Transfer of admissible boundary conditions from a singular point for systems of linear ordinary differential equations

Abstract

A review is given of the results obtained by the authors on the topic formulated in the title. New results are also presented: methods more general than those suggested earlier are described for constructing admissible boundary conditions at a singular point in the case of systems of first-order equations, and generalized systems of second-order equations with pole-type singularities are analysed, special attention being focused on the conditions of boundedness of the solutions at a singular point and to the emission-type conditions at infinity.The paper gives a definition of the admissible boundary condition at a pole of a system of linear ordinary differential equations. A sufficiently wide class of linear relations is specified, which can provide admissible boundary conditions at a singular point. A particular case covered by this class is the practically important case of emission-type boundary conditions at infinity for systems of second-order equations. Methods of transfer of admissible boundary conditions from singularities in solving singular boundary value problems are described. The results on the application of the suggested methods to solving specific applied problems encountered in oceanology, nuclear physics, radiophysics and acoustics are presented.The present paper is closely related to earlier papers [1,2,11] which are not cited in the text. The results to be described below were briefly outlined earlier in [4]. DEFINITION OF ADMISSIBLE BOUNDARY CONDITIONSConsider on the interval [Ο,α] a system of η homogeneous linear ordinary differential equationswhere r is a positive integer. Assume that the matrix A(t) is continuous on the interval [Ο, a], and can be approximated, at small i, by the convergent (or asymptotic) seriesThe point 0 is a singular point of system (1.1). Let us consider the solutions of (1.1) on a half-open interval [Ο,α]. We introduce the concept of admissible boundary condition at a singular point forOriginally published in Russian as the preprint "Transfer of admissible boundary conditions from a singular point for systems of linear ordinary differential equations", Vychislit. Tsentr Akad. Nauk SSSR, Moscow, 1985. Brought to you by | University of Glasgow Library Authenticated Download Date | 6/25/15 7:11 AM

Problems of Optimization and Numerical Stability in Computations

“…Another kind of factorization method, sometimes called method of the transfer of boundary conditions, was investigated in recent years, for example, by ABRAMOV (see [1], [2]) who also briefly mentioned the possibility of showing the suitability of this process for the general case (3.4). See [3]. In [7] and [12] the stability of the differential equations of the factorization method in special cases has been studied.…”

On the computation of Bessel functions of first kind

“…In the neighbourhood of a regular singular point the behaviour of certain linear subspaces of the solutions can be characterized. Namely, using the results of [5], [6], [7] we can formulate the following Theorem. Xflt dr f12 _~_ X2 --p2 = O,…”

On one-dimensional Helmholtz equation

“…The method of boundary condition transfer has also been used in [2,3] for solving one-dimensional problems. Further developments of this technique for systems of ordinary differential equations are presented in [4] and [5]. In [6], the operator method of transferring boundary conditions from infinity was first used for the Helmholtz equation in a half-cylinder with ordinary conditions on the boundary.…”