两类求解y“=f（x,y）具有极小相位延迟的显式方法

本文提出了两类数值积分二阶周期性初值问题ｙ〃＝ｆ（ｘ，ｙ），ｙ（ｘ０）＝ｙ０，ｙ（ｘ０）＝ｙ０具有检小相位延迟的显式两步法。这些方法推广和改进了文献「１」１－「７」中的某些方法。数值试验表明本文中的某些方法优于「１」－「７」中的某些方法。     

对《函数y=a~x和y=loga~x图象的交点》的补充

余炯沛先生在［１］中导出了定理若０＜ａ＜１,则函数ｙ＝ａｘ和ｙ＝ｌｏｇａｘ的图象,当ｅ－ｅ≤ａ＜１时有且仅有一个交点（此交点在直线ｙ＝ｘ上）;当０＜ａ＜ｅ－ｅ时有三个交点,其中一个在直线ｙ＝ｘ上,其余两个在直线ｙ＝ｘ外并且关于直线ｙ＝ｘ对称．文［１］...     

空间曲线在奇异点处的性态

本文讨论了空间曲线ｘ＝ｘ（ｔ），ｙ＝ｙ（ｔ），ｚ＝ｚ（ｔ）上奇异点的性态，结果表明：若［ｘ（ｋ）（ｔ０）］２＋［ｙ（ｋ）（ｔ０）］２＋［ｚ（ｋ）（ｔ０）］２＝０，ｋ＝１，２，…，ｎ－１，而［ｘ（ｎ）（ｔ０）］２＋［ｙ（ｎ）（ｔ０）］２＋［ｚ（ｎ）（ｔ０）］２≠０，则当ｎ为奇数时，曲线在点Ｍ０（ｘ０，ｙ０，ｚ０）是光滑的；当ｎ为偶数时，曲线在点Ｍ０（ｘ０，ｙ０，ｚ０）是不光滑的     

关于函数y=asec~tx-btg~tx的最值

关于函数ｙ＝ａｓｅｃｔｘ－ｂｔｇｔｘ的最值黄俊明（贵州省黔东南州民族林校５５６０００）关于函数ｙ＝ａｓｅｃｔｘ－ｂｔｇｔｘ（ａ，ｂ＞０，ｘ∈（０，π／２），ｔ为常数）的最值，文［１］用与［２］定理对偶的一个不等式，研讨了ｔ＝－ｎ（ｎ∈Ｎ），ｎ（ｎ≥３...     

Riccati微分方程的可积条件

Ｉｎ１９９８,ＺｈａｏＬｉｎｌｏｎｇ［１］ｏｂｔａｉｎｅｄｔｈｅｉｎｔｅｇｒａｂｌｅｃｏｎｄｉｔｉｏｎ：Ｒ＝１αγＰｅ２∫（Ｑ－βＤ）ｄｘ　　　（α,β,γｉｓｃｏｎｓｔ）．（１）ＦｏｒＲｉｃｃａｔｉｅｑｕａｔｉｏｎ：ｙ′＝ｐ（ｘ）ｙ２＋Ｑ（ｘ）ｙ＋Ｒ（ｘ）　　（ＰＲ≠０）．（２）　　Ｈｅｒｅｔｈｅｎｅｗｉｎｔｅｇｒａｂｌｅｃｏｎｄｉｔｉｏｎｓｉｓｇｉｖｅｎ：Ｌ［ｙ０］＝１αγＰｅ２∫（Ｑ＋２ｙ０ｐ－βＤ）ｄｘ．（３）Ｌ［ＡＢ＋ｙ０］＝１αγ（ＡＢ）２Ｌ［ｙ０］ｅ２∫（２ＢＡＬ［ｙ０］＋Ｑ＋２ｙ…     

ON THE NON-EXISTENCE OF LIMIT CYCLES OF CERTAIN QUADRATIC SYSTEMS

§１．Ｆｏｒｔｈｅｓｙｓｔｅｍｘ＝－ｙ＋δｘ＋ｌｘ２＋ｎｙ２＝Ｐ（ｘ，ｙ），ｙ＝ｘ（１＋ａｘ－ｙ）＝Ｑ（ｘ，ｙ），｛（１．１）ｗｅｃａｎｆｉｎｄｉｎ［１］ｔｈｅｆｏｌｏｗｉｎｇ：ＣｏｎｊｅｃｔｕｒｅＩ．Ａｓｓｕｍｅ１ａ＜０，ｎ＞１，ｎ＋ｌ＞０，ｎａ２...     

