Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Cheng Zhang, Qiang Cheng, Da-jun Zhang We provide a direct method for constructing soliton solutions of the sine-Gordon equation in the laboratory coordinates on the half line in the presence of integrable boundary conditions derived in Sklyanin’s classical work (Sklyanin, 1987). Explicit examples, including boundary bound states, are presented.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Ying Shi, Junxiao Zhao In this letter, by a 3-periodic reduction technique, we regain the well-known discrete modified Boussinesq equation and its Lax pairs. It is the first time that the Darboux transformation is successfully constructed for this discrete equation. We then obtain its explicit solutions in terms of Casoratian determinants. The derived solutions are soliton solutions when the seed solutions are taken as v=1 and w=1.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): C.A. Raposo, J.O. Ribeiro, A.P. Cattai In this work we prove global solution for the nonlinear system utt−Δpu+θ= u r−1uθt−Δθ=utwhere Δp is the nonlinear p-Laplacian operator, 2≤p

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Jaime E. Muñoz Rivera, Maria Grazia Naso We prove the exponential stability of the Timoshenko beam with boundary dissipation only on one side of the bending moment, provided the wave speeds of the system are equal.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Wenping Fan, Haitao Qi In this paper, we introduce an efficient Galerkin finite element method for solving the two-dimensional nonlinear time–space fractional Schrödinger equation, with time-dependent potential function, defined on an irregular convex domain. The Caputo time fractional derivative is discretized by the finite difference scheme, while the space fractional derivative defined on the irregular convex domain is dealt with the unstructured mesh finite element method. Given the nonlinear property of the considered equation, an efficient linearized strategy is used. To testify the efficiency of the proposed numerical scheme, two numerical examples are conducted with error and convergence analysis.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Jianhua Zhang, Zewen Wang, Jing Zhao In this paper, a preconditioned symmetric block triangular splitting (PSBTS) iteration method is proposed for solving a class of block two-by-two real linear systems which arise from complex symmetric linear systems. The eigenvalue distributions and the convergence properties of the PSBTS iteration method are established, and the optimal values of the relaxation parameter and the corresponding spectral radius of the PSBTS iteration matrix are obtained, respectively. It is remarkable that the spectral radius of the PSBTS iteration method is smaller than that of the SBTS iteration method under suitable conditions. Finally, two numerical examples are tested for illustrating effectiveness and robustness of the PSBTS iteration method.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Oleg Makarenkov, Anthony Phung For switched systems that switch between distinct globally stable equilibria, we offer closed-form formulas that lock oscillations in the required neighborhood of the equilibria. Motivated by non-spiking neuron models, the main focus of the paper is on the case of planar switched affine systems, where we use properties of nested cylinders coming from quadratic Lyapunov functions. In particular, for the first time ever, we use the dwell-time concept in order to give an explicit condition for non-spiking of linear neuron models with periodically switching current.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): De Tang, Li Ma In this paper, we investigate a reaction–diffusion two-competition species model with advection term under the Dirichlet boundary condition. By employing implicit function theorem, the principal spectral theory and other important formulas, we obtain the existence and uniqueness of spatial nonhomogeneous solutions under some given parameters conditions.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Qiangchang Ju, Xin Xu The small Alfvén number limit of the plane magnetohydrodynamic flows is rigorously proved under appropriate initial conditions. The limit system that we obtained is the compressible Euler equations involving the background magnetic field.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Ferenc Izsák, Béla J. Szekeres A novel algorithm is proposed for computing matrix–vector products Aαv, where A is a symmetric positive semidefinite sparse matrix and α>0. The method can be applied for the efficient implementation of the matrix transformation method to solve space-fractional diffusion problems. The performance of the new algorithm is studied in a comparison with the conventional MATLAB subroutines to compute matrix powers.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Young-Pil Choi, Zhuchun Li We analyze the flocking behavior of a Cucker–Smale-type system with a delayed coupling, where delays are information processing and reactions of individuals. By constructing a system of dissipative differential inequalities together with a continuity argument, we provide a sufficient condition for the flocking behavior showing the velocity alignment between particles as time goes to infinity when the maximum value of time delays is sufficiently small.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Chengliang Li, Changfeng Ma In this paper, we propose an Euler-extrapolated Hermitian/skew-Hermitian splitting (E-HS) iterative method for solving a class of complex symmetric linear systems. The convergence and optimal parameter of the E-HS method are presented. In addition, some spectral properties of the preconditioned matrix are also studied. Numerical experiments verify the effectiveness of the E-HS method either as a solver or a preconditioner.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Jian-Guo Liu Making use of the Hirota’s bilinear form, lump-type solutions for the (2+1)-dimensional generalized fifth-order KdV equation are presented. The interactions between lump-type solutions and double exponential function are discussed. The physical structure and characteristics for these obtained solutions are demonstrated in some three-dimensional plots and corresponding contour plots.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Fang Chen For the generalized saddle-point linear system, we prove the unconditional convergence of the efficient variant of the HSS (EVHSS) iteration method introduced by Zhang (2018).

