Authors:Jaume Giné; Jaume Llibre Pages: 130 - 131 Abstract: Publication date: August 2017 Source:Applied Mathematics Letters, Volume 70 Author(s): Jaume Giné, Jaume Llibre We study the local analytic integrability for real Liénard systems, x ̇ = y − F ( x ) , y ̇ = x , with F ( 0 ) = 0 but F ′ ( 0 ) ≠ 0 , which implies that it has a strong saddle at the origin. First we prove that this problem is equivalent to study the local analytic integrability of the [ p : − q ] resonant saddles. This result implies that the local analytic integrability of a strong saddle is a hard problem and only partial results can be obtained. Nevertheless this equivalence gives a new method to compute the so-called resonant saddle quantities transforming the [ p : − q ] resonant saddle into a strong saddle.

Authors:Abdul-Majid Wazwaz Pages: 1 - 6 Abstract: Publication date: August 2017 Source:Applied Mathematics Letters, Volume 70 Author(s): Abdul-Majid Wazwaz In this work we establish a two-mode modified Korteweg–de Vries equation (TmKdV). We show that multiple soliton solutions exist for specific values of the nonlinearity and dispersion parameters of this equation. We also derive more exact solutions for other values of these parameters. We will use the simplified Hirota’s method, the tanh/coth method to conduct this analysis.

Authors:Andrea N. Ceretani; Natalia N. Salva; Domingo A. Tarzia Pages: 14 - 17 Abstract: Publication date: August 2017 Source:Applied Mathematics Letters, Volume 70 Author(s): Andrea N. Ceretani, Natalia N. Salva, Domingo A. Tarzia This article is devoted to the proof of the existence and uniqueness of the modified error function introduced in Cho and Sunderland (1974). This function is defined as the solution to a nonlinear second order differential problem depending on a real parameter. We prove here that this problem has a unique non-negative analytic solution when the parameter assumes small positive values.

Authors:Juan Huang; Jian Zhang Pages: 32 - 38 Abstract: Publication date: August 2017 Source:Applied Mathematics Letters, Volume 70 Author(s): Juan Huang, Jian Zhang This paper concerns the exact value of cross-constrain problem and strong instability of standing waves in trapped dipolar quantum gases. By analyzing the characters of ground state, we give the relation of the cross-constrain problem and constrain problems. Then the exact value of the cross-constrain problem is obtained in the form of the ground state which is relevant to the frequency. Finally, we show the standing waves are strongly unstable.

Authors:Yu Zhu; Meng Liu Pages: 1 - 7 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Yu Zhu, Meng Liu In this paper, a stochastic service–resource mutualism model is proposed and studied. For each species, the threshold between stochastic permanence and extinction is established. The effects of stochastic perturbations on the permanence and extinction of the resource species and the service species are revealed.

Authors:Fang-Fang Liao Pages: 8 - 14 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Fang-Fang Liao We study the existence of positive periodic solutions for a second order differential equation which is related to the Liebau phenomenon. The proof is based on the fixed point theorem in cones. Our results improve and generalize those in the literature.

Authors:Zhijian Yang; Zhiming Liu Pages: 22 - 28 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Zhijian Yang, Zhiming Liu The paper investigates the upper semicontinuity of global attractors A α of a family of semilinear wave equations with gentle dissipation: u t t − △ u + γ ( − △ ) α u t + f ( u ) = g ( x ) , with α ∈ ( 0 , 1 ∕ 2 ) . It is a continuation of researches in Yang et al. (2016), where the existence of global attractors of the equations has been established. We further show that for any neighborhood U of A α 0 , α 0 ∈ ( 0 , 1 ∕ 2 ) , A α ⊂ U when 0 < α − α 0 ≪ 1 .

