求解初值问题的RKNd方法

本文基于一阶常微分方程所导出的二阶微分方程提出RKNd方法,其内级阶比传统RK方法高一阶.RKNd方法的阶条件由特殊Nystr(o)m树给出.在相同级数下,RKNd方法可达到的最高代数阶比传统的RK方法高.数值实验结果表明RKNd方法比同阶RK方法在计算效率上具有一定的优越性.     

求解初值问题的指数拟合RKNd方法(英文)

本文给出了求解初值问题的指数拟合的RKNd方法.此类方法对于解可以表示为函数{exp(λt),exp(-λt)},λ∈C,或{sin(ωt),cos(ωt)},其中λ=iω,ω∈R之线性组合的微分方程精确成立.数值试验表明,指数拟合RKNd方法的计算效率要优于指数拟合RK方法和RKNd方法.     

一阶贝塞尔函数广义积分的数值计算

物理勘探中,需要计算含一阶贝塞尔函数的广义积分.一种传统的方法是在贝塞尔函数零点之间一次应用一般积分法则积分,最后求和,这种方法收敛比较慢,特别在贝塞尔函数中r值很大的时候.另一种应用广泛的方法是数字滤波技术,该法比第一种方法快,但要求核函数迅速衰减.本文给出了一种新的计算方法,能处理核函数衰减很慢且r很大的问题,方法简单,高效率,精度高.     

第二类端点奇异Fredholm积分方程的分数阶退化核方法

本文针对第二类端点奇异Fredholm积分方程构造基于分数阶Taylor展开的退化核方法，设计了两种计算格式，一是在全区间上使用分数阶Taylor展开式近似核函数，二是在包含奇点的小区间上采用分数阶插值，在剩余区间上采用分段二次多项式插值逼近核函数.讨论了两种退化核方法收敛的条件，并给出了混合插值法的收敛阶估计.数值算例表明对于非光滑核函数分数阶退化核方法有着良好的计算效果，且混合二次插值法比全区间上的分数阶退化核方法有着更广泛的适用范围.     

用摄动配置方法求解含时薛定谔方程

将摄动配置方法应用到含时薛定谔方程,在计算实现的基础上结合摄动配置的特征提出了一类新的数值积分方法,并给出了一个2级2阶和一个3级4阶的辛摄动配置方法对含时薛定谔方程的数值算例.为了检验新的数值积分方法,我们还给出了与两个辛摄动配置格式在理论上等价的辛龙格-库塔方法以及同阶的非辛方法的数值模拟.展示了一些数值结果,并给出了一些分析.     

计算Riemann-Liouville分数阶积分和导数的数值算法

研究计算Riemann-Liouville (RL)分数阶积分和导数的数值算法.首先,分析了RL分数阶积分和导数的定义式,由于定义式中包含一个积分瑕点,使RL分数阶积分和导数难于计算.然后,给出了一种去掉积分瑕点的方法,在此基础上设计出计算RL分数阶积分和导数的数值算法,并证明了此数值算法具有一阶精度.最后,给出了计算实例,计算结果说明提出的算法是有效的.     

求解对流扩散方程的一致四阶紧致格式

目前,许多高精度差分格式,由于未成功地构造与其精度匹配的稳定的边界格式,不得不采用低精度的边界格式.本文针对对流扩散方程证明了存在一致四阶紧致格式,它的边界点的计算格式和内点的计算格式的截断误差主项保持一致,给出了具体内点和边界格式;并分析了此半离散格式的渐近稳定性.数值结果表明该格式是四阶精度;在对流占优情况下,本文边界格式的数值结果比四阶精度的显式差分格式的的数值结果的数值振荡小,取得了不错的效果,理论结果得到了数值验证;驱动方腔数值结果显示,本文对N-S方程的离散格式具有很好的可靠性,适合对复杂流体流动的数值模拟和研究.     

时间分数阶扩散方程的数值解法

分数阶微分方程在许多应用科学上比整数阶微分方程更能准确地模拟自然现象.考虑时间分数阶扩散方程,将一阶的时间导数用分数阶导数α(0<α<1)替换,给出了一种计算有效的隐式差分格式,并证明了这个隐式差分格式是无条件稳定和无条件收敛的,最后用数值例子说明差分格式是有效的.     

三类数列型行列式求值

对于有关教材中的3个行列式,根据其元素的特点,将它们从4阶推广到任意n阶,并给出计算方法;给出并证明一种借助多项式进行行列式计算的方法.     

关于某些循环矩阵的特征值

本文给出了某些循环矩阵特征值的具体形状和降阶计算方法.     

Two-step peer methods with continuous output

The essential feature of peer methods for initial value problems is that all stages have the same order. Hence, dense output is easily available by interpolation. However, continuity of the output requires that the methods possess the FSAL-property (first same as last) well known from one-step methods. An additional advantage of such methods is an increase of the number of stages and the order without increasing the computational effort of the scheme or the number of computing cores for parallel methods. We derive several FSAL peer methods with 3 to 9 stages and compare their numerical efficiency with existing methods in the case of explicit parallel and implicit sequential peer methods.     

