具有磁场效应的非线性Schrdinger方程组的拟谱方法

考察一类具有磁场效应的非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程组的周期初值问题，构造了全离散的Ｆｏｕｒｉｅｒ拟谱格式，利用有界延拓法，证明了其格式的收敛性与稳定性，并给出了误差估计及其算法分析，为对此模型的数值分析提供了理论基础和一个有效的算法．最后，通过数值例子，检验了理论结果的可信性．     

离散的Burgers-Ginzburg-Landau方程组的吸引子

文章通过对空间变量的有限差分方法离散了具有周期边值的Ｂｕｒｇｅｒｓ Ｇｉｎｚｂｕｒｇ Ｌａｎｄａｕ方程组．研究了这个离散方程组初值问题解的适定性．证明了当差分网格足够大时离散方程组存在吸引子，并得到了吸引子的Ｈａｕｓｄｏｒｆｆ维数和分形维数的上界估计．这个上界不会随着网格的加细而无限增大，因此数值分析离散的有限维系统的吸引子可以近似探讨原无限维系统的吸引子．     

一类非线性代数方程组的并行迭代算法

１．引言考虑非线性代数方程组这里,　　　　　　　　　　　　　　　　　　　　　　　　　　　　为连续的对角映射,二者的导函数均存在,但并不一定连续．这类非线性代数方程组具有丰富的实际背景．譬如,Ｓｔｅｆａｎ问题和许多弱非线性椭圆型偏微分方程,就可归结为（１．１）的数值求解问题．根据方程组（１．１）的特殊结构,并利用矩阵多重分裂思想,文ｔＺ］讨论了一类并行非线性Ｇａｕｓｓ－Ｓｅｉｄｅｌ型迭代算法．这类算法具有很好的数值性质和较高的并行效率·在此基础上,运用松弛加速技术,文［８］进一步研究了一类并行多分…     

关于PDAOR算法的广义STEIN—ROSENBERG型定理

1．引言众所周知，许多微分方程（组）经过有限差分或有限元离散，均可归结为大型分块线性代数方程组的数值求解问题，这里n。（5。5N）为给定的N个正整数，满足Zn。＝n．为利用多处理机系统有效而准t’z＝1确地得到JI．n的近似解．诵过合理地分解系统〔1．1），并有机地运用加速超松弛技术，【11提出了一类新的求解大型分块线性代数方程组（1．1）的并行分解型加速超松弛迭代算法，即PDAOR－一算法．这类算法具有很强的并行功能和良好的数值性质．大量数值实验表明，较之经典的AOR算法，PDAOR－一算法具有更快的收敛速度，更大的收…     

具有磁场效应的非线性Schroedinger方程组的拟谱方法

考察一类具有磁场效应的非线性Ｓｃｈｒｏｅｄｉｎｇｅｒ方程组的周期初值问题，构造了全离散地Ｆｏｕｒｉｅｒ拟谱格式，利用有界延拓法，证明了其格式的收敛性与稳定性，并给出了误差估计及其算法分析，为对此模型的数值分析提供了理论基础和一个有效的算法，最后，通过数值例子，检验了理论结果的可信性。     

环形空腔内自然对流问题的的Galerkin方法

本文讨论了环形空腔内自然对流问题所满足的Ｂｏｕｓｓｉｎｅｓｑ方程组－关于涡度ζ、流函数ψ及温度θ的椭圆－抛物非线性耦合方程组用Ｇａｌｅｒｋｉｎ方法对其进行了数值分析，得到了Ｇａｌｅｒｋｉｎ逼近（含半离散和全离散）的最优先验误差估计。     

Cahn-Hilliard方程的拟谱逼近

该文讨论用Ｌｅｇｅｎｄｒｅ拟谱方法数值求解非线性Ｃａｈｎ Ｈｉｌｌｉａｒｄ方程的Ｄｉｒｉｃｈｌｅｔ问题．建立了其半离散和全离散逼近格式，它们保持原问题能量耗散的性质．证明了离散解的存在唯一性，并给出了最佳误差估计．数值实验也证实了我们的结果．     

环形空腔内自然对流问题的Galerkin方法

本文讨论了环形空腔内自然对流问题所满足的Boussinesq方程组——关于涡度ζ、流函数φ及温度θ的椭圆-抛物非线性耦合方程组 用Galerkin方法对其进行了数值分析,得到了Galerkin逼近(含半离散和全离散)的最优先验误差估计。     

一类新的求解双边障碍问题的数值方法

针对双边障碍问题的离散互补形式,提出了一类新的格式将其等价转化为方程组的形式,并采用牛顿迭代法进行求解.实验结果显示所提算法能快速,有效地计算出数值解和接触集.     