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Alessio Fumagalli We consider a Darcy problem for saturated porous media written in dual formulation in presence of a fully immersed inclusion. The lowest order virtual element method is employed to derive the discrete approximation. In the present work we study the effect of cells with cuts on the numerical solution, able to geometrically handle in a more natural way the inclusion tips. The numerical results show the validity of the proposed approach.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): SeakWeng Vong, Chenyang Shi, Dongdong Liu In this paper, we establish new weighted integral inequalities by considering polynomials which are orthogonal in a weighted sense. These inequalities generalize some results established recently. They are applied to study exponential stability of some time-delay systems under the framework of linear matrix inequalities. Numerical tests are given to demonstrate the efficiency of the derived stability conditions.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Yue Hao, Ai-Li Yang, Yu-Jiang Wu Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method and its alternating-directional variant, a parameterized PMHSS (P2MHSS) preconditioner is proposed for a class of block two-by-two systems of linear equations. To optimize the preconditioning effect of the P2MHSS preconditioner, practical formulas for computing the optimal parameters of the P2MHSS preconditioner are derived by employing the scaled norm minimization method. Finally, we test the feasibility and the effectiveness of the P2MHSS preconditioner.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Tengteng Ma, Yu Tian, Qiang Huo, Yong Zhang In this paper, we are interested in the boundary conditions u(0)=au(1),u′(0)=bu′(1) for linear fractional differential equation 0cDtαu(t)+λu(t)=0. Via Laplace transform and inverse Laplace transform, we obtain eigenvalues and eigenfunctions. Furthermore, we study the same boundary conditions for nonlinear fractional differential equation 0cDtαu(t)+f(t,u(t))=0. Combining the obtained eigenvalues and eigenfunctions and the improved Leray–Schauder degree, we prove that there exists at least one nontrivial solution for nonlinear boundary value problem.

