Explicit High-Order Method to Solve Coupled Nonlinear Schrödinger Equations

Models of the coupled nonlinear Schrödinger equations submit various critical physical phenomena with a typical equation for optical fibres with linear refraction. In this article, we will presuppose the Compact Finite Difference method with Runge-Kutta of order 4 (explicit) method, which is sixth-order and fourth-order in space and time respectively, to solve coupled nonlinear Schrödinger equations. Many methods used to solve coupled nonlinear Schrödinger equations are second order in time and need to use extra-technique to rise up to fourth-order as Richardson Extrapolation technique. The scheme obtained is immediately fourth-order in one step. This approach is a conditionally stable method. The conserved quantities and the exact single soliton solution indicate the competence and accuracy of the article’s suggestion schemes. Furthermore, the article discusses the two solitons interaction dynamics.

Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives

In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.

ADJOINT OPERATOR METHOD AND NORMAL FORMS OF HIGHER ORDER FOR NONLINEAR DYNAMICAL SYSTEM

Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.

Two Implicit Runge-Kutta Methods for Stochastic Differential Equation

In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two fully implicit schemes are presented and their stability qualities are discussed. And the numerical report illustrates the better numerical behavior.

Eigenvalue Computation of Regular 4th Order Sturm-Liouville Problems

In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order shooting method based on Magnus expansions (MG4) which use MG4 shooting as the integrator. This method is similar to the SLEUTH (Sturm-Liouville Eigenvalues Using Theta Matrices) method of Greenberg and Marletta which uses the 2nd order Pruess method (also known as the MG2 shooting method) for the integrator. This method often achieves near machine precision accuracies, and some comparisons of its performance against the well-known SLEUTH software package are presented.

空间运动方程快速求解器设计与实现

文章基于四阶经典龙格库塔法(classical Runge-Kutta method of order four,RK-4)和四阶Adams预测校正法(fourth-order Adams predictor-corrector method,Adams-4),提出一种现场可编程逻辑门阵列(field programmable gate array,FPGA)实现的数据路径可动态配置的空间运动方程快速求解器(space motion equation fast solver,SMEFS)。SMEFS采用折叠式结构,借助高效的任务映射和精准的状态管理,通过资源复用和动态配置运算器内部连接关系实现数据路径的动态配置,快速求解空间运动方程,并有效节省硬件资源。采用某型运载火箭的相关数据对SMEFS进行大批量空间运动方程求解的性能评估,实验结果表明SMEFS能够快速可靠地求解发射坐标系下的五自由度空间运动方程,与软件求解的平均加速比为12.765,求解结果最大相对误差小于9×10^(-5,)具备较好的加速效果和较高的计算可靠性。

La、Ce、Yb掺杂对LiNi_(0.5)Mn_(1.5)O_(4)高电压正极材料性能的影响

采用低温燃烧法制备出不同稀土元素掺杂的高电压镍锰酸锂(LiNi_(0.5)Mn1.5O_(4))正极材料,探究了不同掺杂比例(物质的量分数0.5%、1%、2%)和不同掺杂稀土元素(La、Ce、Yb)对样品性能的影响,并通过X射线衍射、拉曼光谱、电子顺磁共振和恒电流间歇滴定等技术探究了其影响机理。从X射线衍射图可以看出,稀土掺杂可以抑制LixNi1-xO杂质相的产生;电感耦合等离子谱结果表明,掺杂进入的稀土元素与设计比例基本相符;从拉曼光谱图可以看出,稀土元素可以使样品的有序相增多,其中Ce掺杂样品的有序相最多;结合电子顺磁共振波谱氧空位测试,发现Ce掺杂诱导了样品中有序相比例增加,从而使样品的稳定性提高;经恒电流间歇滴定技术测试发现,Ce掺杂镍锰酸锂样品的扩散系数比未掺杂样品高了约15倍;在不同掺杂比例上,1%掺杂量时样品性能最佳。在3种最佳掺杂量的稀土元素样品中,Ce掺杂的样品性能最优,首次放电比容量可以达到133.3mAh·g^(-1),比未掺杂样品放电比容量高且首次效率提高了 18%,在1C下循环200次后,容量保持率为102%,比未掺杂样品提高了 8%。

Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

Oxidation of 4-chloro-3-methylphenol using zeolite Y-encapsulated iron(Ⅲ)-,nickel(Ⅱ)-,and copper(Ⅱ)-N,N'-disalicylidene-1,2-phenylenediamine complexes

The degradation of 4-chloro-3-methylphenol（PCMC）using zeolite-encapsulated iron（III）,nickel（II）,and copper（II）complexes of N,N’-disalicylidene-1,2-phenylenediamine as catalysts,in a heterogeneous Fenton-like advanced oxidation process,was studied.The physicochemical properties of the catalysts were determined using powder X-ray diffraction,thermogravimetric analysis,Brunauer–Emmett–Teller surface area analysis,Fourier-transform infrared spectroscopy,elemental analysis,and scanning electron microscopy.The effects of four factors,namely initial H2O2 concentration,catalyst dosage,temperature,and pH,on the degradation of a model organic pollutant were determined.The results show that at low acidic pH,almost complete removal of PCMC was achieved with the iron（III）,nickel（II）,and copper（II）catalysts after 120 min under the optimum reaction conditions：catalyst dosage 0.1 g,H2O2 concentration 75 mmol/L,initial PCMC concentration 0.35mmol/L,and 50 °C.The reusability of the prepared catalysts in PCMC degradation was also studied and a possible catalyst deactivation mechanism is proposed.The possible intermediate products,degradation pathway,and kinetics of PCMC oxidation were also studied.

Solution of Modified Equations of Emden-Type by Differential Transform Method

In this paper the Modified Equations of Emden type (MEE), χ+αχχ+βχ 3 is solved numerically by the differential transform method. This technique doesn’t require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. The current results of this paper are in excellent agreement with those provided by Chandrasekar et al. [1] and thereby illustrate the reliability and the performance of the differential transform method. We have also compared the results with the classical Runge-Kutta 4 (RK4) Method.

CuCo_(2)O_(4)纳米片在有序宏孔电极板上的制备

采用水热法在表面镀镍的有序宏孔电极板(OMEP)的表面和侧壁制备了金属氧化物——CuCo_(2)O_(4)纳米片,获得一种微型且高效的活性电极材料。通过扫描电子显微镜(SEM)观察到,在OMEP上生长的纳米结构由CuCo_(2)O_(4)纳米片组成。与在OMEP上生长的Co_(3)O_(4)纳米材料相比,制备的CuCo_(2)O_(4)/OMEP电极具有更好的电化学性能,在30 mA·cm^(-2)的电流密度下电极比电容为1334 F·g^(-1)(10 F·cm^(-2)),循环5000次后的循环稳定性为86%。这种独特的三维有序多孔结构使电子和离子的输运更加便利,增大了液固界面面积,提高了活性材料的利用效率,同时还减小了电极的质量和尺寸。

4N阶幻立方的构造

介绍了一种新的幻立方构造方法 ,并利用这种方法构造出某些 4阶和

ORDER RESULTS FOR ALGEBRAICALLY STABLEMONO-IMPLICIT RUNGE-KUTTA METHODS

It is well known that mono-implicit Runge-Kutta methods have been applied in the efficient numerical solution of initial or boundary value problems of ordinary differential equations. Burrage (1994) has shown that the order of an s-stage monoimplicit Runge-Kutta method is at most s+1 and the stage order is at most 3. In this paper, it is shown that the order of an s-stage mono-implicit Runge-Kutta method being algebraically stable is at most min((s) over tilde, 4), and the stage order together with the optimal B-convergence order is at most min(s, 2), where [GRAPHICS]

ORDER PROPERTIES AND CONSTRUCTION OF SYMPLECTIC RUNGE-KUTTA METHODS

The main results of this paper are as follows: (1) Suppose an s stage Runge-Kutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least 2p + l(1 less than or equal to p less than or equal to s - 1) provided that the simplifying conditions C(p) (or D(p) with non-zero weights) and B(2p + l) hold, where 1 0, 1, 2. (2) Suppose an s stage Runge-Kutta method is consistent, irreducible and non-confluent, and satisfies the simplifying conditions C(p) and D(p) with 0 < p <less than or equal to> s. Then this method is symplectic if and only if either p = s or the nonlinear stability matrix M of the method has an (s - p) x (s - p) chief submatrix (M) over cap = 0. (3) Using the results (1) and (2) as bases, we present a general approach for the construction of symplectic Runge-Kutta methods, and a software has been designed, by means of which, the coefficients of s stage symplectic Runge-Kutta methods satisfying C(p),D(p) and B(2p + l) can be easily computed, where 1 I P less than or equal to s, 0 less than or equal to l less than or equal to 2, s less than or equal to 2p + l less than or equal to 2s.