带常系数的Cauchy型奇异积分方程的快速方法

Petrov-Galerkin 方法是研究Cauchy型奇异积分方程的最基本的数值方法. 用此方法离散积分方程可得一系数矩阵是稠密的线性方程组. 如果方程组的阶比较大, 则求解此方程组所需要的计算复杂度则会变得很大. 因此, 发展此类方程的快速数值算法就变成了必然. 该文将就对带常系数的Cauchy型奇异积分方程给出一种快速数值方法. 首先用一稀疏矩阵来代替稠密系数矩阵, 其次用数值积分公式离散上述方程组得到其完全离散的形式,然后用多层扩充方法求解此完全离散的线性方程组. 证明经过上述过程得到方程组的逼进解仍然保持了最优阶, 并且整个过程所需要的计算复杂度是拟线性的. 最后通过数值实验证明结论.     

A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions

In this paper, we propose a new numerical algorithm for solving linear and non linear fractional differential equations based on our newly constructed integer order and fractional order generalized hat functions operational matrices of integration. The linear and nonlinear fractional order differential equations are transformed into a system of algebraic equations by these matrices and these algebraic equations are solved through known computational methods. Further some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm. The results obtained, using the scheme presented here, are in full agreement with the analytical solutions and numerical results presented elsewhere.     

Chebyshev polynomial approximation for high‐order partial differential equations with complicated conditions

In this article, a new method is presented for the solution of high‐order linear partial differential equations (PDEs) with variable coefficients under the most general conditions. The method is based on the approximation by the truncated double Chebyshev series. PDE and conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Chebyshev coefficients, via Chebyshev collocation points. Combining these matrix equations and then solving the system yields the Chebyshev coefficients of the solution function. Some numerical results are included to demonstrate the validity and applicability of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On macromodeling of nonlinear systems with time periodic coefficients

Some new techniques for reduced order (macro) modeling of nonlinear systems with time periodic coefficients are discussed in this paper. The dynamical evolution equations are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the new set of equations become time-invariant. The techniques presented here reduce the order of this transformed system and all original states are obtained via the appropriate transformations. This macromodel preserves the desired stability and bifurcation characteristics of the original large-scale system and due to relatively few states; it is suitable for simulation and controller design.In this work, methodologies based on linear and nonlinear projections as well as ‘time periodic invariant manifold’ idea are presented. The invariant manifold technique yields a ‘reducibility condition’ that determines when an accurate nonlinear order reduction is possible. A comparative study of these order reduction methods is also included. These techniques are compared by means of time traces and Poincaré maps. A numerical error analysis is also included and advantages and limitations are discussed by means of a practical example.     

A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains

The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included.     

Relaxation approximation to bed-load sediment transport

In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit–explicit Runge–Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems.     

A New Approach for Solving a Class of Delay Fractional Partial Differential Equations

In this article, a new numerical approach has been proposed for solving a class of delay time-fractional partial differential equations. The approximate solutions of these equations are considered as linear combinations of Müntz–Legendre polynomials with unknown coefficients. Operational matrix of fractional differentiation is provided to accelerate computations of the proposed method. Using Padé approximation and two-sided Laplace transformations, the mentioned delay fractional partial differential equations will be transformed to a sequence of fractional partial differential equations without delay. The localization process is based on the space-time collocation in some appropriate points to reduce the fractional partial differential equations into the associated system of algebraic equations which can be solved by some robust iterative solvers. Some numerical examples are also given to confirm the accuracy of the presented numerical scheme. Our results approved decisive preference of the Müntz–Legendre polynomials with respect to the Legendre polynomials.     

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

In the current study, an approximate scheme is established for solving the fractional partial differential equations (FPDEs) with Volterra integral terms via two‐dimensional block‐pulse functions (2D‐BPFs). According to the definitions and properties of 2D‐BPFs, the original problem is transformed into a system of linear algebra equations. By dispersing the unknown variables for these algebraic equations, the numerical solutions can be obtained. Besides, the proof of the convergence of this system is given. Finally, several numerical experiments are presented to test the feasibility and effectiveness of the proposed method.     

一类三维拟线性双曲型方程交替方向有限元法

对一类一般的三维拟线性双曲型方程通过转化二阶时间导数得到关于一阶时间导数的耦合方程组,然后进行离散得到交替方向有限元格式,应用微分方程先验估计的理论和技巧得到了最优阶H~1-模和L~2-模误差估计,并给出了数值算例,数值结果和理论分析得到很好的吻合.     

Galerkin alternating-direction method for a kind of second-order quasi-linear hyperbolic problems

A kind of second-order quasi-linear hyperbolic equation is firstly transformed into a first-order system of equations, then the Galerkin alternating-direction procedure for the system is derived. The optimal order estimates in H¹ norm and L² norm of the procedure are obtained respectively by using the theory and techniques of priori estimate of differential equations. The numerical experiment is also given to support the theoretical analysis. Comparing the results of numerical example with the theoretical analysis, they are uniform.     

Numerical solutions of two-dimensional nonlinear integral analysis

In this paper, the approximate solutions for two different type of two-dimensional nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.