Abstract: Publication date: December 2018Source: Applied Mathematics Letters, Volume 86Author(s): Juan J. Nieto, José M. Uzal For given nonlinear differential equations it may occur that there are no periodic solutions. By introducing impulses at prescribed instants, periodic solutions may appear. We consider some impulsive nonlinear first order differential equations having periodic solutions and extend previous results to a larger class of nonlinearities.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Hui Zhang, Xiaoyun Jiang, Moli Zhao, Rumeng Zheng In this paper, Fourier spectral approximation for the time fractional Boussinesq equation with periodic boundary condition is considered. The space is discretized by the Fourier spectral method and the Crank–Nicolson scheme is used to discretize the Caputo time fractional derivative. Stability and convergence analysis of the numerical method are proven. Some numerical examples are included to testify the effectiveness of our given method. Based on the presented numerical results, the Fourier spectral method is shown to be effective for solving the time fractional Boussinesq equation.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Shimin Guo, Liquan Mei, Zhengqiang Zhang, Yutao Jiang In this letter, we consider the numerical approximation of a two-dimensional distributed-order time–space fractional reaction–diffusion equation. The time- and space-fractional derivatives are considered in the senses of Caputo and Riesz, respectively. By using the composite mid-point quadrature, the original fractional problem is approximated by a multi-term time–space fractional differential equation. Then the multi-term Caputo fractional derivatives are discretized by the L2-1σ formula. We apply the Legendre–Galerkin spectral method for the spatial approximation. Two numerical experiments with smooth and non-smooth initial conditions, respectively, are performed to illustrate the robustness of the proposed method. The results show that: our scheme can arrive at the spectral accuracy (resp. algebraic accuracy) in space for the problem with smooth (resp. non-smooth) initial condition. For both of these two cases, our scheme can lead to the second-order accuracies in time. Additionally, the convergence rates in both spatial and temporal distributed-order variables are two.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Peng Zhao, Engui Fan, Temuerchaolu Without solving the equations, we show that soliton solutions of the relativistic Lotka–Volterra hierarchy can be derived from those of the Lotka–Volterra hierarchy. The proof is based on the reductions of a discrete spectral problem.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Fei Wang, Huayi Wei This work aims at studying the virtual element method (VEM) to solve a simplified friction problem, which is a typical elliptic variational inequality of the second kind. An optimal error estimate is derived in the H1 norm for the lowest-order VEM. A numerical example is reported to demonstrate the theoretically predicted convergence order.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Mengmeng Li, JinRong Wang This paper derives a representation of a solution to the initial value problem for a linear fractional delay differential equation with Riemann–Liouville derivative. We apply the method of variation of constants to obtain the representation of a solution via a delayed Mittag-Leffler type matrix function.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Ping Li, Lei Wang, Liang-Qian Kong, Xin Wang, Ze-Yu Xie We study the linear stability analysis and nonlinear excitations on a continuous wave background for the fifth-order nonlinear Schrödinger equation. We find that there are a modulation stability (MS) quasi-elliptic ring and a MS curve in the modulation instability (MI) regime. The concrete expressions of different types of nonlinear waves are obtained by using Darboux transformation. And we present the relations and phase diagram between MI and these waves. We discover that the antidark (AD) soliton and nonrational W-shaped soliton can only exist on the resonance line of outside of the MS quasi-elliptic ring and the right of the MS curve. We perform numerical simulation to test the stability of the AD soliton. By introducing the perturbation energy, we further show that solitons in the MI regime are caused by both the fourth- and fifth-order effects.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): K.D. Chu, D.D. Hai, R. Shivaji We study the existence of positive solutions to the non-cooperative pq-Laplacian system −Δpu=λf(u,v)inΩ,−Δqv=λg(u,v)inΩ,with Dirichlet boundary conditions on a bounded domain Ω with smooth boundary ∂Ω, where f,g:(0,∞)×(0,∞)→R are allowed to have infinite semipositone structure at u=0 or v=0. Our approach is based on the Schauder fixed point theorem.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Haiyong Wang In this paper, we present a new and sharper bound for the Legendre coefficients of differentiable functions and then provide a new bound on the approximation error of the truncated Legendre series in the uniform norm. An illustrative example is provided to demonstrate the sharpness of our new results.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Chun Li In this paper, we study the existence of subharmonic solutions with prescribed minimal period for a class of second-order Hamiltonian systems with even potentials. By using the variational methods, we obtain some new existence theorems which improve some recent results in the literature.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Jiu Liu, Tao Liu, Hui-Lan Pan By using the method of Nehari manifold, we obtain the existence of a positive ground state solution for the following non-autonomous Kirchhoff type equation −(a+b∫RN ∇u 2dx)△u+V(x)u=uN+2N−2,x∈RN,u∈D1,2(RN),N=3,4,where a>0,b>0 and V satisfies some appropriate assumptions.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): JunCai Pu, HengChun Hu Many mixed lump–soliton solutions for the (3+1)-dimensional integrable equation are obtained by using the Hirota bilinear method. These mixed lump–soliton solutions with arbitrary constants are given out analytically and graphically.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Liguang Xu, Zhenlei Dai, Danhua He This paper is devoted to the investigation of the ultimate boundedness of impulsive stochastic delay differential equations. The pth moment global exponential ultimate boundedness criteria are obtained based on the Itô formula and the successful construction of suitable Lyapunov functionals, and the estimated exponential convergence rate and the ultimate bound are provided as well. It is shown that unbounded stochastic delay differential equations can be turned into bounded ones by impulses.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Dorian Popa, Ioan Rasa, Adrian Viorel In the present work, we deal with the logistic equation and its stability with respect to perturbations. In fact, for perturbations below a certain threshold, we provide an estimate for the difference between solutions of the exact and perturbed models, which scales linearly with the magnitude of the perturbation. This actually proves the conditional Ulam stability of the logistic equation.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Hideaki Matsunaga, Rina Suzuki This paper deals with a system of rational difference equations xn+1=ayn+bcyn+d,yn+1=axn+bcxn+d,n=0,1,2,…,where a, b, c, d are real numbers with c≠0 and ad−bc≠0. We establish a representation formula of solutions of the system and classify global behavior of solutions when no initial values belong to the forbidden set of the system.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): J.A. Ezquerro, M.A. Hernández-Verón We introduce auxiliary points to obtain domains of global convergence for Newton’s method without imposing convergence conditions on the local solution of equation to solve, as local convergence results do.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Peng Hu, Haijun Xiao This paper is concerned with the delay dependent dissipativity for a class of nonlinear neutral delay differential equations. By taking into account the influence of the delay, a new dissipativity criterion is derived by using a generalized Halanay’s inequality, which is less conservative than those in the existing literature in some cases. At last, two examples are provided to validate our theoretical results.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Zhe Jiao, Yong Xu A linear wave equation with acoustic boundary conditions (ABC) on a portion of the boundary and Dirichlet conditions on the rest of the boundary is considered. The ABC contain a memory damping with respect to the normal displacement of the boundary point. In this paper, we establish polynomial energy decay rates for the wave equation by using resolvent estimates.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Hui Wang Based on the Hirota bilinear method, the lump solution of a (2+1)-dimensional Burgers equations is presented through symbolic computation with Maple, which is rationally localized in all directions in the space. Then the interaction solution between lump solution and one stripe solution is obtained and the result shows that the lump soliton will be drowned or swallowed by the stripe soliton, which can also be called the fission or fusion phenomenon. Furthermore, by the interaction between lump solution and a pair of resonance stripe solitons, a rogue wave phenomenon is revealed.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Martin Stynes Various new definitions of fractional-order derivatives seek to replace the singular kernel of the Caputo definition by some continuous function. It is shown here that the use of any continuous kernel often places a severe and unnatural restriction on the data that can be used in problems formulated using these new definitions.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Francesco Salvarani, Ana Jacinta Soares In this note, we rigorously prove the relaxation limit of the Maxwell–Stefan system to a system of heat equations when all binary diffusion coefficients tend to the same positive value.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): N.I. Mahmudov We introduce a discrete delayed exponential depending on sequence of matrices. This discrete matrix gives a representation of a solution to the Cauchy problem for a discrete linear system with pure delay with sequence of matrices. We discard the commutativity condition used in recent works related to the representation of solutions for discrete delay linear systems.

