Authors:Piyanuch Siriwat; Sergey V. Meleshko Pages: 1 - 6 Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): Piyanuch Siriwat, Sergey V. Meleshko The group analysis method is applied to the extended Green–Naghdi equations. The equations are studied in the Eulerian and Lagrangian coordinates. The complete group classification of the equations is provided. The derived Lie symmetries are used to reduce the equations to ordinary differential equations. For solving the ordinary differential equations the Runge–Kutta methods were applied. Comparisons between solutions of the Green–Naghdi equations and the extended Green–Naghdi equations are given.

Authors:Meiirkhan Borikhanov; Mokhtar Kirane; Berikbol T. Torebek Pages: 14 - 20 Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): Meiirkhan Borikhanov, Mokhtar Kirane, Berikbol T. Torebek In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana–Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana–Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial–boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.

Authors:Guofeng Che; Haibo Chen Pages: 21 - 26 Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): Guofeng Che, Haibo Chen In this paper, we study the following Schrödinger equation − Δ u + V λ ( x ) u + μ ϕ u p − 2 u = f ( x , u ) + β ( x ) u ν − 2 u , in R 3 , ( − Δ ) α 2 ϕ = μ u p , in R 3 , where μ ≥ 0 is a parameter, α ∈ ( 0 , 3 ) , ν ∈ ( 1 , 2 ) and p ∈ [ 2 , 3 + 2 α ) . V λ is allowed to be sign-changing and ϕ u p − 2 u is a Hartree-type nonlinearity. We require that V λ = λ V + − V − with V + having a bounded potential well Ω whose depth is controlled by λ . Under some mild conditions on V λ ( x ) and f ( x , u ) , we prove that the above system has at least two nontrivial solutions. Specially, our results cover the general Schrödinger equations and the Schrödinger–Poisson equations.

Authors:Arnaud Heibig Pages: 27 - 34 Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): Arnaud Heibig We prove well-posedness for a transport-diffusion problem coupled with a wave equation for the potential. We assume that the initial data are small. A bilinear form in the spirit of Kato’s proof for the Navier–Stokes equations is used, coupled with suitable estimates in Chemin–Lerner spaces. In the one dimensional case, we get well-posedness for arbitrarily large initial data by using Gagliardo–Nirenberg inequalities.

Authors:Aymen Laadhari Pages: 35 - 43 Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): Aymen Laadhari We present a computational framework based on the use of the Newton and level set methods to model fluid–structure interaction problems involving elastic membranes freely suspended in an incompressible Newtonian flow. The Mooney–Rivlin constitutive model is used to model the structure. We consider an extension to a more general case of the method described in Laadhari (2017) to model the elasticity of the membrane. We develop a predictor–corrector finite element method where an operator splitting scheme separates different physical phenomena. The method features an affordable computational burden with respect to the fully implicit methods. An exact Newton method is described to solve the problem, and the quadratic convergence is numerically achieved. Sample numerical examples are reported and illustrate the accuracy and robustness of the method.

Authors:Fanchao Kong; Zhiguo Luo Pages: 44 - 49 Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): Fanchao Kong, Zhiguo Luo In this paper, a system of neutral functional differential equations in critical case is investigated. Under some suitable assumptions, we prove that every bounded solution of the considered system tends to zero vector as t → + ∞ . An example and its numerical simulations are carried out to illustrate the theoretical results.

Authors:J. Vanterler da C. Sousa; E. Capelas de Oliveira Pages: 50 - 56 Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): J. Vanterler da C. Sousa, E. Capelas de Oliveira Using the ψ -Hilfer fractional derivative, we present a study of the Hyers–Ulam–Rassias stability and the Hyers–Ulam stability of the fractional Volterra integro-differential equation by means of fixed-point method.

Authors:Jinge Yang Pages: 57 - 62 Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): Jinge Yang This paper studies the Cauchy problem for a nonlocal diffusion equation u t = J ∗ u − u + u p . We determine the second critical exponent, namely the optimal decay order for initial data at space infinity to distinguish global and blow-up solutions in the “super Fujita” range. Similar to the critical Fujita exponent obtained very recently by Alfaro (2017), we find that the second critical exponent also relies heavily on the behavior of the Fourier transform of the kernel function J .