再谈抛物线的阿基米德三角形的性质

过圆锥曲线弦的两端的切线与弦围成的三角形称为阿基米德三角形．弦叫做这三角形的底边,其他两边叫做这三角形的腰,两腰的公共端点叫做这三角形的顶点．文［１］,［２］给出了阿基米德三角形的三条性质,本文提供另外一些性质．引理［３］　自抛物线ｙ２＝２ｐｘ（ｐ＞０）外一点Ｔ（ｘ０,ｙ０）引两切线,切点弦所在直线的方程为ｙ０ｙ－ｐ（ｘ＋ｘ０）＝０．性质１　设抛物线ｆ（ｘ,ｙ）＝ｙ２－２ｐｘ＝０（ｐ＞０）的阿基米德三角形的顶点为Ｔ（ｘ０,ｙ０）（ｘ０≠０）,底边为Ｐ１Ｐ２,两腰为ＴＰ１,ＴＰ２,∠Ｐ１ＴＰ２＝α…     

Some Properties of a Class Self—homomorphism of BCI—algebras

ＢｙａＢＣＩ－ａｌｇｅｂｒａｗｅｍｅａｎａｎａｌｇｅｂｒａ（Ｘ；，０）ｏｆｔｙｐｅ（２，０）ｓａｔｉｓｆｙｉｎｇｔｈｅａｘｉｏｍｓ：（１）（（ｘｙ）（ｘｚ））（ｚｙ）＝０；（２）（ｘ（ｘｙ））ｙ＝０；（３）ｘｘ＝０；（４）ｘｙ＝ｙｘ＝０ｘ＝ｙｆｏｒａｎｙｘ，ｙａｎｄｚｉｎＸ．ＦｏｒａｎｙＢＣＩ－ａｌｇｅｂｒａＸ，ｔｈｅｒｅｌａｔｉｏｎ≤ｄｅｆｉｎｅｄｂｙｘ≤ｙｉｆａｎｄｏｎｌｙｉｆｘｙ＝０ｉｓａｐａｒｔｉａｌｏｒｄｅｒｏｎＸ［１］．ＩｎａｎｙＢＣＩ－ａｌｇｅｂｒａＸ，…     

STUDY ON THE UNIQUENESS OF LIMIT CYCLE OF THE QUADRATIC SYSTEM BY USING THE QUADRATIC CURVE WITHOUT CONTACT(CONTINUED 1)

１ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＲｅｓｕｌｔｓＣｏｎｓｉｄｅｒｔｈｅｑｕａｄｒａｔｉｃｓｙｓｔｅｍ［１］ｄｘｄｔ＝－ｙ＋δｘ＋ｌｘ２＋ｍｘｙ＋ｎｙ２＝Ｐ２（ｘ，ｙ），ｄｙｄｔ＝ｘ（１＋ａｘ＋ｂｙ）＝Ｑ２（ｘ，ｙ），（ｎ≥０，ａｂ≠０）Ｅ２ｗｉｔｈｏｕ...     

一类二阶非线性微分方程的求积问题

讨论了二阶非线性微分方程ｙ″＋ｐ（ｘ）ｙ′＋Ｗ＇（ｙ）／Ｗ（ｙ）ｙ＇＾２＝Ｑ（ｘ）１／Ｗ（ｙ）Ｆ［ｘ，（ｙ＇Ｗ（ｙ）＾ａ］的可积性问题，提供了可积的一些充分条件，在Ｆ为某些特殊类型时，给出能解公式。     

EXPLICIT TWO-STEP HIGH-ACCURACY METHODS FOR y" = f(x,y)

In this paper, an almost P-stable two-step sixth-order Hybrid method with phase-lag of order infinity and a class explicit eighth-order Obreckoff methods with phase-lag of order 10-24 are developed for the numerical integration of the special second-order periodic initial-value problems. These methods have the advantage of higher algebraic order and considerably smaller phase-tag compared with some methods in [1-6]. Numerical examples indicate that these new methods are more accurate than methods developed by [1-6].     

HIGH-ACCURACY EXPLICIT TWO-STEP METHODS WITH MINIMAL PHASE-LAG FOR y″= f（t,y）

In this paper, two families of high accuracy explicit two-step methods with minimal phase-lag are developed for the numerical integration of special second-order periodic initial-value problems. In comparison with some methods in [1-4,6], the advantage of these methods has a higher accuracy and minimal phase-lag. The methods proposed in this paper can be considered as a generalization of some methods in [1,3,4]. Numerical examples indicate that these new methods are generally more accurate than the methods used in [3,6]. second order periodic initial-value problems, phase-lag, local truncation error     

High-Accuracy P-Stable Methods with Minimal Phase-Lag for $y "= f(t,y) ^*$

In this paper, we develop a one-parameter family of P-stable sixth-order and eighth-order two-step methods with minimal phase-lag errors for numerical integration of second order periodic initial value problems: $$ y''=f(t,y), \quad y(t_0)=y_0, \quad y'(t_0)=y'_0. $$ We determine the parameters so that the phase-lag (frequency distortion) of these methods are minimal. The resulting methods are P-stable methods with minimal phase-lag errors. The superiority of our present P-stable methods over the P-stable methods in [1-4] is given by comparative studying of the phase-lag errors and illustrated with numerical examples.     

