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The basis property of eigenfunctions in the problem of a nonhomogeneous damped string

Rzepnicki Ł.

Opuscula Mathematica

2017

Vol. 37, no. 1

141--165

The equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator iA. We prove that the set of root vectors of the operator A forms a basis of subspaces in a certain Hilbert space H. Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator A is a Riesz basis for H.

Generating the exponentially stable C0-semigroup in a nonhomogeneous string equation with damping at the end

Rzepnicki Ł.

Opuscula Mathematica

2013

Vol. 33, no. 1

151-162

Small vibrations of a nonhomogeneous string of length one with left end fixed and right one moving with damping are described by the one-dimensional wave equation [formula] where ρ is the density of the string and h is a complex parameter. This equation can be rewritten in an operator form as an abstract Cauchy problem for the closed, densely defined operator B acting on a certain energy space H. It is proven that the operator B generates the exponentially stable C0-semigroup of contractions in the space H under assumptions that Re h > 0 and the density function is of bounded variation satisfying 0 < m ≤ ρ(x) for a.e. x ∈ [0; 1].