Authors:Won-Kwang Park Pages: 63 - 71 Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): Won-Kwang Park In this paper, we develop a fast imaging technique for small anomalies located in homogeneous media from scattering parameter data measured at dipole antennas. Based on the representation of scattering parameters when an anomaly exists, we design a direct sampling method (DSM) for imaging an anomaly and establishing a relationship between the indicator function of DSM and an infinite series of Bessel functions of integer order. Simulation results using synthetic data at f = 1 GHz of angular frequency are illustrated to support the identified structure of the indicator function.

Authors:Junxiong Jia; Bangyu Wu Pages: 1 - 7 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Junxiong Jia, Bangyu Wu This paper is concerned with Carleman estimates for some anisotropic space-fractional diffusion equations, which are important tools for investigating the corresponding control and inverse problems. By employing a special weight function and the nonlocal vector calculus, we prove a Carleman estimate and apply it to build a stability result for a backward diffusion problem.

Authors:Jishan Fan; Bessem Samet; Yong Zhou Pages: 8 - 11 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Jishan Fan, Bessem Samet, Yong Zhou In this paper, we first prove global well-posedness of weak solutions for an epitaxial growth model with L 2 initial data in any dimension d . Then, we establish a regularity criterion of strong solutions with dimension d ≥ 3 .

Authors:Xingqiu Zhang; Qiuyan Zhong Pages: 12 - 19 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Xingqiu Zhang, Qiuyan Zhong In this paper, different height functions of the nonlinear term on special bounded sets together with Leggett–Williams and Krasnosel’skii fixed point theorems are employed to establish the existence of triple positive solutions for a class of higher-order fractional differential equations with integral conditions. The singularities are with respect not only to the time but also to the space variables.

Authors:Sun-Hye Park; Mi Jin Lee; Jum-Ran Kang Pages: 20 - 26 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Sun-Hye Park, Mi Jin Lee, Jum-Ran Kang In this work we consider a viscoelastic wave equation of the form u t t − Δ u + ∫ 0 t g ( t − s ) Δ u ( s ) d s + h ( u t ) = u p − 2 u with Dirichlet boundary condition. There are much literature on the blow-up result of solutions for the wave equation with damping term having polynomial growth near zero. However, to my knowledge, there is no blow-up result of solutions for the viscoelastic wave equation without polynomial growth near zero assumption on the damping term. This work is devoted to study a finite time blow-up result of solution with nonpositive initial energy as well as positive initial energy without imposing any restrictive growth near zero assumption on the damping term.

Authors:Yulei Cao; Jiguang Rao; Dumitru Mihalache; Jingsong He Pages: 27 - 34 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Yulei Cao, Jiguang Rao, Dumitru Mihalache, Jingsong He The ( 2 + 1 ) -dimensional [ ( 2 + 1 ) d ] Fokas system is a natural and simple extension of the nonlinear Schrödinger equation (see Eq. (2) in Fokas, 1994). In this letter, we introduce its PT -symmetric version, which is called the ( 2 + 1 ) d nonlocal Fokas system. The N -soliton solutions for this system are obtained by using the Hirota bilinear method whereas the semi-rational solutions are generated by taking the long-wave limit of a part of exponential functions in the general expression of the N -soliton solution. Three kinds of semi-rational solutions, namely (1) a hybrid of rogue waves and periodic line waves, (2) a hybrid of lump and breather solutions, and (3) a hybrid of lump, breather, and periodic line waves are put forward and their rather complicated dynamics is revealed.

Authors:Alexander Zlotnik; Raimondas Čiegis Pages: 35 - 40 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Alexander Zlotnik, Raimondas Čiegis The stability bounds and error estimates for a compact higher order Numerov–Crank–Nicolson scheme on non-uniform spatial meshes for the 1D time-dependent Schrödinger equation have been recently derived. This analysis has been done in L 2 and H 1 mesh norms and used the non-standard “converse” condition h ω ≤ c 0 τ , where h ω is the mean spatial step, τ is the time step and c 0 > 0 . Now we prove that such condition is necessary for some families of non-uniform meshes and any spatial norm. Also computational results for zero and non-zero potentials show unacceptably wrong behavior of numerical solutions when τ decreases and this condition is violated.

Authors:Yu Ping Wang; V.A. Yurko Pages: 41 - 47 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Yu Ping Wang, V.A. Yurko The inverse spectral problem for Dirac operators is studied. By using the result on the Weyl m -function, we show that the Hochstadt–Lieberman-type and Borg-type theorem for Dirac operators except for one arbitrary eigenvalue hold.