共轭算子法和非线性动力系统的高阶规范形

规范形理论是研究非线性动力系统退化分含的强有力的方法．在本文里我们利用共轭算子法计算了具有幂零线性部分和不具有Z2-对称性的非线性动力系统的2阶、3阶和4阶规范形，讨论了几种余维3退化分含情况下的普适开析问题及其一些全局特性．

用能量自洽法研究双原子分子离子XY^+的五阶解析势能函数

将用于研究双原子分子离子XY+解析势能函数的能量自洽法(energy consistent method for ion XY+,ECMI)从三阶推广到五阶,并将双原子分子离子XY+的三阶解析势能函数ECMI(3)推广为五阶的解析势能函数ECMI(5),研究CO+的B2Σ+态和SO+的b4Σ-态的解析势能函数.结果表明,CO+的B2Σ+态和SO+的b4Σ-态的新解析势能函数ECMI(5)势与基于实验的RKR数据符合得很好,优于常用的中性双原子分子势能函数Morse势、HMS势和双原子分子离子XY+的ECMI(3)势在这两个离子状态下的表现.不仅如此,ECMI势给出了对原子分子碰撞等诸多研究非常重要的正确理解极限和全程势能数据.

Runge-Kutta Discontinuous Galerkin Method Using WENO-Type Limiters:Three-Dimensional

This paper further considers weighted essentially non-oscillatory(WENO)and Hermite weighted essentially non-oscillatory(HWENO)finite volume methods as limiters for Runge-Kutta discontinuous Galerkin(RKDG)methods to solve problems involving nonlinear hyperbolic conservation laws.The application discussed here is the solution of 3-D problems on unstructured meshes.Our numerical tests again demonstrate this is a robust and high order limiting procedure,which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.

Stability of Stochastic Logistic Model with Ornstein-Uhlenbeck Process for Cell Growth of Microorganism in Fermentation Process

Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uhlenbeck process. First, this research proves that the stochastic logistic model with Ornstein-Uhlenbeck process has a positive solution. After that, it also introduces the sufficient conditions for stochastically stability of stochastic logistic model for cell growth of microorganism in fermentation process for positive equilibrium point by using Lyapunov function. In addition, this research establishes the sufficient conditions for zero solution as mentioned in Appendix A due to the cell growth of microorganism &mu;max, which cannot be negative in fermentation process. Furthermore, for numerical simulation, current research uses the 4-stage stochastic Runge-Kutta (SRK4) method to show the reality of the results.

Stability Analysis and Order Improvement for Time Domain Differential Quadrature Method

The differential quadrature method has been widely used in scientific and engineering computation.However,for the basic characteristics of time domain differential quadrature method,such as numerical stability and calculation accuracy or order,it is still lack of systematic analysis conclusions.In this paper,according to the principle of differential quadrature method,it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the important V-transformation feature.Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method,it has been proved that the differential quadrature method is A-stable and s-stage s-order method.On this basis,in order to further improve the accuracy of the time domain differential quadrature method,a class of improved differential quadrature method of s-stage 2s-order has been proposed by using undetermined coefficients method and Pad´e approximations.The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.

Numerical Simulation of Two-Dimensional Dendritic Growth Using Phase-Field Model

In this article, we study the phase-field model of solidification for numerical simulation of dendritic crystal growth that occurs during the casting of metals and alloys. Phase-field model of solidification describes the physics of dendritic growth in any material during the process of under cooling. The numerical procedure in this work is based on finite difference scheme for space and the 4th-order Runge-Kutta method for time discretization. The effect of each physical parameter on the shape and growth of dendritic crystal is studied and visualized in detail.