Abstract: Publication date: November 2018Source: Applied Mathematics Letters, Volume 85Author(s): Fajun Yu We present the generalized Gross–Pitaevskii (GP(p,q)) equation with two kinds of specific space–time modulated potentials, including the generalized Rosen–Morse potential, the combination of harmonic and Gaussian (harmonic–Gaussian) potential. Some analytical bright soliton solutions are derived from the generalized GP(p,q) equation via the complex similarity transformation and the generalized stationary nonlinear Schrödinger (NLS) equation. The soliton solutions can be reduced to Jacobi elliptic and spikon-like solutions by choosing the different parameters, and describe some physically relevant phenomenons. Furthermore, some stabilities of the obtained matter-wave solutions are addressed numerically. We find that some solutions are stable and can be observed over a broad range of parameters through analyzing the effect of the phase noise on these solutions. The obtained results may give a certain theoretical guiding significance of relative experiments in Bose–Einstein condensates.

Abstract: Publication date: Available online 6 July 2018Source: Applied Mathematics LettersAuthor(s): Jiaxiang Cai, Chuanzhi Bai, Haihui Zhang We give a systematic method for discretizing Hamiltonian partial differential equations. An application of the method to the Hamiltonian form of the coupled Schrödinger-KdV equations yields a temporal first-order conservative scheme. Temporal second- and fourth-order schemes are developed by employing the composition method. All the schemes are decoupled and exactly conserve three invariants simultaneously. Numerical results show good performance of the schemes and verify the theoretical results.

Abstract: Publication date: Available online 6 July 2018Source: Applied Mathematics LettersAuthor(s): Xiuying Li, Boying Wu In this letter, a new reproducing kernel collocation method is presented for solving nonlocal fractional boundary value problems with non-smooth solutions. The method is based on the reproducing kernel basis function and the polynomial basis function. The present method can avoid finding a reproducing kernel satisfying nonlocal boundary conditions. The effectiveness of the numerical scheme is illustrated through two numerical examples.