Authors:Xi-Yang Xie; Ze-Hui Yan Pages: 48 - 53 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Xi-Yang Xie, Ze-Hui Yan Under investigation in this paper is a Kundu–Eckhaus equation with variable coefficients, which models the propagation of the ultra-short femtosecond pulses in an optical fiber. Bright one- and two-soliton solutions for this equation are constructed, based on the bilinear forms obtained. Then, by the aid of the solutions, propagation of the one solitons and collisions between the two solitons are illustrated in figures, and with the help of the asymptotic analysis on the two-soliton solutions, the collisions are proved to be elastic. Influences of r ( x ) and m ( x ) on the solitons are also be analyzed, where r ( x ) and m ( x ) are respectively the group velocity dispersion and nonlinearity parameter: When they are both chosen as the constants, it can be found that the one solitons propagate with unvarying velocities and amplitudes, and shapes of the two solitons are maintained before and after the collisions; r ( x ) and m ( x ) are found to affect the velocities and amplitudes of the solitons, respectively.

Authors:Xiaowei An; Zhen He; Xianfa Song Pages: 59 - 63 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Xiaowei An, Zhen He, Xianfa Song Using the solutions of an elliptic system and an ODE system, under certain conditions, we get the explicit solution to the following initial–boundary value problem of Gierer-Meinhardt model u t = d 1 Δ u − a 1 u + u p v q + δ 1 ( x , t ) , x ∈ Ω , t > 0 v t = d 2 Δ v − a 2 v + u r v s + δ 2 ( x , t ) , x ∈ Ω , t > 0 ∂ u ∂ η = ∂ v ∂ η = 0 ( or u = v = 0 ) , x ∈ ∂ Ω , t > 0 u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω . Here p > 1 , s > − 1 , d 1 , d 2 , q , r > 0 and a 1 , a 2 ≥ 0 are constants, while δ 1 ( x , t ) and δ 2 ( x , t ) are nonnegative continuous functions.

Authors:Yunfeng Jia Pages: 64 - 70 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Yunfeng Jia We consider a reaction–diffusion population model with predator–prey-dependent functional response. Firstly, we discuss the conditions which ensure the model has a unique positive constant solution. Secondly, we investigate the dynamical properties of the model, including the large time behaviors of the nonconstant solutions and the local and the global asymptotic stability of the positive constant solution.

Authors:Jian-Guo Liu; Li Zhou; Yan He Pages: 71 - 78 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Jian-Guo Liu, Li Zhou, Yan He The multiple exp-function method is utilized for solving the multiple soliton solutions for the new ( 2 + 1 ) -dimensional Korteweg–de Vries equation, which include one-soliton, two-soliton, and three-soliton type solutions. The physical phenomena of these obtained multiple soliton solutions are analyzed and illustrated in figures by selecting appropriate parameters.

Authors:Wen Zhang; Jian Zhang; Weijin Jiang Pages: 79 - 87 Abstract: Publication date: June 2018 Source:Applied Mathematics Letters, Volume 80 Author(s): Wen Zhang, Jian Zhang, Weijin Jiang This paper is concerned with the nonlinear Dirac–Poisson system − i ∑ k = 1 3 α k ∂ k u + ( V ( x ) + a ) β u + ω u − ϕ u = F u ( x , u ) , − Δ ϕ = 4 π u 2 , in R 3 where V is an external potential and F is a superlinear nonlinearity modeling various types of interactions. Existence and multiplicity of stationary solutions are obtained for the system with periodicity condition via variational methods.

Authors:Chein-Shan Liu; Jiang-Ren Chang Pages: 138 - 145 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Chein-Shan Liu, Jiang-Ren Chang An inverse source problem for the recovery of an unknown space–time dependent source term of a time-fractional Burgers equation is solved in the paper. By using the prescribed boundary data, a sequence of boundary functions is derived, which together with the zero element constitute a linear space. An energy boundary functional equation is derived in the linear space, of which the time-dependent energy is preserved for each energy boundary function. The iterative algorithm used to recover the unknown source with energy boundary functions as the bases is developed, which is robust and convergent fast.

Authors:Jitsuro Sugie; Kazuki Ishibashi Pages: 146 - 154 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Jitsuro Sugie, Kazuki Ishibashi The purpose of this paper is to provide an oscillation theorem that can be applied to half-linear differential equations with time-varying coefficients. A parametric curve by the coefficients is focused in order to obtain our theorem. This parametric curve is a generalization of the curve given by the characteristic equation of the second-order linear differential equation with constant coefficients. The obtained theorem is proved by transforming the half-linear differential equation to a standard polar coordinates system and using phase plane analysis carefully.

