On the spectrum of an integro-differential equation arising in viscoelasticity theory

Шамаев, А. С.; Шумилова, В. В.

doi:10.1007/s10958-012-0712-8

Cited by 5 publications

(3 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this case, the spectrum of problem (2.1), (2.2) is completely real for l ∈ [l 1 , l 2 ] and also contains a complex part for l ∈ (π/ √ ρ 0 M 1 , l 1 ) and l > l 2 [12]. However, keep in mind that that the presence of three positive roots of Eq.…”

Section: Equation (28) Has Only Real Roots Ifmentioning

confidence: 93%

“…Denote by P k the set of roots of the cubic equation (2.8) for fixed k ∈ N. Two cases are possible [12]: (1) Eq. (2.8) has one real root λ 1k and two complex-conjugate roots z 1k ± z 2k i and (2) Eq.…”

Section: Substituting (25) In (23) and (24) We Obtain Formentioning

confidence: 99%

“…Using geometric constructions and the Viéte theorem, it can be shown [12] that independently of the number of positive roots of Eq. (2.11)…”

Section: Equation (28) Has Only Real Roots Ifmentioning

confidence: 99%

See 2 more Smart Citations

Spectrum of one-dimensional oscillations in the combined stratified medium consisting of a viscoelasticmaterial and a viscous compressible fluid

Шамаев¹,

Шумилова²

2013

Fluid Dyn

Self Cite

View full text Add to dashboard Cite

Two averaged (effective) models corresponding to transverse and longitudinal oscillations in a periodic combined stratified medium are constructed. The medium considered consists of alternating layers of an isotropic viscoelastic material and a viscous compressible fluid. The dependence of the qualitative characteristics of the spectra of the averaged models on the linear dimensions of the stratified medium specimen is investigated.Summing the results obtained, we can assert that for l ≤ π/ √ ρ 0 M 1 the spectrum of problem (2.1), (2.2) is the union of three series {λ 1k }, {λ 2k }, and {λ 3k } (k ∈ 1, 2,... ) of real numbers. Now consider the case of l > π/ √ ρ 0 M 1 . Here, the spectrum of problem (2.1), (2.2) is the union of a series {λ 1k } (k ∈ 1, 2,... ) of real numbers and two series {η 2k } and {η 3k } (k ∈ 1, 2,... ) each of which may contain not more than d 0 = [l √ ρ 0 M 1 /π] complex numbers ([ ⋅ ] is the integer part of the corresponding number). Namely, if for the given k the roots of Eq. (2.8) are real, then η 2k = λ 2k and η 3k = λ 3k ; on the FLUID DYNAMICS Vol.

show abstract

Section: Equation (28) Has Only Real Roots Ifmentioning

confidence: 93%

Section: Substituting (25) In (23) and (24) We Obtain Formentioning

confidence: 99%