新题征展(3)

题组新编１．（１）设Ｍ＝｛ｘ｜ｆ（ｘ）＝０｝、Ｎ＝｛ｘ｜ｇ（ｘ）＝０｝,则｛ｘ｜ｆ（ｘ）·ｇ（ｘ）＝０｝为（　　）;（Ａ）Ｍ　（Ｂ）Ｎ　（Ｃ）Ｍ∪Ｎ　（Ｄ）以上都不对（２）设ｆ（ｘ）＝ｘ－１ｘ＋３,ｇ（ｘ）＝ｘ＋３ｘ－１,则集合｛ｘ｜ｆ（ｘ）·ｇ（ｘ）＝０｝＝　　;（３）设函数ｆ（ｘ）、ｇ（ｘ）的定义域依次是Ｆ、Ｇ,且Ｍ＝｛ｘ｜ｆ（ｘ）＝０｝、Ｎ＝｛ｘ｜ｇ（ｘ）＝０｝,则｛ｘ｜ｆ（ｘ）·ｇ（ｘ）＝０｝＝　　．２．（１）设ｍ、ｋ∈Ｎ,则Ｃｎｎ＋Ｃｎｎ＋１＋Ｃｎｎ＋２＋…＋Ｃｎｎ＋ｋ＝　　;（２）求…     

方程f_1(x)·f_2(x)=0的解集纠错

下列四种中学数学教学的权威文献都涉及到方程ｆ１（ｘ）·ｆ２（ｘ）＝０的解集问题．（１）高级中学课本《代数》上册（必修）在引入并集的概念和解释其应用时写道：方程（ｘ２－４）·（ｘ２－１）＝０的解集，可以从求方程ｘ２－４＝０的解集与方程ｘ２－１＝０的解集...     

复合函数求导法则的又一证明

复合函数的求导法则是求导运算的重要法则;对于ｙ＝ｆ（ｕ）,ｕ＝ｇ（ｘ）,复合函数ｙ＝ｆ〔ｇ（ｘ）〕的求导法则的证明有一个很自然的想法：ΔｙΔｘ＝ΔｙΔｕ·ΔｕΔｘ,ｌｉｍΔｘ→０ΔｙΔｘ＝ｌｉｍΔｕ→０ΔｙΔｕ·ｌｉｍΔｘ→０ΔｕΔｘ;但是,当Δｘ→０时,Δｕ可能等于０,此时ΔｙΔｕ没有意义,所以上面很直接的想法行不通;一般的证明采取另外的方法［１］,［２］;本文仍从上面直观的想法出发,加以改进,得到了又一个证明;定理　若ｙ＝ｆ（ｕ）在ｕ可导,函数ｕ＝ｇ（ｘ）在ｘ可导,则复合函数ｙ＝ｆ〔ｇ（ｘ）…     

具有三个转向点方程渐近解的完全表达式

本文研究二阶线性常微分方程　　ｄ２ ｙｄｘ２ ＋ ［λ２ｑ１（ｘ） ＋ λｑ２（ｘ,λ）］ｙ ＝ ０ ,其中　　ｑ１（ｘ） ＝ （ｘ － μ１）（ｘ － μ２）（ｘ － μ３）ｆ（ｘ） ,　　ｆ（ｘ） ≠０,μ１ ＜ μ２ ＜ μ３ ,λ为大参数,即具有三个转向点的方程·　而　　ｑ２（ｘ,λ） ＝ ∑∞ｉ＝０ｇｉ（ｘ）λ－ ｉ　　（ 此时ｇ０（ｘ） ０）·　本文使用ＪＬ函数得到方程在转向点附近形式一致有效渐近解的完全表达式·     

ON THE EXISTENCE OF ALMOST GLOBAL WEAK SOLUTION TO MULTIDIMENSIONAL VLASOV-POISSON EQUATION

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｏｗｉｎｇｄｄｉｍｅｎｓｉｏｎａｌＶｌａｓｏｖＰｏｉｓｏｎｓｙｓｔｅｍ，ｄ＝２，３，ｔｆ＋ｖ·ｘｆ－Ｅ·ｖｆ＝０，ｆ（０，ｘ，ｖ）＝ｆ０（ｘ，ｖ），Ｅ（ｔ，ｘ）＝ｃ（ｄ）∫ｘ－ｙ｜ｘ...     