Authors:Jian-Guo Liu; Yu Tian; Jian-Guo Hu Pages: 162 - 168 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Jian-Guo Liu, Yu Tian, Jian-Guo Hu The ( G ′ ∕ G ) -expansion approach is an efficient and well-developed approach to solve nonlinear partial differential equations. In this paper, the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation is investigated by using this approach, which describes the (2+1)-dimensional interaction of the Riemann wave propagated along the y -axis with a long wave propagated along the x -axis and can be considered as a model for the incompressible fluid. With the aid of symbolic computation, a family of exact solutions are obtained in forms of the hyperbolic functions and the trigonometric functions. When the parameters are selected special values, non-traveling wave solutions are also presented, and these gained solutions have abundant structures. The figures corresponding to these solutions are illustrated to show the particular localized excitations and the interactions between two solitary waves.

Authors:Won-Kwang Park Pages: 169 - 175 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Won-Kwang Park Various studies have confirmed the possibility of identifying the location of a set of small inhomogeneities via a direct sampling method; however, when their permeability differs from that of the background, their location cannot be satisfactorily identified. However, no theoretical explanation for this phenomenon has been verified. In this study, we demonstrate that the indicator function of the direct sampling method can be expressed by the Bessel function of order one of the first kind and explain why the exact locations of inhomogeneities cannot be identified. Numerical results with noisy data are exhibited to support our examination.

Authors:Yuanyuan Abstract: Publication date: July 2018 Source:Applied Mathematics Letters, Volume 81 Author(s): Yuanyuan Li In this paper, we investigate the quasilinear elliptic equations involving multiple Hardy terms with Dirichlet boundary conditions on bounded smooth domains Ω ⊂ R N N ≥ 3 , and prove the multiplicity of solutions by employing Ekeland’s variational principle.

Authors:Jen-Yuan Chen; David R. Kincaid; Bei-Ru Lin Pages: 1 - 5 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Jen-Yuan Chen, David R. Kincaid, Bei-Ru Lin We propose a new variant of Newton’s method based on Simpson’s three-eighth rule. It can be shown that the new method is cubically convergent.

Authors:Zhongzhou Lan; Bo Gao Pages: 6 - 12 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Zhongzhou Lan, Bo Gao Under investigation in this paper is a ( 2 + 1 ) -dimensional Heisenberg ferromagnetic spin chain equation with time-dependent coefficients. Based on the symbolic computation, Lax pair and infinitely many conservation laws are constructed. Multi-soliton solutions are derived by virtue of the Darboux transformation. Propagation and interaction properties of the solitons are discussed. Amplitude of the soliton is determined by the spectral parameter and wave number, while the velocity is related to both these parameters and the time-dependent coefficients. Elastic interactions between the two solitons are displayed, and their amplitudes keep unchanged after the interaction except for the phase shifts.

Authors:Meng Liu; Yu Zhu Pages: 13 - 19 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Meng Liu, Yu Zhu A budworm growth model perturbed by both white noises and regime switchings is proposed and analyzed. It is proven that there is a threshold. If this threshold is positive, then the model has a unique ergodic stationary distribution; if this threshold is negative, then the zero solution of the model is stable. The results show that both white noises and regime switchings can change the stability of the model greatly. Several numerical simulations based on realistic data are also introduced to illustrate the main results.

Authors:Wenjie Hu; Yinggao Zhou Pages: 20 - 26 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Wenjie Hu, Yinggao Zhou The present work is devoted to the stability and attractivity analysis of a nonlocal delayed reaction–diffusion equation (DRDE) with a non-monotone bistable nonlinearity that describes the population dynamics for a two-stage species with Allee effect. By the idea of relating the dynamics of the nonlinear term to the DRDE and some stability results for the monostable case, we describe some basin of attractions for the DRDE. Additionally, existence of heteroclinic orbits and periodic oscillations are also obtained. Numerical simulations are also given at last to verify our theoretical results.