Authors:Nikolay A. Kudryashov Pages: 29 - 34 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Nikolay A. Kudryashov Continuum model corresponding to the generalization of both the Fermi–Pasta–Ulam and the Frenkel–Kontorova models is considered. This generalized model can be used for the description of nonlinear dislocation waves in the crystal lattice. Using the Painlevé test we analyze the integrability of this equation. We find that there exists an integrable case of the partial differential equation for nonlinear dislocations. Exact solutions of nonlinear dislocation equation are presented.

Authors:Yuchan Wang; Jijun Liu Pages: 42 - 48 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Yuchan Wang, Jijun Liu When electronic devices are in operation, the sharp change of the temperature on devices surface can be considered as an indicator of devices faults. Based on this engineering background, we consider an inverse problem for 1-dimensional heat conduction model, with the aim of detecting the nonsmooth heat dissipation coefficient from measurable temperature on the device surface. We establish the uniqueness for the nonsmooth dissipation coefficient and prove the convergence property of the minimizer of the regularizing cost functional for the inverse problem theoretically. Then a double-iteration scheme minimizing the data-match term and the regularization term alternatively is proposed to implement the reconstructions.

Authors:Lixing Han Pages: 49 - 54 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Lixing Han Multilinear systems of equations arise in various applications, such as numerical partial differential equations, data mining, and tensor complementarity problems. In this paper, we propose a homotopy method for finding the unique positive solution to a multilinear system with a nonsingular M-tensor and a positive right side vector. We analyze the method and prove its convergence to the desired solution. We report some numerical results based on an implementation of the proposed method using a prediction–correction approach for path following.

Authors:Liangwei Wang Pages: 55 - 60 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Liangwei Wang In this paper, we consider the Cauchy problem of the evolution p -Laplacian equation, and reveal the fact that there exists an equivalence relation between the ω -limit set of solutions and the ω -limit set of initial values. This relation can be used to prove different asymptotic behaviors of solutions, and two examples are given at the end of this paper.

Authors:Xiaowei Liu; Jin Zhang Pages: 61 - 66 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Xiaowei Liu, Jin Zhang In this paper we analyze a streamline diffusion finite element method (SDFEM) for a model singularly perturbed convection diffusion problem. By decreasing the standard stabilization parameter properly near the exponential layers, we obtain uniform estimates in a new norm which is stronger than the usual ε -energy norm and weaker than the standard streamline diffusion norm. Through the analysis we know about the influences of the stabilization parameter on the SDFEM solutions.

Authors:Lifang Pei; Dongyang Shi Pages: 67 - 74 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Lifang Pei, Dongyang Shi A nonconforming finite element method (FEM for short) is proposed and analyzed for a plate contact problem by employing the Bergan’s energy-orthogonal plate element. Because the shape function and its first derivatives of this element are discontinuous at the element’s vertices, which is quite different from the conventional finite elements used in the existing literature, some novel approaches, including interpolation operator splitting and energy orthogonality, are developed to present a new error analysis for deriving an optimal estimate of order O ( h ) . At last, some numerical results are also provided to confirm the theoretical analysis.

Authors:Vo Anh Khoa; Tran The Hung Pages: 75 - 81 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Vo Anh Khoa, Tran The Hung This Note derives regularity bounds for a Gevrey criterion when the Cauchy problem of elliptic equations is solved by regularization. When utilizing the regularization, one knows that checking such criterion is basically problematic, albeit its importance to engineering circumstances. Therefore, coping with that impediment helps us improve the use of some regularization methods in real-world applications. This work also considers the presence of the power-law nonlinearities.

Authors:Xiaowei An; Xianfa Song Pages: 82 - 86 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Xiaowei An, Xianfa Song In this paper, using a delicate application of general Sobolev inequality, we establish the lower bound for the blowup time of the solution to a quasi-linear parabolic problem, which improves the result of Theorem 2.1 in Bao and Song (2014).