利用单调函数的性质证明一类不等式

利用单调函数的性质证明一类不等式苏万春（吉林省永吉三中）由单调函数定义知：若函数ｆ（ｘ）在区间Ｍ上是增函数，则对于Ｍ上的任意两个不同的自变量的值ｘ１和ｘ２都有（ｘ１－ｘ２）·［ｆ（ｘ１）－ｆ（ｘ２）］＞０（反之亦然）；若函数ｆ（ｘ）在区间Ｍ上是减函数...     

关于f~n（x）＝x的讨论

关于ｆ~ｎ（ｘ）＝ｘ的讨论张伟年（中国科学院成都计算所数理中心６１００４１）首先从一简单的中学数学题谈起．问题ｘ和ｃ为实数，求：（ａ）满足ｆ（ｆ（ｘ））＝ｘ的ｃ值；（ｂ）满足ｆ（ｆ（ｘ））＝ｘ的ｘ值．解由ｆ（ｆ（ｘ））＝ｘ，ｘ≠０得显然当２ｘ－１＝０?..     

重视常规方法解题——从今年一道高考题引发的思考

重视常规方法解题——从今年一道高考题引发的思考桂元今年高考理科（２４）题是这样的：“设二次函数ｆ（ｘ）＝ａｘ２＋ｂｘ＋ｃ（ａ＞０），方程ｆ（ｘ）－ｘ＝０的两个根ｘ１，ｘ２满足０＜ｘ１＜ｘ２＜１ａ．（Ⅰ）当ｘ∈（０，ｘ１）时，证明ｘ＜ｆ（ｘ）＜ｘ１；（...     

对《关于一类函数最小值问题的一种处理方法》的一点补充

读了《数学通讯》１９９９年第４期刊登的刘汉顶老师的《关于一类函数最小值问题的一种处理方法》一文后深受启发,本文特给出不能应用均值不等式处理“和一定,积最大”另一类函数的最大值问题的处理方法,以此作为前文的一个补充．定理　设初等函数ｆ（ｘ）在区间Ｉ上恒有０＜ｆ（ｘ）＜ｄ,ｄ为正常数,则当且仅当｜２ｆ（ｘ）－ｄ｜取最小值时,函数ｇ（ｘ）＝ｆ（ｘ）·［ｄ－ｆ（ｘ）］取最大值．证　（１）若２ｆ（ｘ）＝ｄ在Ｉ上有解,则结论显然成立．（２）若２ｆ（ｘ）＝ｄ在Ｉ上无解,则由初等函数的连续性知：２ｆ（ｘ）＞ｄ或…     

概率方法解拟线性偏微分方程

本文用随机分析方法证明了拟线性抛物型方程ｕｔ＋ｆ（ｕ）ｕｘ、ｕｘｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）在ｕ０有界可测，ｆ连续且ｆ＞０条件下，其解当→０时收敛于拟线性方程ｕｔ＋ｆ（ｕ）ｕｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）的熵解，即论证了“沾性消失法”解此方程的正确性，１９５７年Ｏｌｅｉｎｉｋ曾用差分方法解决了此问题。这里用概率方法重新获得此结果。     

求解微分方程初值问题的一种弧长法

对于连续介质力学问题中导出的微分方程初值问题,常常具有解奇异性,如不连续、Ｓｔｉｆ性质或激波间断·本文通过在相应空间,引入一个或数个弧长参数变量,克服解的奇异性·对于常微分方程组引入弧长参数变量后,奇异性得以消除和削弱,应用一般的解常微分方程组的方法（如Ｒｕｎｇｅ＿Ｋｕｔａ法）求解·对于偏微分方程引入弧长参数变量后,在相应的空间离散成常微分方程组,用解奇异性常微分方程组相同的方法即可求解·本文给出了两个算例     

解非线性方程的二阶敛速指数迭代法

1．gi言文［1，2］中利用ODE方法［'］给出解非线性方程在卜6I内的根x"的两个非线性迭代法其中'w由文［2］中（5）式确定．令h-1方法（2）具有M阶敛速，方法（3）是线性收敛的．它们是李雅普诺夫渐近稳定性和文［4］中Lambert提出的解Stiff方程的非线性方法相结合的结果．Lllbll't在每个小区间【Ln，Ln＋1］上用一个有理函数月O一句（I十利来逼近微分方程的解z二"I，＊。）；＊。Ek；q，使得对I＿，J。）一J＿，"乙十；，J。）＝。＿＋i，l'（Ln，10）一人，而tim0（7；00）一0"．那么我们能否在每个小区间【Ln，Ln＋1］上用一个指…     