EFFICIENT SIXTH ORDER P-STABLE METHODS WITH MINIMAL LOCAL TRUNCATION ERROR FOR y"=f(x,y)

1. IntroductionWe consider a class of direct hybrid methods proposed in [11 for solving the second orderinitial value problemy" = f(t,y), y(0),y'(0) given (1.1)The basic method has the formandHere t. = nh and we define t.l.. = t. I aih, i = 1, 2 and n=0,1…     

A CLASS OF EXPLICIT RATIONAL SYMPLECTIC INTEGRATORS

In this paper, a class of rational explicit symplectic integrators for one-dimensional oscillatory Hamiltonian problems are presented. These methods are zero-dissipative, and of first algebraic order and high phase-lag order. By means of composition technique, we construct second and fourth order methods with high phase-lag order of this type. Based on our ideas, three applicable explicit symplectic schemes with algebraic order one, two and four are derived, respectively. We report some numerical results to illustrate the good performance of our methods.     

A robust trigonometrically fitted embedded pair for perturbed oscillators

A new kind of trigonometrically fitted embedded pair of explicit ARKN methods for the numerical integration of perturbed oscillators is presented in this paper. This new pair is based on the trigonometrically fitted ARKN method of order five derived by Yang and Wu in [H.L. Yang, X.Y. Wu, Trigonometrically-fitted ARKN methods for perturbed oscillators, Appl. Numer. Math. 9 (2008) 1375–1395]. We analyze the stability properties, phase-lag (dispersion) and dissipation of the higher-order method of the new pair. Numerical experiments carried out show that our new embedded pair is very competitive in comparison with the embedded pairs proposed in the scientific literature.     

Explicit One-Step P-Stable Methods for Second Order Periodic Initial Value Problems

In this paper, we present an explicit one-step method for solving periodic initial value problems of second order ordinary differential equations. The method is P-stable, and of first algebraic order and high phase-lag order. To improve the algebraic order, we give a composition second order scheme with the proposed method and its adjoint. We report some numerical results to illustrate the efficiency of our methods.     

解y"=g(x,y)初值问题含参数线性多步方法的相容阶和收敛阶

1 引 言对于直接积分二阶常微分方程的初值问题 y"=g(x,y) y (x_0)=y_0,y'(x_0)=y"_0,x_0 x T,(1)     

解初值问题y″=g(x，y)的含参数块方法

1引言对于二阶常微分方程的初值问题y″=g(x,y),y(x_0)=y_0,y′(x_0)=y_0′,x_0(?)x(?)T(1)的数值解法的研究引起人们的广泛兴趣．对于直接积分(1),自从1976年J．D．Lambert和I．A．Waston提出二阶P-稳定方法和1978年G．Dahlquist证明P-稳定常系数线性多步方法的最高相容阶不超过2的重要结论以来,截止目前,已积累了许多高于2阶的P-稳定方法．例如,修正的Numerov方法,混合法(特殊形式RK的方法),多导法,Obrechkoff方法,显式RKN方法,单隐方法和对角隐式RKN方法等(顺便指出,文献[5,16]中所说的高阶方法的相容阶均不超过4)．所有这些方法,有些相     

解初值问题y"=g(x,y)的含参数块方法

1引言对于二阶常微分方程的初值问题y″=g(x,y),y(x_0)=y_0,y′(x_0)=y_0′,x_0(?)x(?)T(1)的数值解法的研究引起人们的广泛兴趣．对于直接积分(1),自从1976年J．D．Lambert和I．A．Waston提出二阶P-稳定方法和1978年G．Dahlquist证明P-稳定常系数线性多步方法的最高相容阶不超过2的重要结论以来,截止目前,已积累了许多高于2阶的P-稳定方法．例如,修正的Numerov方法,混合法(特殊形式RK的方法),多导法,Obrechkoff方法,显式RKN方法,单隐方法和对角隐式RKN方法等(顺便指出,文献[5,16]中所说的高阶方法的相容阶均不超过4)．所有这些方法,有些相