New development in Poincaré's problem of irregular integrals

In connection with non-Fuchsian equations Poincaré has made an important conclusion: It is impossible to obtain explicit expressions of irregular integrals .To elucidate the essence of Poincaré's problem, we establish correspondence theorem. Irregular integrals are analytic functions of new kind, possessing tree structure; part of which can be represented by conventional recursive series, while its remaining part is expressed by the so-called tree series, not subjecting to any recursive relation at all.In contrast to the numerical solution calculated by infinite determinant of classical theory (Hill-Poincaré-von Koch), our method yields naturally exact analytic solution in explicit form. The method proposed may be used to construct a unifying theory for general equations with variable coefficients, having various kinds of singularities as singular lines.The significance of Poincaré conjecture is discussed, the tree series obtained belong to higher automorphic functions.     

The spectral problem of Poiseuille flow

For Poiseuille flow the Orr-Sommerfeld equation is solved exactly. Regular solutions are obtained, thereby eigenvalue equation can be analyzed analytically and explicitly. The bifurcation solutions will be discussed subsequently.     

关于Lamé-Helmholtz方程的新解法和椭球波动函数

双周期系数方程虽在数理方法中具有重要意义,但Lamé—Helmholtz方程的解至今仍来求出,因为Arscott和M?glich的双重级数展开法,Malurkar的非线性积分方程都无法进一步处理 本文的主要结果是由原方程导出一组线性微积分方程,利用积分变换,直接求得四类椭球波动函数,ε_ci(sna),ε_si(sna)(i=1,2,3,4),它的特例就是熟知的Lamé函数E_ci(sna),E_si(sna),推广Riemann P函数思想,引进D函数来表示其变换规律。     

Eigenvalue problem for integro-differential equation of supersonic panel,flutter

The dynamic stability of a thin plate in supersonic flow based on 2-dimensional linear theory leads to the study of a new problem in mathematical physics: complex eigenvalue prob-lem for a non-self-adjoint fourth-order integro-differential equation of Volterra’s type.Exact solutions of the aeroelastic system is obtained. In contrast to various approximate analyses, our critical curve agrees satisfactorily with experimental data, being free from divergence in the low supe’rsonic region. Moreover, we observe some notable physical behaviors: (1) mutual separation of flutter and vacuum frequency spectrums, (2) degeneracy of critical Mach number. The present method may be generalized in solving the supersonic flutter for 3-dimensional airfoil model as well as blade cascade in turbo-generator.     

NEW SOLUTIONS OF NOVOZHILOV’S EQUATION OF TOROIDAL SHELLS

New solutions are obtained for Novozhilov’s equation of toreidal shells having generalslenderness ratio 0<α<1(α=α/R).In contrast to the results by continued fractiontechnique,the exponents and expansion coefficients of our series solutions are all closed andexplicit.The series satisfies shell equation identically.Convergence proof is alsodemonstrated.Explicit expressions for boundary effect and monodromy indices are also given.Finally,we discuss the possibility of applying the present method to solve the fundamentalsystem of equations for elastic shells with rotational symmetry.     

CORRESPONDENCE FUNCTION D(z)AND GENERALIZED IRREGULAR EQUATIONS

Extending Riemann’s idea of P function(using equation’s parameters to represent thefunction defined by the equation).we introduce correspondence functions (?)(z) to describeregular and irregular integrals in a unifying way.By explicit solution discuss monodromy group of non-Fuchsian equations.The explicitexpressions of exponent and expansion coefficients for Floquet solution are obtained.Method of correspondence functions permits us to obtain systematically the solutionsof generalized irregular equations. having regular,irregular poles,essential.algebraic,transcendental.logarithmic singularities as well as singular line.The representation of basic set of solutions by (?) function makes it possible to enlargethe scope of investgation of analytic theory.The significance of Poincaré’s conjecture is discussed,as(?)functions are automorphic.     

Novozhilov环壳方程的新解

当环壳的细度比α(=α/R)取一般值时0<α<1求得Novozhilov方程的新解.与连分法的结果相比,这里级数解的指标和展开系数全是封闭的显式.直接验证,此级数解满足方程.给出收敛性证明.边缘效应的范围和单值群的指标也都用显式表述.最后讨论本法适用于求解旋成薄壳的基本方程组.     

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.     

超声速流中有限长柱壳蒙皮颤振的精确线性理论

本文讨论了有限长柱壳超声速颤振的线性问题的数学基础,包括问题的提法以及气动力和颤振振型的精确求法;所采用的方法是作者在前文[11]中提出的方法的推广. 考虑到壳面气动力的非定域性以及壳体的精确力短理论,根据严格的线性理论,将柱壳蒙皮颤振问题化成了讨论一个新型的八阶非自共轭微分积分方程以及相应的八个边界条件所构成的复固有值问题:     

NEW DEVELOPMENT IN POINCAR’S PROBLEM OF IRREGULAR INTEGRALS

In connection with non-Fuchsian equations Poincaré has made an importantconclusion; It is impossible to obtain explicit expressions of irregular integrals(?).To elucidate the essence of Poincaré’s problem. we establish correspondence theorem.Irregular integrals are analytic functions of new kind, possessing tree structure, part ofwhich can be represented by conventional recursive series.while its remaining part isexpressed by the so-called tree series, not subjecting to any recursive relation at all.In contrast to the numerical solution calculated by infinite determinant of classicaltheory (Hill-Poincaré-von Koch), our method yields naturally exact, analytic solution inexplicit form. The method proposed may be used to construct a unifying theory for generalequations with variable coefficients. having various kinds of singularities as singular lines.The significance of Poincaré conjecture is discussed, the tree series obtained belong tohigher automorphic functions.