Note on exact solutions for a class of nonlinear diffusion equations

An effective characterization is given for a class of generalized nonlinear diffusion equations with power law dependent terms. Further, a new auxiliary equation ansatz is derived. Consequently, new exact traveling wave trigonometric function, solitary-like and Weierstrass elliptic solutions to a subclass are obtained by means of an auxiliary equation method and a generalized Riccati equation expansion method.     

Analytical and numerical solutions of mathematical biology models: The Newell-Whitehead-Segel and Allen-Cahn equations

In this paper, we combine the unified and the explicit exponential finite difference methods to obtain both analytical and numerical solutions for the Newell-Whitehead-Segel–type equations which are very important in mathematical biology. The unified method is utilized to obtain various solitary wave solutions for these equations. Numerical solutions of the specific case studies are investigated by using the explicit exponential finite difference method ensures the accuracy and reliability of the proposed scheme. After obtaining the approximate solutions, convergence analysis and error estimation (the error norms and absolute errors) are presented by comparing these results with the analytical obtained solutions and other methods in the literature through tables and graphs. The obtained analytical and numerical results are in good agreement.     

应用改进的简单方程法求非线性数学物理方程的精确解

应用改进的简单方程法求得Cahn-Allen方程和Jimbo-Miwa方程的精确解,这些解包括双曲函数解、三角函数解.当对双曲函数解中的参数取特殊值时,可以得到了孤立波解.当对三角函数解中的参数取特殊值时,可以得到对应的周期波函数解.实践证明,简单方程法对于研究非线性数学物理方程具有非常广泛的应用意义.     

Analysis of a Diagonal Form of the Fast Multipole Algorithm for Scattering Theory

Discretisation of the integral equations of acoustic scattering yields a system of linear equations with full coefficient matrices. In recent years a number of fast algorithms for the solution of this system have been proposed. In this paper we present a complete analysis for a fast multipole method for the Helmholtz equation. A one-level diagonal form of the multipole method is applied to a hypersingular integral equation arising from 2d scattering theory. The error of the approximation is analysed and the results used to establish the complexity of the method.     

The variational iteration method for analytic treatment for linear and nonlinear ODEs

In this work, the variational iteration method (VIM) is used for analytic treatment of the linear and nonlinear ordinary differential equations, homogeneous or inhomogeneous. The method is capable of reducing the size of calculations and handles both linear and nonlinear equations, homogeneous or inhomogeneous, in a direct manner. However, for concrete problems, a huge number of iterations are needed for a reasonable level of accuracy.     

Analytic and approximate solutions of the space- and time-fractional telegraph equations

The Adomian decomposition method is used to obtain analytic and approximate solutions of the space-and time-fractional telegraph equations. The space- and time-fractional derivatives are considered in the Caputo sense. The analytic solutions are calculated in the form of series with easily computable terms. Some examples are given. The results reveal that the Adomian method is very effective and convenient.     

New solutions for the modified generalized Degasperis-Procesi equation

In this paper, using three distinct computational methods we obtain some new exact solutions for the generalized modified Degasperis-Procesi equation (mDP equation) u_t-u_xxt+(b+1)u²u_x=bu_xu_xx+uu_xxx. We show the graph of some of the new solutions obtained here with the aim to illustrate their physical relevance. Mathematica is used. Finally some conclusions are given.     

On approximation in the fictitious domain method for a bidimensional transport equation

In this article a numerical method for solving a two‐dimensional transport equation in the stationary case is presented. Using the techniques of the variational calculus, we find the approximate solution for a homogeneous boundary‐value problem that corresponds to a square domain D₂. Then, using the method of the fictitious domain, we extend our algorithm to a boundary value problem for a set D that has an arbitrary shape. In this approach, the initial computation domain D (called physical domain) is immersed in a square domain D₂. We prove that the solution obtained by this method is a good approximation of the exact solution. The theoretical results are verified with the help of a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010