Authors:Chunhong Zhang; Zhisu Liu Pages: 87 - 93 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Chunhong Zhang, Zhisu Liu In this paper, we study the following Kirchhoff type problem with critical growth − M ∫ Ω ∇ u 2 d x △ u = λ u 2 u + u 4 u in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R 3 , M ∈ C ( R + , R ) and λ > 0 . We prove the existence of multiple nontrivial solutions for the above problem, when parameter λ belongs to some left neighborhood of the eigenvalue of the nonlinear operator − M ( ∫ Ω ∇ u 2 d x ) △ .

Authors:Huanying Xu; Xiaoyun Jiang; Bo Yu Pages: 94 - 100 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Huanying Xu, Xiaoyun Jiang, Bo Yu This work numerically investigates the space fractional Navier–Stokes equations obtained through replacing Laplacian operator in Navier–Stokes equations by Riesz fractional derivatives. The pressure-driven flow between two parallel plates is solved with the finite difference method of fractional differential equations. The influence of pertinent parameters on the characteristics of fluid flow is analyzed with graphics. Further, the Levenberg–Marquardt algorithm is also proposed to estimate model parameters.

Authors:Hai-Qiang Zhang; Rui Hu; Meng-Yue Zhang Pages: 101 - 105 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Hai-Qiang Zhang, Rui Hu, Meng-Yue Zhang In this Letter, the Darboux transformation is first applied to the defocusing Sasa–Satsuma equation which describes the propagation dynamics of ultrashort light pulses in the normal dispersion regime of optical fibers. The N -fold iterative transformation is expressed in terms of the determinants. This transformation enables us to construct explicit analytical dark soliton solutions from non-vanishing background. In illustration, the single-dark soliton solution is presented from the once-iterated formula.

Authors:Yuri Kondratiev; Yuri Kozitsky Pages: 106 - 112 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Yuri Kondratiev, Yuri Kozitsky The Markov dynamics is studied of an infinite system of point entities placed in R d , in which the constituents disperse and die, also due to competition. Assuming that the dispersal and competition kernels are continuous and integrable we show that the evolution of states of this model preserves their sub-Poissonicity, and hence the local self-regulation (suppression of clustering) takes place. Upper bounds for the correlation functions of all orders are also obtained for both long and short dispersals, and for all values of the intrinsic mortality rate.

Authors:Guoqiang Zhang; Zhenya Yan; Yong Chen Pages: 113 - 120 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Guoqiang Zhang, Zhenya Yan, Yong Chen We investigate the defocusing integrable nonlocal nonlinear Schrödinger (nNLS) equation using the loop group method and perturbation expansion idea. The associated N -fold Darboux transformation is presented in terms of the simple determinants. As the application of the N -fold Darboux transformation, we find the higher-order rational solitons of the nNLS equation, which exhibit many new types of soliton structures. Moreover, we also numerically study the stability of some rational solitons. The results are useful to explain the corresponding wave phenomena in nonlocal wave models.

Authors:Stylianos Dimas; Igor Leite Freire Pages: 121 - 125 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Stylianos Dimas, Igor Leite Freire We study a family of PDEs, which was derived as an approximation of an extended Lotka–Volterra system, from the point of view of symmetries. Also, by performing the self adjoint classification on that family we offer special cases possessing non trivial conservation laws. Using both classifications we justify the particular cases studied in the literature, and we give additional cases that may be of importance.

Authors:Jozef Džurina; Irena Jadlovská Pages: 126 - 132 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Jozef Džurina, Irena Jadlovská The purpose of this paper is to study the second-order half-linear delay differential equation r ( t ) y ′ ( t ) α ′ + q ( t ) y α ( τ ( t ) ) = 0 under the condition ∫ ∞ r − 1 ∕ α ( t ) d t < ∞ . Contrary to most existing results, oscillation of the studied equation is attained via only one condition. A particular example of Euler type equation is provided in order to illustrate the significance of our main results.

