a Lazer–Leach type condition.

Authors:Yanfang Gao; Zhiyong Wang Pages: 59 - 64 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Yanfang Gao, Zhiyong Wang This paper concerns the semi-relativistic Hartree equation i ∂ t u = − Δ + m 2 u − ( ⋅ − 1 ∗ u 2 ) u in R 3 . We prove the concentration results for finite time blow-up solutions with general H x 1 ∕ 2 ( R 3 ) data, and show the relation between the concentration rate and the blow-up order.

Authors:Shou-Fu Tian Pages: 65 - 72 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Shou-Fu Tian In this paper, we consider the asymptotic behavior of a weakly dissipative modified two-component Dullin–Gottwald–Holm (mDGH2) system. We derive the asymptotic behavior of the solution at infinity expressed in exponential and algebraic forms. It is shown that there are some results about these properties of the strong solutions in L ∞ -space.

Authors:Li Zou; Zong-Bing Yu; Xiu-Bin Wang Pages: 73 - 79 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Li Zou, Zong-Bing Yu, Xiu-Bin Wang Under investigation in this work is an extend Kadomtsev–Petviashvili (eKP) equation, which appears in the study of multi-component plasmas. By considering Bell’s polynomials, an effective and straightforward way is presented to succinctly derive its bilinear form and soliton solutions. Moreover, the homoclinic breather limit method is employed to construct the breather wave and rogue wave solutions of the equation. Finally, the dynamic behaviors of breather waves, rogue waves and solitary waves are discussed with graphic analysis. It is hoped that our results can be useful for explaining and enriching the dynamic behavior of these KP-type equations.

Authors:Nguyen Huy Tuan; Tran Dong Xuan; Nguyen Anh Triet; Daniel Lesnic Pages: 80 - 86 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Nguyen Huy Tuan, Tran Dong Xuan, Nguyen Anh Triet, Daniel Lesnic We study, for the first time in the literature on the subject, the Cauchy problem for a semilinear fractional elliptic equation. Under an a priori assumption on the solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic type is established.

Authors:Adnan Khaliq; Mujeeb ur Rehman Pages: 95 - 102 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Adnan Khaliq, Mujeeb ur Rehman In this paper by using the variational methods for a class of impulsive differential equation of fractional order with non-instantaneous impulses, we setup sufficient conditions for the existence and uniqueness of weak solutions. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of a functional. Main results of the present work are established by using Lax–Milgram Theorem.

Authors:Eduardo Hernández; Donal O’Regan Pages: 103 - 109 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Eduardo Hernández, Donal O’Regan We introduce a new type of non-local conditions, which we call state dependent non-local conditions, and we study existence and uniqueness of solutions for abstract differential equation subjected to this class of conditions. The non-local condition proposed generalizes several types of non-local conditions studied in the literature. Some examples are given to illustrate our theory.

Authors:Min Li; Juan-Juan Shui; Tao Xu Pages: 110 - 115 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Min Li, Juan-Juan Shui, Tao Xu In this paper, we analyze the generation mechanism of rogue waves for the discrete nonlinear Schrödinger (DNLS) equation from the viewpoint of structural discontinuities. First of all, we derive the analytical breather solutions of the DNLS equation on a new nonvanishing background through the Darboux transformation (DT). Via the explicit expressions of group and phase velocities, we give the parameter conditions for existence of the velocity jumps, which are consistent with the derivation of rogue waves via the generalized DT. Furthermore, to verify such statement, we apply the Taylor expansion to the breather solutions and find that the first-order rogue wave can be obtained at the velocity-jumping point. Our analysis can help to enrich the understanding on the rogue waves for the discrete nonlinear systems.

Authors:Boris N. Chetverushkin; Alexander A. Zlotnik Pages: 116 - 122 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Boris N. Chetverushkin, Alexander A. Zlotnik We deal with a linear parabolic initial–boundary value problem and its hyperbolic perturbation with a small parameter ε > 0 in front of the 2nd order time derivative. We derive bounds for the perturbation error of the orders O ( ε ) and O ( ε ) in several norms in dependence with smoothness of data (without a priori conditions on solutions) but for non-smooth coefficients and keeping the free terms in equations as distributions. They essentially complement or improve some previously known bounds. In addition, we discuss regularizations of the initial time derivative.

Authors:Liangcai Mei; Yuntao Jia; Yingzhen Lin Pages: 123 - 129 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Liangcai Mei, Yuntao Jia, Yingzhen Lin In this paper, a new algorithm is presented to solve the impulsive delay initial value problems. This is the first time to propose the simplified reproducing kernel method (SRKM for short) to solve the impulsive delay differential equations. Then the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The proposed method is proved to be stable and is not less than second order convergence. The algorithm is employed successfully in some numerical examples.

Authors:Bin Wang; Ting Li; Yajun Wu Pages: 130 - 139 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Bin Wang, Ting Li, Yajun Wu It is well known that for gradient systems in Euclidean space or on a Riemannian manifold, the energy decreases monotonically along solutions. In this letter we derive and analyse functionally fitted energy-diminishing methods to preserve this key property of gradient systems. It is proved that the novel methods are energy-diminishing and can achieve damping for very stiff gradient systems. We also show that the methods can be of arbitrarily high order and discuss their implementations. A numerical test is reported to illustrate the efficiency of the new methods in comparison with three existing numerical methods in the literature.

Authors:N. Bondarenko; V. Yurko Pages: 140 - 144 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): N. Bondarenko, V. Yurko We consider second-order functional differential operators with a constant delay. Properties of their spectral characteristics are obtained and a nonlinear inverse problem is studied, which consists in recovering the operators from their spectra. We establish the uniqueness and develop a constructive algorithm for solution of the inverse problem.

Authors:Shujing Gao; Deming Zhong; Yan Zhang Pages: 145 - 150 Abstract: Publication date: September 2018 Source:Applied Mathematics Letters, Volume 83 Author(s): Shujing Gao, Deming Zhong, Yan Zhang A stochastic switching Susceptible–Exposed–Infective–Removed ( S E I R ) epidemic model with continuous and impulsive control schemes is proposed and investigated, in which a general incidence function is considered. The existence and uniqueness of the global positive solution is achieved. By using the Lyapunov–Razumikhin approach, sufficient conditions are developed which guarantee the hybrid system is stochastically stable. The results of simulating our new S E I R model indicate that the hybrid control schemes could be an effective strategy to eradicate the disease.