Authors:Jianhua Chen; Xianhua Tang; Bitao Cheng Pages: 27 - 33 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Jianhua Chen, Xianhua Tang, Bitao Cheng In this paper, we study the following quasilinear Schrödinger equation − Δ u + u − Δ ( u 2 ) u = h ( u ) , x ∈ R N , where N ≥ 3 , 2 ∗ = 2 N N − 2 , h is a continuous function. By using a change of variable, we obtain the existence of ground state solutions. Unlike the condition lim u → ∞ ∫ 0 u h ( s ) d s u 4 = ∞ , we only need to assume that lim u → ∞ ∫ 0 u h ( s ) d s u 2 = ∞ .

Authors:Longjie Xie Pages: 34 - 42 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Longjie Xie By using the Krylov estimate, we prove the exponential ergodicity of the invariant measure for stochastic Langevin equation with singular coefficients, where the classical Lyapunov condition cannot be verified.

Authors:Zhiyong Wang; Jihui Zhang Pages: 43 - 50 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Zhiyong Wang, Jihui Zhang In this paper, we are concerned with the existence of periodic solutions for the following non-autonomous second order Hamiltonian systems u ̈ ( t ) + ∇ F ( t , u ( t ) ) = 0 , a.e. t ∈ [ 0 , T ] , u ( 0 ) − u ( T ) = u ̇ ( 0 ) − u ̇ ( T ) = 0 , where F : R × R N → R is T -periodic ( T > 0 ) in its first variable for all x ∈ R N . When potential function F ( t , x ) is either locally in t asymptotically quadratic or locally in t superquadratic, we show that the above mentioned problem possesses at least one T -periodic solutions via the minimax methods in critical point theory, specially, a new saddle point theorem which is introduced in Schechter (1998).

Authors:Djillali Bouagada; Samuel Melchior; Paul Van Dooren Pages: 51 - 57 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Djillali Bouagada, Samuel Melchior, Paul Van Dooren We present an efficient algorithm to compute the H ∞ norm of a fractional system. The algorithm is based on the computation of level sets of the maximum singular value of the transfer function, as a function of frequency. Numerical examples are given to illustrate the new method.

Authors:Liguang Xu; Wen Liu Pages: 58 - 66 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Liguang Xu, Wen Liu This paper concerns with the ultimate boundedness problem for impulsive fractional delay differential equations. Based on the impulsive fractional differential inequality, the boundedness of Mittag-Leffler functions, and the successful construction of suitable Lyapunov functionals, some algebraic criteria are derived for testing the global ultimate boundedness of the equations, and the estimations of the global attractive sets are provided as well. One example is also given to show the effectiveness of the obtained theoretical results.

Authors:Shi Sun; Ziping Huang; Cheng Wang Pages: 67 - 72 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Shi Sun, Ziping Huang, Cheng Wang The purpose of this paper is to study the weak Galerkin finite element method for a class of quasilinear elliptic problems. The weak Galerkin finite element scheme is proved to have a unique solution with the assumption that guarantees the corresponding operator to be strongly monotone and Lipschitz-continuous. An optimal error estimate in a mesh-dependent energy norm is established. Some numerical results are presented to confirm the theoretical analysis.

Authors:Guanggang Liu; Yong Li; Xue Yang Pages: 73 - 79 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Guanggang Liu, Yong Li, Xue Yang In this paper we consider a class of super-linear second order Hamiltonian systems. We use Morse theory to obtain the existence and multiplicity of rotating periodic solutions, which might be periodic, subharmonic or quasi-periodic ones.

Authors:A.M. Khludnev; V.V. Shcherbakov Pages: 80 - 84 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): A.M. Khludnev, V.V. Shcherbakov The note is concerned with a model of linear elastostatics for a two-dimensional inhomogeneous anisotropic body weakened by a single straight crack. On the crack faces, nonpenetration conditions/Signorini conditions are imposed. Relying upon a higher regularity result in Besov spaces for the displacement field in a neighborhood of the crack tip, we prove that the energy release rate is actually independent of the choice of a subsequent crack path (among the possible continuations of class H 3 ).

Authors:Moon-Jin Kang Pages: 85 - 91 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Moon-Jin Kang This article provides a rigorous justification on a hydrodynamic limit from the Vlasov–Poisson system with strong local alignment to the pressureless Euler–Poisson system for repulsive dynamics.

