Multiple expansions in series of eigen- and associated functions of boundary eigenvalue problems nonlinearly dependent on the spectral parameter; Part I

1990 ◽  
Vol 17 (1-2) ◽  
pp. 83-99 ◽  
Author(s):  
Gerhard Freiling
Keyword(s):  
Spectral Parameter ◽  
Eigenvalue Problems ◽  
Associated Functions ◽  
In Series ◽  
Boundary Eigenvalue

General boundary eigenvalue problems for systems of differential equations with multiple roots of the characteristic equation

1988 ◽  
Vol 108 (1-2) ◽  
pp. 45-56 ◽  
Author(s):  
Gerhard Freiling
Keyword(s):  
Differential Equations ◽  
Characteristic Equation ◽  
Eigenvalue Problems ◽  
Necessary Conditions ◽  
General Boundary ◽  
Multiple Roots ◽  
Systems Of Differential Equations ◽  
Associated Functions ◽  
Boundary Eigenvalue

SynopsisWe consider a class of non-self adjoint multipoint eigenvalue problems. Using necessary conditions for the regularity of these problems, we obtain a theorem on the expansion of certain functions into a series of eigen- and associated functions.

Bifurcation of Eigenvalues of Nonselfadjoint Differential Operators in Nonconservative Stability Problems

2002 ◽  
Author(s):  
Oleg N. Kirillov ◽  
Alexander P. Seyranian
Keyword(s):  
Parameter Space ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Initial Point ◽  
Potential Force ◽  
Arbitrary Length ◽  
Eigen Values ◽  
Associated Functions ◽  
Linear Differential ◽  
Eigenfunctions And Associated Functions

In the present paper eigenvalue problems for non-selfadjoint linear differential operators smoothly dependent on a vector of real parameters are considered. Bifurcation of eigenvalues along smooth curves in the parameter space is studied. The case of multipleeigen value with Keldysh chain of arbitrary length is considered. Explicit expressions describing bifurcation of eigen-values are found. The obtained formulae use eigenfunctions and associated functions of the adjoint eigenvalue problems as well as the derivatives of the differential operator taken at the initial point of the parameter space. These results are important for the stability theory, sensitivity analysis and structural optimization. As a mechanical application the extended Beck’s problem of stability of an elastic column under action of potential force and tangential follower force is considered and discussed in detail.

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