Authors:Marc Bakry Pages: 133 - 137 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Marc Bakry We construct a reliable goal-oriented a posteriori error estimate for the oscillating single layer equation in acoustics. It is based on the computation of the residual corresponding to the primal problem and a reconstruction of the local error for the solution of the dual problem following the Dual-Weighted-Residual approach. We give a numerical example in 2D where the target is the far field value in some direction.

Authors:Aymen Laadhari Pages: 138 - 145 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Aymen Laadhari In this letter, we present a computational framework based on the use of the Newton and level set methods and tailored for the modeling of bubbles with surface tension in a surrounding Newtonian fluid. We describe a fully implicit and monolithic finite element method that maintains stability for significantly larger time steps compared to the usual explicit method and features substantial computational savings. A suitable transformation avoids the introduction of an additional mixed variable in the variational problem. An exact tangent problem is derived and the nonlinear problem is solved by a quadratically convergent Newton method. In addition, we consider a generalization to the multidimensional case of the Kou’s and McDougall’s methods, resulting in a faster convergence. The method is benchmarked against known results with the aim of illustrating its accuracy and robustness.

Authors:Jae-Myoung Kim Pages: 153 - 160 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Jae-Myoung Kim We give a weak − L p Serrin type regularity criteria of weak solutions to the 3D viscoelastic Navier–Stokes equations with damping, which is described as an incompressible non-Newtonian fluid in the whole space R 3 .

Authors:Hui Wang; Yun-Hu Wang Pages: 161 - 167 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Hui Wang, Yun-Hu Wang By using the truncated Painlevé expansion and consistent Riccati expansion (CRE), we investigate a dissipative ( 2 + 1 )-dimensional Ablowitz–Kaup–Newell–Segur (AKNS) equation. Through the truncated Painlevé expansion, its nonlocal symmetry and Bäcklund transformation (BT) are presented. Then the nonlocal symmetry is localized to the corresponding nonlocal group by an enlarged system. Based on the CRE method proposed by Lou (2013), the AKNS equation is proved CRE solvable, and the soliton-cnoidal wave interaction solutions are explicitly given.

Authors:Song Liu; Xiang Wu; Yan-Jie Zhang; Ran Yang Pages: 168 - 173 Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Song Liu, Xiang Wu, Yan-Jie Zhang, Ran Yang In this paper, the asymptotical stability of Riemann–Liouville fractional neutral systems is investigated. Applying Lyapunov direct method, we present new sufficient conditions on asymptotical stability in terms of linear matrix inequality (LMI) which can be easily solved. The advantage of our employed method is that one may directly calculate integer-order derivatives of the Lyapunov functions. Finally, two simple examples are given to show that the proposed method is computationally flexible and efficient.

Authors:Tonghua Zhang; Tongqian Zhang; Xinzhu Meng Pages: 1 - 7 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Tonghua Zhang, Tongqian Zhang, Xinzhu Meng In this paper, we dedicate ourselves to the study of a diffusive model for unstirred membrane reactors with maintenance energy and subject to a homogeneous Neumann boundary condition. It shows that the unique constant steady state is globally asymptotically stable when it exists. This result further implies the non-existence of any spatial patterns.

Authors:Anmin Mao; Lijuan Yang; Aixia Qian; Shixia Luan Pages: 8 - 12 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Anmin Mao, Lijuan Yang, Aixia Qian, Shixia Luan The aim of this paper is to study the following Schrödinger–Poisson problem ( SP ) − △ u + V ( x ) u + ε ϕ ( x ) u = λ f ( u ) , in R 3 , − △ ϕ = u 2 , lim x → + ∞ ϕ ( x ) = 0 , in R 3 , u > 0 , in R 3 . We prove existence and concentration of positive solutions to system ( SP ) for suitable range of ε and λ .

