Poincaré非正则积分问题的新进展

关于非Fuchs型方程,Poincaré曾经作出过重要的论断:没有方法可以求出非正则积分的显示表述.为了阐明这一论断的实质,我们证明对应定理:非正则积分是类具有树结构的新型解析函数,其中一部份解是通常的递推级数,而另一部则是不遵循递推关系的“树级数”.与经典理论(Hill-Poincaré-von Koch)计算无穷行列式的数值解不同,本法自然地给出严格解析解的显示表式.本法可以建立统一的解析理论以讨论一般变系数方程,包括有奇线在内的多种奇点.由于树级数具有自守性.我们讨论Poincaré猜测的意义.     

非Fuchs型方程的新理论——树级数解的表现定理(Ⅱ)

本文的主要结果是证明表现定理:非正则积分是类新颖解析函数,它表成Taylor-Fourier混合型树级数,其中Fourier级数的每一系数本身都是Taylor级数,而所有Taylor系数则是方程参数的常项树级数,每一系数的高阶修正项具有树结构的无穷繁衍性. 证明此树级数解在原方程的系数定义域中解析,收敛条件是方程的结构因子小于1,直接代入可以验证树级数解逐代满足已知方程. 与经典理论相对比,本法的优点不仅可以给出非正则积分的显式,从而解决Poincaré问题,并能统一处理具有多种奇点的方程,扩大解析理论的研究范围. 利用树图法可得非正则积分的严格解析表述.据此易证树级解的收敛性,并满足方程. 树级数具有自守性,这与Poincaré猜测完全符合.     

对应函数D(z)和广义非正则方程

推广Riemann P函数的思想(用方程的参数表示方程所定义的函数),引入(?)函数统一表示正则积分和非正则积分.利用显式解讨论非Fuchs型方程的单值群.得到Floquet解的指标展开系数的显式.根据对应函数法统一研究广义非正则方程的求解问题,包括具有正则和非正则极点,本性奇点,代数,对数和超越奇点以及奇线的方程.利用(?)函数表示基本解系,从而推广解析理论的研究范围.指出(?)函数的自守性,并讨论Poincaré猜测的意义.     

拓扑定理及其在超线性脉冲方程中的应用

本文研究一类含脉冲的非保守超线性方程的周期振动问题.通过详细刻画跳跃映射的角度函数,并应用相平面分析方法,本文证明方程的Poincaré映射有无穷多个不动点,而这些不动点对应了方程的无穷多个2π周期解.由于本文考虑的Poincaré映射不是同胚,所以,本文重新构造并证明了一个拓扑定理用于替代经典的Poincaré-Birkhoff扭转不动点定理.     

广义双正则函数的非线性带位移边值问题

本文研究Ｃｌｉｆｆｏｒｄ分析中的广义双正则函数及其一类非线性带Ｈａｓｅｍａｎ位移的边 值问题．通过积分变换将边值问题转化成积分方程问题，借助于积分方程理论和不动点 定理证明了边值问题解的存在性并给出了解的积分表示式．     

线形振子散射或辐射问题的严格解

本文应用时序展开法和Wiener-Hopf技术求得圆管(或圆柱)状振子的脉冲散射或辐射问题的严格解。在单脉冲激励下,振子的电流响应是一脉冲系列。系列中每一个脉冲依次用时序递推积分方程来确定。递推方程有两类:一类是全区间积分方程。它可以用富氏积分变换来求解;另一类是带约束的半区间积分方程。除了应用富氏变换外,还采用了Wiener-Hopf技术来求解。本文求得两类递推方程的严格显式解并验证了解的正确性。对于长振子,在引入描写相邻脉冲间关系的传输函数后,代表总电流的无穷级数可以求和,并得到形式十分简单的总体解,后一结果可以推广到连续波情形电流响应的计算。     

在指数增长的函数类中的奇异积分方程与Riemann边值问题

提出并讨论了在指数增长的函数类中带有卷积核与Cauchy核的奇异积分方程,通过Fourier变换及文章所给出的引理,将奇异积分方程转化为一类推广的两条平行直线上的Riemann边值问题,并在正则型的情况给出了方程的可解条件及方程的显式解,特别讨论了解在结点的性态.     

Smoluchowski方程的对称与显式解析解

研究了一类带齐次核函数的偏微分一积分Smoluchowski方程.利用发展了的李群分析方法给出了带齐次核函数的Smoluchowski方程的决定方程的通解、对称、最优子李代数系统、约化的常微分—积分方程、群不变解和显式解析解.     

多复变解析函数的一个非线性边值问题

该文讨论多复变解析函数的一个非线性边值问题；利用积分方程方法和Schauder不动点定理证明了问题解的存在性和积分表示式以及线性情况下解的唯一性.     

Clifford分析中双正则函数的Taylor展式及其性质

首先借助实Clifford分析中双正则函数的累次积分的换序公式,给出了双正则函数的Cauchy积分公式,然后由特征边界的Cauchy积分公式,得到了双正则函数的Taylor展式,并由此给出了双正则函数的唯一性定理,柯西不等式和Weierstrass定理.     

任意形状弹性薄板弯曲的精确解*

本文提供了一个求解任意形状弹性薄板弯曲的新方法,在求得了极坐标系中弹性薄板弯曲微分方程的精确解后,将解代入薄板的边界条件,利用Fourier级数将边界方程展开,可确定出各待定常数,所得结果是精确的。     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

A Series Solution of the Unsteady Von Kármán Swirling Viscous Flows

A new analytic technique is applied to solve the unsteady viscous flow due to an infinite rotating disk, governed by a set of two fully coupled nonlinear partial differential equations deduced directly from the exact Navier-Stokes equations. The system of coupled nonlinear partial differential equations is replaced by a sequence of uncoupled systems of linear ordinary differential equations. Different from all other previous analytic results, our series solution is accurate and valid for all time in the whole spatial region. Accurate expressions for skin friction coefficients are given, which are valid for all time. Such kind of series solutions have not been reported, to the best of our knowledge.      

Approximate solutions of linear Volterra integral equation systems with variable coefficients

In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions.     

An improved approximate analytic solution for Riccati equations over extended intervals

In this paper, we present a new effective approach based on our previous method for solving Riccati equations [Vahidi and Didgar, An improved method for determining the solution of Riccati equations, Neural Comput & Applic DOI 10.1007/s00521-012-10645, 2012]. The proposed technique combines transformation of variables with the Laplace Adomian decomposition method, which is a variant of the classic Adomian decomposition method. The method examined practicality for several specific examples of the Riccati equation. Numerical results show that the proposed method is more efficient and accurate than other applied methods over extended intervals, and stands in good agreement to the exact solutions. The obtained results provide a rapidly convergent series, from which we achieve a high degree of accuracy using only a few terms of the recursion scheme. Thus we have developed a natural sequence of rational approximations as a technique of solution continuation for the Riccati equation.     

On a method of calculation of hypersingular integrals

We propose a method for calculation of one-dimensional and two-dimensional hypersingular integrals, which allows to obtain a solution in the analytic form in a series of cases. Based on this method, we construct an approximate method for calculation of one-dimensional and twodimensional hypersingular integrals having singularities on rather complicated manifolds. We give estimates of errors of suggested algorithms.     

Bi-Hamiltonian systems of natural form

We propose a new method for constructing integrable systems of natural form. In this method, integrals of motion are solutions of an overdetermined system of algebraic and partial differential equations obtained from the compatibility condition for Poisson tensors polynomial in the momenta and from the condition that the bi-Lagrangian distribution corresponding to the integrals of motion is invariant under the action of the recursion operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 161–182, November, 2006.