On the transfer of boundary conditions for systems of ordinary linear differential equations (a variant of the dispersive method)

Абрамов, А. А.

doi:10.1016/0041-5553(63)90156-3

Cited by 34 publications

(56 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Since we will return to this equation and its variants repeatedly, let us recall several results from [5,6]. Remark 2.…”

Section: Preliminariesmentioning

confidence: 98%

See 1 more Smart Citation

Transfer of boundary conditions for DAEsof index 1

Balla¹,

März²

1996

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

In this paper, the concept of Abramov's method for transferring boundary conditions posed for regular ordinary differential equations (ODEs) is applied to index-1 differential algebraic equations (DAEs). Having discussed the reduction of inhomogeneous problems to homogeneous ones and analyzed the underlying ideas of Abramov's method, we consider boundary value problems for index-1 linear DAEs both with constant and varying leading matrices. We describe the relations defining the subspaces of solutions satisfying the prescribed boundary conditions at one end of the interval. The index-1 DAEs which realize the transfer are given and their properties are studied. The results are reformulated for inhomogeneous index-1 DAEs as well.

show abstract

“…Since we will return to this equation and its variants repeatedly, let us recall several results from [5,6]. Remark 2.…”

Section: Preliminariesmentioning

confidence: 98%

“…Each method has its own advantages and disadvantages. In [5], the following choice is proposed. Take the (matrix) solution of the problem…”

Section: Preliminariesmentioning

confidence: 99%

Transfer of boundary conditions for DAEsof index 1

Balla¹,

März²

1996

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…Choosing the columns of the identity matrix of order as the vector , we transfer the first condition in (2.4) over the interval for all choices of . These transfers can be performed simultaneously if, for instance, the orthogonal transfer algorithm is used (see [3]). In this process, the additional calculations caused by using vectors for the same are small compared to the compu tational costs of transferring a single condition.…”

Section: 2mentioning

confidence: 99%

“…Suppose that, for transferring the boundary conditions, the method proposed in [3] is used. As shown in [4], the numerical stability of problem (1.1), (2.4) ensures that the resulting system of linear algebraic equations (2.7) is not ill conditioned and can be solved in a numerically stable way.…”

Section: 2mentioning

confidence: 99%

Nonlinear eigenvalue problem for a system of ordinary differential equations subject to a nonlocal condition

Абрамов

Yukhno

2012

Comput. Math. and Math. Phys.

View full text Add to dashboard Cite

For a system of linear ordinary differential equations supplemented with a nonlocal condi tion specified by the Stieltjes integral, the problem of calculating the eigenvalues belonging to a given bounded domain in the complex plane is examined. It is assumed that the coefficient matrix of the sys tem and the matrix function in the Stieltjes integral are analytic functions of the spectral parameter. A numerically stable method for solving this problem is proposed and justified. It is based on the use of an auxiliary boundary value problem and formulas of the argument principle type. The problem of calculating the corresponding eigenfunctions is also treated.

show abstract

“…Of special interest are problems on cylindrical shells of varying curvature. Such problems are solved using various analytic and numerical methods [1,9,10,24,[29][30][31][32]. The natural vibrations of a semi-infinite momentless cylindrical shell damped with distance from its free edge are examined in [5,12,16,17].…”

mentioning

confidence: 99%

Vibrations of a corrugated orthotropic cylindrical shell with free edges

Gulgazaryan

2006

Int Appl Mech

View full text Add to dashboard Cite

The natural vibrations of a corrugated elastic orthotropic cylindrical shell with a directrix perpendicular to its edges that are free are examined Keywords: corrugated elastic orthotropic cylindrical shell, natural vibrations, momentless stress state, absence of bending stiffness Introduction. Many important research and technology problems involve study of the vibrations of and wave propagation in deformed solid media. These are in particular problems of seismic prospecting, aircraft construction, shipbuilding, instrument making, dynamics of power-engineering structures, etc. Rayleigh's work [28] was the first to study elastic surface waves. He discovered elastic waves propagating along the free boundary of a half-space and having amplitude quickly decreasing with depth. Such waves observed in elastic bodies of various geometries are commonly referred to as Rayleigh waves. Waves localized near the free boundary of a semi-infinite plate and waves propagating along a semi-infinite cylindrical shell and decaying with distance from the free end are of Rayleigh type too [7]. In a semi-infinite plate, according to a Kirchhoff-Love hypothesis, there exist independent planar and bending waves localized along the free boundary [2,3,6,21,22]. If the plate is curved, these two waves appear coupled, giving rise to new two (preferential bending and preferential tangential) oscillations localized at the edge [13][14][15]. Of special interest are problems on cylindrical shells of varying curvature. Such problems are solved using various analytic and numerical methods [1,9,10,24,[29][30][31][32]. The natural vibrations of a semi-infinite momentless cylindrical shell damped with distance from its free edge are examined in [5,12,16,17]. The natural vibrations of a cantilever corrugated orthotropic momentless cylindrical shell are studied in [18].The present paper is concerned with the natural vibrations of a corrugated (and closed) orthotropic momentless cylindrical shell with free edges. It is assumed that the generatrices are perpendicular to the edges and the squared curvature of the directrix can be expanded into a series: R k r r km km m m m -= ¥ = + + ae è ç ç ö ø ÷ ÷ å 2 2 0 1 2 / cos sin b r b ,where b is the directed arc length of the directrix. Note that the class of cylindrical shells under consideration includes closed cylindrical shells with free edges and arbitrary smooth directrices. In this case, k n s = 2 0 p / , where s is the total length of the directrix, n N 0 Î . We have derived dispersion equations and established asymptotic relationships between these equations and the dispersion equations for a shell structure consisting of countably identical open orthotropic circular cylindrical shells and an orthotropic strip plate and the dispersion equations for a corrugated semi-infinite orthotropic momentless cylindrical shell with free edge. We will present approximate values of the dimensionless natural frequencies and damping factors for cylindrical shells having lengths l = 5 and l = 15, directrices y a cx b c...

show abstract

On the transfer of boundary conditions for systems of ordinary linear differential equations (a variant of the dispersive method)

Cited by 34 publications

References 0 publications

Transfer of boundary conditions for DAEsof index 1

Transfer of boundary conditions for DAEsof index 1

Nonlinear eigenvalue problem for a system of ordinary differential equations subject to a nonlocal condition

Vibrations of a corrugated orthotropic cylindrical shell with free edges

Contact Info

Product

Resources

About