Authors:Wei-Hua Luo; Ting-Zhu Huang; Xian-Ming Gu; Yi Liu Pages: 13 - 19 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Wei-Hua Luo, Ting-Zhu Huang, Xian-Ming Gu, Yi Liu In this study, based on barycentric rational interpolation functions, three collocation methods are presented for a class of nonlinear parabolic partial differential equations (PDEs). Corresponding algebraic equations are obtained. Numerical examples involving the Fitzhugh–Nagumo equation, Heat equation and Runge’s example are reported to illustrate the effectiveness of the proposed techniques.

Authors:G.E. Chatzarakis; H. Péics Pages: 20 - 26 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): G.E. Chatzarakis, H. Péics This paper is concerned with the oscillatory behavior of first-order differential equations with several non-monotone delay arguments and non-negative coefficients. A sufficient condition, involving lim sup , which guarantees the oscillation of all solutions is established. Also, an example illustrating the significance of the result is given.

Authors:Xiao-Diao Chen; Jiaer Shi; Weiyin Ma Pages: 27 - 32 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Xiao-Diao Chen, Jiaer Shi, Weiyin Ma The root-finding problem of a univariate nonlinear equation is a fundamental and long-studied problem, and has wide applications in mathematics and engineering computation. This paper presents a fast and robust method for computing the simple root of a nonlinear equation within an interval. It turns the root-finding problem of a nonlinear equation into the solution of a set of linear equations, and explicit formulae are also provided to obtain the solution in a progressive manner. The method avoids the computation of derivatives, and achieves the convergence order 2 n − 1 by using n evaluations of the function, which is optimal according to Kung and Traub’s conjecture. Comparing with the prevailing Newton’s methods, it can ensure the convergence to the simple root within the given interval. Numerical examples show that the performance of the derived method is better than those of the prevailing methods.

Authors:Julia Elyseeva Pages: 33 - 39 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Julia Elyseeva In this paper we investigate oscillations of conjoined bases of linear Hamiltonian differential systems related via symplectic transformations. Both systems are considered without controllability (or normality) assumptions and under the Legendre condition for their Hamiltonians. The main result of the paper presents new explicit relations connecting the multiplicities of proper focal points of Y ( t ) and the transformed conjoined basis Y ̃ ( t ) = R − 1 ( t ) Y ( t ) , where the symplectic transformation matrix R ( t ) obeys some additional assumptions on the rank of its components. As consequences of the main result we formulate the generalized reciprocity principle for the Hamiltonian systems without normality. The main tool of the paper is the comparative index theory for discrete symplectic systems generalized to the continuous case.

Authors:Xiu-Bin Wang; Shou-Fu Tian; Chun-Yan Qin; Tian-Tian Zhang Pages: 40 - 47 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Xiu-Bin Wang, Shou-Fu Tian, Chun-Yan Qin, Tian-Tian Zhang In this paper, the homoclinic breather limit method is employed to find the breather wave and the rational rogue wave solutions of the ( 2 + 1 )-dimensional Ito equation. Moreover, based on its bilinear form, the solitary wave solutions of the equation are also presented with a detailed derivation. The dynamic behaviors of breather waves, rogue waves and solitary waves are analyzed with some graphics, respectively. The results imply that the extreme behavior of the breather solitary wave yields the rogue wave for the ( 2 + 1 )-dimensional Ito equation.

Authors:Hui Liu; Hongjun Gao Pages: 48 - 54 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Hui Liu, Hongjun Gao The incompressible Navier–Stokes equations with damping are considered in this paper. Using Fourier splitting method, we will prove the L 2 decay of weak solutions for β > 2 with any α > 0 . Finally, we show the asymptotic stability of the solution to the system for β > 3 with any α > 0 and α ≥ 1 2 as β = 3 .

Authors:Hong-Qian Sun; Ai-Hua Chen Pages: 55 - 61 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Hong-Qian Sun, Ai-Hua Chen In this paper, we study the lump solutions and their dynamics of the ( 3 + 1 ) -dimensional Jimbo–Miwa equation and two extended Jimbo–Miwa equations via bilinear forms. Based on the obtained lump solutions, the lump–kink solution which contains interaction between a lump and a kink wave of the ( 3 + 1 ) -dimensional Jimbo–Miwa equation is also obtained.

