Zur numerischen Lösung gewöhnlicher Differential-gleichungen mit Splines in einem Sonderfall

In an earlier paper [1] a general procedure has been presented to obtain polynomial spline approximations for the solution of the initial value problem for ordinary differential equations. In this paper the general procedure is described by an equivalent one step method. Furthermore two convergence theorems are proved for a special case which is not included in the general convergence or divergence theory given in [1].     

On the Galerkin Method Based on a Particular Class of Scaling Functions

The aim of this paper is to provide a large class of scaling functions for which the convergence analysis for the Galerkin method developed in [9] is applicable, whereas in that paper the only scaling functions considered for practical applications are B-splines and a few of the orthonormal Daubechies scaling functions. The functions considered here, were recently introduced in [12] where it was proved that they satisfy many properties making them interesting for the applications. In particular, here we show that the use of these functions has some advantages with respect to other basis functions.     

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…     

变双障碍问题解的抽象稳定性

本文考虑了主部为非线性变双障碍问题解的抽象稳定性 (连续依赖性 ) .由于采用了弱收敛原理和文 [2 ]中取检验函数的技巧 ,我们的证明无需像 [1 ]那样应用Minty引理 .     

Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line

We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm-Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re-made survey of our results obtained during the period 1953-58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].     

Splineapproximationen von beliebigem Defekt zur numerischen Lösung gewöhnlicher Differentialgleichungen. Teil III

Summary In the first part [1] a general procedure is presented to obtain polynomial spline approximations for the solutions of initial value problems for ordinary differential equations; furthermore a divergence theorem is proved there. Sufficient conditions for convergence of the method are given in the second part [2]. The remaining case which has not been considered in [1] and [2] is treated in the present paper. In this special case the procedure is equivalent to an unstable two-step method with special initial values; nevertheless, convergence can be proved. Finally,A ₀-stability of the method as well as the influence of rounding errors are investigated.

     

A Trilayer Difference Scheme for One-Dimensional Parabolic Systems

In order to obtain the numerical solution for a one-dimensional parabolic system, an unconditionally stable difference method is investigated in [1]. If the number of unknown functions is M, for each time step only M times of calculation are needed. The rate of convergence is $O(\tau+h^2)$. On the basis of [1], an alternating calculation difference scheme is presented in [2]; the rate of the convergence is $O(\tau^2+h^2)$. The difference schemes in [1] and [2] are economic ones. For the $\alpha$-$th$ equation, only $U_{\alpha}$ is an unknown function; the others $U_{\beta}$ are given evaluated either in the last step or in the present step. So the practical calculation is quite convenient. The purpose of this paper is to derive a trilayer difference scheme for one-dimensional parabolic systems. It is known that the scheme is also unconditionally stable and the rate of convergence is $O(\tau^2+h^2)$.     

关于(F,ρ)-不变凸性函数多目标规划的充分性条件

本文给出了一类更一般的广义凸性函数的定义:(F,ρ)-不变凸性函数,并论证了其多目标规划关于有效解的充分性条件.本文的一些主要结论是对文献[2]～[5]中相应结论的改进和推广.     

Mesures de Gibbs invariantes et mesures d'equilibre

We are interested here in the characterization on a symbolic space, of invariant Gibbs measures as equilibrium measures. The first result in this topic was obtained by Lanford and Ruelle (see for example [6]).This problem involves different objects that can all be defined by using the only amenability of the translation group and the only continuity of the local specification. We therefore tried to state our theorems in this general frame. Among the elements of our proof, there is the use of the information gain introduced by H. Föllmer [1] and some arguments similar to those of C.J. Preston in [5]. But the amenability techniques that we widely develop in [2], [4] and [7] are decisive tools for getting the result.The corresponding problem for subshifts is not considered in the present paper so symbolic spaces are product spaces.     

亚纯函数微分多项式f~kQ[f]+P[f]的零点分布

文[4]对简单形式的微分多项式fkf’+a的零点分布进行了讨论,文[1]对一般形式的微分多项式fkQ[f]+P[f]的零点分布进行了讨论.但由于极点给证明带来的困难,这些工作主要是对整函数来做的.本文证明了任一满足δ(∞,f)>k+2ΓQ+3ΓP+2/2k+2ΓQ+1的超越亚纯函数f,微分多项式fkQ[f]+P[f]在不含f,Q[f]极点和P[f]零、极点的可数个圆盘并集之外有无穷多个零点,其中k≥3Γp+2,而ΓQ,ΓP分别是f的微分多项式Q[f],P[f]的权.文[1]和[2,4,6]中的结论是本文结论的特殊情况.     

On a differential-difference game of escape : PMM vol. 40, n 6, 1976, pp. 995–1002

A nonlinear escape problem for conflict-controlled systems described by differential equations with a lagging argument is considered. The sufficient escape conditions which are realized in the class of piece wise-constant functions are obtained. The paper relates to the researches in [1 – 8] and is a continuation of [9, 10].     

APPROXIMATION BY VECTOR-VALUED FUNCTIONS

ABSTRACT

The theory of H-sets was first propounded by L. Collatz [3] and [4]. This concept has been shown to be useful in the study of uniform approximation, and we here consider the form H-sets take in this setting of vector-valued functions and prove a general characterization theorem. A similar exposition for the linear real-valued case can be found in [1].     