Authors:Lin Liu; Fawang Liu Pages: 92 - 99 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Lin Liu, Fawang Liu A novel investigation about the boundary layer flow of fractional Maxwell fluid over a stretching sheet with variable thickness is presented. By introducing new variables, the irregular boundary changes as a regular one. Solutions of the governing equations are obtained numerically where the L1-scheme is applied. Dynamic characteristicswith the effects of different parameters are shown by graphical illustrations. Three kinds of distributions versus power law parameter are presented, including monotonically increasing in nearly linear form at y =1, increasing at first and then decreasing at y =1.4 and monotonically decreasing in nearly linear form at y =2.

Authors:Deyue Zhang; Fenglin Sun; Linyan Lu; Yukun Guo Pages: 100 - 104 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Deyue Zhang, Fenglin Sun, Linyan Lu, Yukun Guo This paper concerns a boundary value problem of Laplace’s equation, which is solved by determining the unknown coefficients in the expansion of harmonic polynomials. A regularization method is proposed to tackle the resulting ill-posed linear system. The stability and convergence results are provided and a validating numerical experiment is presented.

Authors:Shuai Yang; Liping Wang; Shuqin Zhang Pages: 105 - 110 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Shuai Yang, Liping Wang, Shuqin Zhang The conformable derivative is used to develop the Swartzendruber model for description of non-Darcian flow in porous media. The proposed conformable Swartzendruber models are solved employing the Laplace transform method and validated on the basis of water flow in compacted fine-grained soils. The results of fitting analysis present a good agreement with experimental data. Furthermore, sensitivity analyses are carried out to illustrate the effects of related parameters on the conformable Swartzendruber models.

Authors:J. Deteix; D. Yakoubi Pages: 111 - 117 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): J. Deteix, D. Yakoubi The incremental projection scheme and its enhanced version, the rotational projection scheme are powerful and commonly used approaches producing efficient numerical algorithms for solving the Navier–Stokes equations. However, the much improved rotational projection scheme cannot be used on models with non-homogeneous viscosity, imposing the use of the less accurate incremental projection. This paper presents a projection method for the Navier–Stokes equations for fluids having variable viscosity, giving a consistent pressure and increased accuracy in pressure when compared to the incremental projection. The accuracy of the method will be illustrated using a manufactured solution.

Authors:Zhenya Yan Pages: 123 - 130 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Zhenya Yan We use two families of parameters { ( ϵ x j , ϵ t j ) ϵ x j , t j = ± 1 , j = 1 , 2 , … , n } to first introduce a unified novel hierarchy of two-family-parameter equations (simply called Q ϵ x n → , ϵ t n → ( n ) hierarchy), connecting integrable local, nonlocal, novel mixed-local–nonlocal, and other nonlocal vector nonlinear Schrödinger (VNLS) equations. The Q ϵ x n → , ϵ t n → ( n ) system with ( ϵ x j , ϵ t j ) = ( ± 1 , 1 ) , j = 1 , 2 , … , n is shown to possess Lax pairs and infinite number of conservation laws. Moreover, we also analyze the P T symmetry of the Hamiltonians with self-induced potentials. The multi-linear forms and some symmetry reductions are also studied. In fact, the used two families of parameters can also be extended to the general case { ( ϵ x j , ϵ t j ) ϵ x j = e i θ x j , ϵ t j = e i θ t j , θ x j , θ t j ∈ [ 0 , 2 π ) , j = 1 , 2 , … , n } to generate more types of nonlinear equations. The novel two-family-parameter (or multi-family-parameter for higher-dimensional cases) idea can also be applied to other local nonlinear evolution equations to find novel integrable and non-integrable nonlocal and mixed-local–nonlocal systems.

Authors:Xi-An Li; Wei-Hong Zhang; Yu-Jiang Wu Pages: 131 - 137 Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Xi-An Li, Wei-Hong Zhang, Yu-Jiang Wu For solving a class of complex symmetric linear system, we first transform the system into a block two-by-two real formulation and construct a symmetric block triangular splitting (SBTS) iteration method based on two splittings. Then, eigenvalues of iterative matrix are calculated, convergence conditions with relaxation parameter are derived, and two optimal parameters are obtained. Besides, we present the optimal convergence factor and test two numerical examples to confirm theoretical results and to verify the high performances of SBTS iteration method compared with two classical methods.

Abstract: Publication date: May 2018 Source:Applied Mathematics Letters, Volume 79 Author(s): Li Ma In this paper, we investigate a two-cooperative species model with reaction cross-diffusion under the Dirichlet boundary value condition. By applying Lyapunov–Schmidt reduction method and some important formulas, we obtain the existence of coexistence solutions under some given conditions.