Authors:S.A. Khuri; A. Sayfy Pages: 68 - 75 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): S.A. Khuri, A. Sayfy The aim of this article is to extend the Variational Iteration Method (VIM) for the proper treatment of boundary value problems (BVPs). This is achieved by introducing a modified correction functional, expressed in terms of two Lagrange multipliers, that takes both endpoints of the interval into consideration. The settings of the generalized VIM are given for three classes of BVPs. The numerical results of two examples confirm the validity and high accuracy of this novel generalization of the VIM.

Authors:Jaume Llibre; Claudia Valls Pages: 76 - 79 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Jaume Llibre, Claudia Valls In this paper we provide a lower bound for the maximum number of limit cycles surrounding the origin of systems ( x ̇ , y ̇ = x ̈ ) given by a variant of the generalized Riccati equation x ̈ + ε x 2 n + 1 x ̇ + b x 4 n + 3 = 0 , where b > 0 , b ∈ R , n is a non-negative integer and ε is a small parameter. The tool for proving this result uses Abelian integrals.

Authors:Xiuying Li; Haixia Li; Boying Wu Pages: 80 - 86 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Xiuying Li, Haixia Li, Boying Wu In this letter, a high order numerical scheme is proposed for solving variable order fractional functional differential equations. Firstly, the problem is approximated by an integer order functional differential equation. The integer order differential equation is then solved by the reproducing kernel method. Numerical examples are given to demonstrate the theoretical analysis and verify the efficiency of the proposed method.

Authors:Minghua Chen; Weihua Deng Pages: 87 - 93 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Minghua Chen, Weihua Deng This paper focuses on providing the high order algorithms for the space–time tempered fractional diffusion-wave equation. The designed schemes are unconditionally stable and have the global truncation error O ( τ 2 + h 2 ) , being theoretically proved and numerically verified.

Authors:Pouria Ramazi; Hildeberto Jardón-Kojakhmetov; Ming Cao Pages: 94 - 100 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Pouria Ramazi, Hildeberto Jardón-Kojakhmetov, Ming Cao For continuously differentiable vector fields, we characterize the ω limit set of a trajectory converging to a compact curve Γ ⊂ R n . In particular, the limit set is either a fixed point or a continuum of fixed points if Γ is a simple open curve; otherwise, the limit set can in addition be either a closed orbit or a number of fixed points with compatibly oriented orbits connecting them. An implication of the result is a tightened-up version of the Poincaré–Bendixson theorem.

Authors:Jakub Wiktor Both; Manuel Borregales; Jan Martin Nordbotten; Kundan Kumar; Florin Adrian Radu Pages: 101 - 108 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Jakub Wiktor Both, Manuel Borregales, Jan Martin Nordbotten, Kundan Kumar, Florin Adrian Radu We study the iterative solution of coupled flow and geomechanics in heterogeneous porous media, modeled by a three-field formulation of the linearized Biot’s equations. We propose and analyze a variant of the widely used Fixed Stress Splitting method applied to heterogeneous media. As spatial discretization, we employ linear Galerkin finite elements for mechanics and mixed finite elements (lowest order Raviart–Thomas elements) for flow. Additionally, we use implicit Euler time discretization. The proposed scheme is shown to be globally convergent with optimal theoretical convergence rates. The convergence is rigorously shown in energy norms employing a new technique. Furthermore, numerical results demonstrate robust iteration counts with respect to the full range of Lamé parameters for homogeneous and heterogeneous media. Being in accordance with the theoretical results, the iteration count is hardly influenced by the degree of heterogeneities.

