Nonlinear spectral problem for a self-adjoint vector differential equation

2017 ◽  
Vol 53 (7) ◽  
pp. 900-907
Author(s):  
A. A. Abramov ◽  
L. F. Yukhno
Keyword(s):  
Differential Equation ◽  
Spectral Problem ◽  
Nonlinear Spectral Problem ◽  
Vector Differential Equation

scholarly journals Spherical curves and quadratic relationships for special functions

1985 ◽  
Vol 27 (1) ◽  
pp. 111-124
Author(s):  
Even Mehlum ◽  
Jet Wimp
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Position Vector ◽  
Third Order ◽  
Generalized Hypergeometric Functions ◽  
Transcendental Functions ◽  
Horn Functions ◽  
Vector Differential Equation

AbstractWe show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.

scholarly journals Trapped modes around freely floating bodies in a two-layer fluid channel

2014 ◽  
Vol 470 (2170) ◽  
pp. 20140396 ◽  
Author(s):  
Filipe S. Cal ◽  
Gonçalo A. S. Dias ◽  
Juha H. Videman
Keyword(s):  
Spectral Problem ◽  
Water Waves ◽  
Adjoint Operator ◽  
Trapped Modes ◽  
Floating Bodies ◽  
Floating Structures ◽  
Nonlinear Spectral Problem ◽  
Time Harmonic ◽  
Fluid Channel ◽  
Elimination Scheme

Unlike the trapping of time-harmonic water waves by fixed obstacles, the oscillation of freely floating structures gives rise to a complex nonlinear spectral problem. Still, through a convenient elimination scheme the system simplifies to a linear spectral problem for a self-adjoint operator in a Hilbert space. Under symmetry assumptions on the geometry of the fluid domain, we present conditions guaranteeing the existence of trapped modes in a two-layer fluid channel. Numerous examples of floating bodies supporting trapped modes are given.

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