Darstellung skalarer und vektorwertiger Distributionen aus D′Lp durch Randwerte holomorpher Funktionen

In the present paper we show that every distribution T in D′_Lp (D′_Lp(E) in the vector-valued case) has a representation as boundary value of a holomorphic function T (with values in E). This problem was considered in [1], [5], [12]. The representing functions are characterized by growth conditions. It is shown that the conditions in [1] and [5] don't extend to the vector-valued case while the conditions in [12] do.     

Effect of random irregularities of the creep of reinforced layered plastics

The authors investigate the creep of inhomogeneous materials consisting of a large number of stiff orthotropic elastic layers alternating with layers of linear isotropic viscoelastic material. The elastic layers are assumed to be almost plane; the functions describing the irregularities (curvature) form a random field. The averaged characteristics of the medium are found together with the variation of the averaged displacements and strains in time. An analogous problem was previously considered in [1, 6] on the assumption that the binder layers are elastic. The present paper is based on the equations of [1] and the elastic-viscoelastic correspondence principle [4]. When the correlation scales of the irregularities are small as compared with the dimensions of the body and the characteristic distances over which the averaged parameters of the stress-strain state vary appreciably is considered in detail. A relation is established between the creep functions for simple cases of the state of stress and the parameters characterizing the properties of the components, the properties of the random field of initial irregularities, etc. The development of perturbations with different wave numbers is investigated. The theory is used to describe the creep of reinforced layered plastics.Mekhanika Polimerov, Vol. 2, No. 5, pp. 755–762, 1966     

Solvability of an initial boundary value problem for the euler equations in twodimensional domain with corners

The aim of this paper is to prove the existence and uniqueness of local solutions of some initial boundary value problems for the Euler equations of an incompressible fluid in a bounded domain Ω ? R ² with corners. We consider two cases of a nonvanishing normal component of velocity on the boundary. In three-dimensional case such problems have been considered in papers [12], [13], [14]. Similar problems in domains without corners have been considered in [2]–[6], [11]. In this paper the relation between the maximal corner angle of the boundary and the smoothness of the solutions is shown. The paper consists of four sections. In section 1 two initial boundary value problems for the Euler equations are formulated. In section 2 the existence and uniqueness of solutions of the Laplace equation in twodimensional domain with corners for the Dirichlet and Neumann problems is proved in the Sobolev spaces. In sections 3 and 4 we prove the existence and uniqueness of solutions of problems formulated in section 1, using the method of successive approximations.     

Dynamical stability of an elastic column and the phenomenon of bifurcation

The article studies the stability of rectilinear equilibrium shapes of a non-linear elastic thin rod (column or Timoshenko's beam), the ends of which are pressed. Stability is studied by means of the Lyapunov direct method with respect to certain integral characteristics of the type of norms in Sobolev spaces. To obtain equations of motion, a model suggested in [16] is used. Furta [6] solved the problem of stability for all values of the parameter except bifurcational ones. When values of the system's parameter become bifurcational, the study of stability is more complicated already in a finite-dimensional case. To solve a problem like that, one often has to use a procedure of solving the singularities described in [1], for example. In this paper a change of variables is made which, in fact, is the first step of the procedure mentioned. To prove instability, we use a Chetaev function which can be considered as an infinite-dimensional analogue of functions suggested in [14, 9]. The article also investigates a linear problem on the stability of adjacent shapes of equilibrium when the parameter has supercritical values (post-buckling).     

On Meromorphic Starlike Multivalent Functions

In this paper we introduce and study some new subclasses of meromorphic starlike multivalentfunctions.Inclusion relations are established,Integral transforms of functions in these classes are alsoconsidered.In particular,our results include or improve several results due to Mogra et al.[2],Mogra[3],Goel and Sohe[4]and Bajpai[5].     

On the theory of fugoid motions

In 1891 Zhukovslii in his paper “On soaring of birds” [1] solved the problem of the motion of a body of high lift — drag ratio in an atmosphere of constant density. In [2] this problem was considered in greater detail, but the basic assumption of a constant density was made here as well. There have recently appeared numerous papers concerning the analytical solution of the problem of entry into the atmosphere with orbital and escape velocities [3 to 5]. But these studies were concerned primarily with the problems of ballistic entry and entry with low lift — drag ratio. In considering oscillatory states, the authors limited their treatment to small angles between the trajectory and local horizon. In the present paper we consider the problem without imposing any limitations on the slope of the trajectory or initial velocity. The case examined will be that of a hypothetical glider spacecraft of sufficiently high lift — drag ratio. It is interesting to note that the solution of this problem reduces to the solution of Zhukovskii's problem, but for an atmosphere of variable density. The associated trajectories are termed “fugoid”. All of our assumptions about the parameters of such a glider are of a particular hypothetical character.     

Existence and multiplicity of solutions for elliptic problems with indefinite discontinuous nonlinearities

In this paper we study the critical points for a locally Lipschitz functional that in some sense will be solutions of an elliptic problem with indefinite discontinuous nonlinearities. We should mention that our results were inspired by the work of Ambrosetti-Badiale [3], Arcoya-Calahorrano [5], Alama-Tarantello [1] and Chang [8]. For the problem studied in [3] we introduce indefinite nonlinearities as in [1] and [6]. To obtain the existence and multiplicity of solutions we use the critical points theory developed by Chang. Applications for Plasma Physics are considered with nonlinearities that change sign. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)