NEW THEORY FOR EQUATIONS OF NON-FUCHSIAN TYPE REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅰ)

In the analytic theory of differential equations the exact explicit analytic solution has not been obtained for equations of the non-Fuchsian type (Poincare's problem).The new theory proposed in this paper for the first time affords a qeneral method of finding exact analytic expression for irregular integrals. By discarding the assumption of formal solution of classical theory, our method consists in deriving a correspondence relation from the equation itself and providing the analytic structure of irregular integrals naturally by the residue theorem. Irregular integrals are made up of three parts: noncontracted part, represented by ordinary recursion series, all-and semi-contracted part by the so-called tree series. Tree series solutions belong to analytic function of the new kind with recursion series as the special case only. The purpose of our present paper consists of the establishment of a general theory for the irregular integrals. For this it is needed to elucidate the essence of Poincare's prob     

NEW THEORY FOR EQUATIONS OF NON-FUCHSIAN TYPE——REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅱ)

Our main result consists in proving the representation theorem. Irregular integral is a new type of analytic function, represented by a compound Taylor-Fourier tree series, in which each coefficient of the Fourier series is a Taylor series, while the Taylor coefficients are tree series in terms of equations parameters, higher order correction terms to each coefficient having tree structure with inexhaustible proliferation. The solution obtained is proved to be convergent absolutely and uniformly in the region defined by coefficient functions of the original equation, provided the structure parameter is less than unity. Direct substitution shows that our tree series solution satisfies the equation explicity generation by generation. As compared with classical theory our method not only furnishes explicit expression of irregular integral, leading to the solution of Poincaré problem, but also provides possibility of extending the scope of investigation for analytic theory to equations with various kinds o     

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 solutions of Novozhilov's equation of toroidal shells

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

NEW THEORY FOR EQUATIONS OF NON-FUGHSIAN TYPE REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅰ)

In the analytic theory of differential equations the exact explicit analytic solution has not been obtained for equations of the non-Fuchsian type (Poincare's problem). The new theory proposed in this paper for the first time affords a general method of finding exact analytic expres-sion for irregular integrals.By discarding the assumption of formal solution of classical theory,our method consists in deriving a cor-respondence relation from the equation itself and providing the analytic structure of irregular integrals naturally by the residue theorem. Irregular integrals are made up of three parts: noncontracted part,represented by ordinary recursion series,all-and semi-contracted part by the so-called tree series. Tree series solutions belong to analytic function of the new kind with recursion series as the special case only.     

New method of solving Lame-Helmholtz equation and ellipsoidal wave functions

Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lamé-Helmholtz equation still remains to be tackled. Arscott and Moglich method of double-series expansion as well as Malurkar nonlinear integral equation are incapable of reaching the final explicit solution.Our main result consists in obtaining analytic expressions for ellipsoidal wave functions of four species (i=1,2,3,4) including the well known Lam(α) functions E_ci(snα),E_z1(snα) as special cases. This is effected by deriving two Integra-differential equations with variable coefficients and solving them by integral transform. Generalizing Riemann’s idea of P function, we introduce D function to express their transformation properties.     

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 problem 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 supersonic 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.     

Poiseuille流的谱问题

对Poiseuille流问题的Orr-Sommerfeld方程严格求解.得到的正规解不含外来奇点.从而谱方程可作解析的显式分析.本文结果可以进一步讨论分岔解.     

NEW METHOD OF SOLVING LAME-HELMHOLTZ EQUATION AND ELLIPSOIDAL WAVE FUNCTIONS

Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lamé-Helmholtz equation still remains to be tackled. Arscott and M(?)glich method of double-series expansion as well as Malurkar nonlinear integral equation are incapable of reaching the final explicit solution. Our main result consists in obtaining analytic expressions for ellipsoidal wave functions of four species including the well known Lame functions E_(ci)(sna),E_(si)(sna) as special cases.This is effected by deriving two integro-differential equations with variable coefficients and solving them by integral transform. Generalizing Riemann's idea of P function, we introduce D function to express their transformation properties o     

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.