Abstract: Publication date: Available online 6 July 2018Source: Applied Mathematics LettersAuthor(s): H. Ding, C. Li By constructing new numerical approximation formulas for the left and right Riemann–Liouville tempered fractional derivatives of order α∈(0,1), a numerical algorithm is proposed to solve the space tempered fractional convection equation. The algorithm is shown to be unconditionally stable and convergent with order Oτ2+h2. Finally, numerical example is provided to verify the theoretical analysis.

Abstract: Publication date: Available online 4 July 2018Source: Applied Mathematics LettersAuthor(s): Lingju Kong By using the critical point theory, the existence of at least three homoclinic solutions is established for a higher order difference equation defined on Z with p-Laplacian and containing both advance and retardation. Some known results in the literature are extended and complemented.

Abstract: Publication date: Available online 30 June 2018Source: Applied Mathematics LettersAuthor(s): Jishan Fan, Dan Liu, Yong Zhou In this paper, we use the bootstrap argument to prove the uniform-in-k global strong solutions of the 2D magnetic Bénard problem in a bounded domain. Here k is the heat conductivity coefficient.

Abstract: Publication date: Available online 30 June 2018Source: Applied Mathematics LettersAuthor(s): Chein-Shan Liu, Dongjie Liu By using the multiquadric radial basis function (MQ-RBF) method for solving the mixed boundary value problem as well as the Cauchy problem of the Laplace equation in an arbitrary plane domain, an energy gap functional (EGF) is proposed to choose the shape parameters. Upon minimizing the EGF we can pick up the optimal shape parameter and hence achieve the best accuracy of numerical solution. The performance of the minimizing EGF (MEGF) is assessed by numerical tests.

Abstract: Publication date: Available online 30 June 2018Source: Applied Mathematics LettersAuthor(s): Mihály Pituk, Christian Pötzsche It is known that if the coefficient matrix of a linear autonomous difference equation is nonnegative and primitive, then the solutions starting from nonnegative nonzero initial data are strongly ergodic. The strong ergodic property of the nonnegative solutions was earlier extended to equations with asymptotically constant coefficients. In this paper, we present a generalization of the previous results by showing that the nonnegative solutions satisfy a similar ergodic property also in some cases when the coefficient matrices are not asymptotically constant.

Abstract: Publication date: Available online 23 June 2018Source: Applied Mathematics LettersAuthor(s): Wanru Chang, C.S. Chen, Wen Li The availability of the closed-form particular solution for a given differential equation based on a chosen basis function is crucial for solving partial differential equations using the method of particular solutions. In general, the derivation of such a closed-form particular solution is by no means trivial, particularly for higher order partial differential equations. In this paper we give a simple algebraic procedure to avoid the direct derivation of the closed-form particular solutions for fourth order partial differential equations. One numerical example is given to demonstrate the effectiveness of our proposed approach.

Abstract: Publication date: Available online 20 June 2018Source: Applied Mathematics LettersAuthor(s): Si-Liu Xu, Milivoj R. Belić, Guo-Peng Zhou, Jun-Rong He, Xue-Li We demonstrate three-dimensional (3D) vortex solitary waves in the (3+1)D nonlinear Gross–Pitaevskii equation (GPE) with spatially modulated nonlinearity and a trapping potential. The analysis is carried out in spherical coordinates, providing for novel localized solutions, and the 3D vortex solitary waves are built that depend on three quantum numbers. Our analytical findings are corroborated by a direct numerical integration of the original equations. It is demonstrated that the vortex solitons found are stable for the quantum numbers n≤2, l≤2 andm=0,1, independent of the propagation distance.

Abstract: Publication date: Available online 30 May 2018Source: Applied Mathematics LettersAuthor(s): Zhong-Zhou Lan In this paper, a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation is investigated, which can describe an inhomogeneous Heisenberg ferromagnetic spin chain with the octuple-dipole interaction or alpha helical protein with higher-order excitations and interactions under the continuum approximation. Bilinear forms, one- and two-soliton solutions are derived by virtue of the Hirota method with an auxiliary function. Propagation and interaction properties of the solitons are discussed: Parabolic, cubic and periodic solitons are presented. Solitonic amplitudes are only related to the wave numbers, while solitonic velocities are related to both the wave numbers and variable coefficients. Interactions between the two parallel solitons are discussed.