Authors:Lan Zhu; Ting-Zhu Huang; Liang Li Pages: 109 - 116 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Lan Zhu, Ting-Zhu Huang, Liang Li This work presents a high-order hybridizable discontinuous Galerkin (HDG) method based on hybrid mesh to solve the two-dimensional time-harmonic Maxwell’s equations. The hybrid mesh consists of unstructured triangular mesh in the domain with complex geometries and structured quadrilateral mesh in the rest domain. The coupling of different meshes is natural in HDG framework. Numerical simulations show that the HDG method on hybrid mesh converges at the optimal rate. One can save computation costs by employing hybrid mesh due to the reduction in number of degrees of freedom (DOFs).

Authors:Haibo Yu; Hao Xu; Mingxuan Zhu Pages: 117 - 121 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Haibo Yu, Hao Xu, Mingxuan Zhu This paper considers the global regularity to the 3D generalized MHD equations with the fractional dissipation and magnetic diffusion ( − Δ ) α for 0 < α < 5 ∕ 4 . Let μ , ν , u and b denote the viscosity coefficient, magnetic diffusivity, velocity field and magnetic field, respectively. It is shown that ‖ ( u , b ) ‖ H s ( s > 5 ∕ 2 − α ) is globally bounded as long as μ − ν ( μ + ν ) − 1 and H ̇ 5 2 − 2 α -norm on ( u 0 − b 0 ) or ( u 0 + b 0 ) sufficiently small, which generalizes the previous result about large solutions in He et al. (2014).

Authors:Nikolaos S. Papageorgiou; Calogero Vetro; Francesca Vetro Pages: 122 - 128 Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro We consider a semilinear elliptic problem with Robin boundary condition and an indefinite and unbounded potential. The reaction term is a Carathéodory function exhibiting linear growth near ± ∞ . We assume that double resonance occurs with respect to any positive spectral interval. Using variational tools and critical groups, we show that the problem has a nontrivial smooth solution.

Authors:Fatma Abstract: Publication date: August 2017 Source:Applied Mathematics Letters, Volume 70 Author(s): Fatma Karakoç We investigate oscillation about the positive equilibrium point of a population model with piecewise constant argument. By using linearized oscillation theory for difference equations a necessary and sufficient condition for the oscillation is obtained. Moreover it is showed that every nonoscillatory solution approaches to the equilibrium point as t tends to infinity.

Authors:Ting Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Ting Su A Modified 2+1-Dimensional Coupled Burgers equation and its Lax pairs are proposed. A Darboux transformation for the Modified 2+1-Dimensional Coupled Burgers equation and its Lax pairs is established with the help of the gauge transformation between spectral problems. As an application of the Darboux transformation, we give some explicit solutions of the Modified 2+1-Dimensional Coupled Burgers equation.

Authors:Kai Liu Abstract: Publication date: July 2017 Source:Applied Mathematics Letters, Volume 69 Author(s): Kai Liu In this note, we shall consider the norm continuity of a class of solution semigroups associated with linear functional differential equations of neutral type with time lag r > 0 in Hilbert spaces. The norm continuity plays an important role in the analysis of asymptotic stability of the system under consideration by means of spectrum approaches. We shall show that for a square integrable neutral delay term and an unbounded infinitesimal generator A multiplied by a square integrable weight function in the distributed delay term, the associated solution semigroup of the system is norm continuous at every t > r .

Authors:Caisheng Chen Abstract: Publication date: June 2017 Source:Applied Mathematics Letters, Volume 68 Author(s): Caisheng Chen In this paper we prove the Liouville type theorem for stable solution of the p -Laplace equation (0.1) − Δ p u = f ( x ) u q − 1 u , x ∈ R N , where f ( x ) is a continuous and nonnegative function in R N ∖ { 0 } such that f ( x ) ≥ a 0 x a as x ≥ R 0 > 0 , where a > − p and a 0 > 0 . The results hold true for 1 ≤ p − 1 < q = q c ( p , N , a ) . Here q c is a new exponent, which is infinity in low dimension and is always larger than the classical critical one.