Zero Dissipative, Explicit Numerov-Type Methods for Second Order IVPs with Oscillating Solutions

New explicit, zero dissipative, hybrid Numerov type methods are presented in this paper. We derive these methods using an alternative which avoids the use of costly high accuracy interpolatory nodes. We only need the Taylor expansion at some internal points then. The method is of sixth algebraic order at a cost of seven stages per step while their phase lag order is fourteen. The zero dissipation condition is satisfied, so the methods possess an non empty interval of periodicity. Numerical results over some well known problems in physics and mechanics indicate the superiority of the new method.     

Numerical treatment of a finite strain viscoplasticity model based on a double multiplicative split

A viscoplastic material model of overstress type is analyzed. The model is based on the kinematic assumption of a double multiplicative split in order to simulate a nonlinear kinematic hardening. We investigate theoretically and numerically two implicit time–stepping methods, which were used lately in engineering literature to integrate the evolution equations in the context of multiplicative plasticity/viscoplasticity. A common feature of both methods is that the plastic incompressibility constraint is exactly satisfied. Moreover, we show that both methods preserve the symmetry of internal variables, which is inherent to the problem. According to testing results, both methods are equivalent regarding robustness, accuracy, and computational efficiency. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

任意形状凸起地形对平面SH波的散射

将具有任意形状的凸起地形对称态SH波散射问题转换为契合问题加以研究,利用求解弹性波动问题的复变函数与保角映射方法,在包括任意形状凸起边界在内的一个区域中,构造一个在凸起边界上应力自由,其他部分位移和应力均为任意的驻波解,然后再将这个驻波解与其余下的区域中的散射波解在两个区域结合面上完成契合过程,由此决定出这两个区域中的驻波和散射波解答,最后对圆弧形和半椭圆形凸起进行了数值计算,并将计算结果与有限元法的数值解进行了比较。     

等式与界约束非线性优化的信赖域增广Lagrangian算法

１．引 言本文讨论如下非线性约束优化问题：其中； 是Ｒｎ→Ｒ的可微函数，　　　　　　．记 问题（１．１）是非线性约束优化问题中的一类重要类型，事实上任一个非线性等式与不等式约束优化均可引入松驰变量转化为（１．１）的形式．因此（１．１）的求解是人们讨论的热点问     

迟滞型材料阻尼转轴的分岔

应用平均法研究迟滞型材料阻尼转轴的分岔.首先用Hamilton原理推导出复数形式的转轴运动微分方程,然后用平均法求出各阶模态主共振时的平均方程,并分析定常解的稳定性,最后用奇异性理论分析正常运动和失稳运动响应(异步涡动)的分岔.研究表明,一定参数条件下,转轴在通过各阶临界转速(主共振)时,可能会因受到冲击而失稳(Hopf分岔).正常运动响应在不平衡量较大时有滞后和跳跃现象,而失稳运动响应是一类余维数较高的非对称分岔.由于内阻尼的非线性,响应随转速增加时还可能产生二次Hopf分岔,对应原系统的双调幅运动.做好动平衡及提高外阻尼水平是避免这种大幅值自激振动的有效措施.     

A curve-fitting genetic algorithm for a styling application

Fitting curves in computer-aided geometric design is generally regarded as an optimisation problem. Depending on the application, the conditions to be satisfied can make the problem difficult to solve using classic methods, and for this reason, stochastic methods, such as genetic algorithms appear to be appropriate. This article considers a curve fitting problem, with the objective of generating shapes with specific curvature variations for use in the design of car bodies. To this end, a particular curve model was developed and implemented within a genetic algorithm. The main characteristics of this algorithm are described and its promising results are presented. The conclusion will show that this technique can be used as an alternative method in the design of car bodies.     

固体的离散型变分原理——有限元离散分析的变分原理

本文提出一类新的固体的离散型变分原理.它是从有限元离散分析的实际出发,考虑到元素的边界为可动边界,并且由于分片构造待解函数,使待解函数在元素的交界处具有各种间断性.由此,我们利用数学中的具有各种间断性的可动边界的变分方法,基于一阶变分为零的驻值条件上,建立了固体的离散型变分原理.离散型变分原理消除了元素交界处所导入的误差.它概括了古典与非古典变分原理.本文得到的待解函数应满足的交界方程,是有限元的收敛性(包括非保形元素在内)的必要条件,它开拓了待解函数应满足协调性的收敛性要求.     

The conversion of Fredholm integral equations to equivalent Cauchy problems— II. computation of resolvents

A new method is given for computing the resolvent of a large class of Fredholm integral equations. The technique is based on converting the integral equation satisfied by the resolvent to a family of two point boundary value problems. The application of invariant imbedding then gives an equivalent Cauchy problem satisfied by the resolvent kernel. The procedure is compared to previous ones based on the Bellman—Krein equation. It is shown that our method requires fewer equations to integrate if the number of output points on each axis exeeds the bank of the kernel.     

Introducing internal model to robust output synchronization of FitzHugh–Nagumo neurons in external electrical stimulation

A novel internal model control method is proposed for the robust output synchronization of FizHugh–Nagumo (FHN) neurons under external electrical stimulation. The output synchronization problem can be converted into a robust stabilization problem of an augmented system consisting of the original given plant and an internal model. Hence, the design procedures include the design of a satisfied internal model and a proper state-feedback stabilizer for this augmented system. The stability analysis of the resulting closed-loop system leads to semiglobal Lyapunov asymptotic stability of the robust output synchronization achieved for some appointed initial condition in the state space and for all possible values of the uncertain parameter vector. Finally, the simulation results demonstrate the validity of the proposed method.