A linear integro-differential equation for viscoelastic rods and plates
Author:
Kenneth B. Hannsgen
Journal:
Quart. Appl. Math. 41 (1983), 75-83
MSC:
Primary 45K05; Secondary 73F99
DOI:
https://doi.org/10.1090/qam/700662
MathSciNet review:
700662
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Abstract: It is proved that the resolvent kernel of a certain integrodifferential equation in Hilbert space is absolutely integrable on $\left ( {0,\infty } \right )$. The equation arises in the linear theory of isotropic viscoelastic rods and plates.
- D. R. Bland, The theory of linear viscoelasticity, International Series of Monographs on Pure and Applied Mathematics, Vol. 10, Pergamon Press, New York-London-Oxford-Paris, 1960. MR 0110314
- Ralph W. Carr and Kenneth B. Hannsgen, A nonhomogeneous integro-differential equation in Hilbert space, SIAM J. Math. Anal. 10 (1979), no. 5, 961–984. MR 541094, DOI https://doi.org/10.1137/0510089
- Ralph W. Carr and Kenneth B. Hannsgen, Resolvent formulas for a Volterra equation in Hilbert space, SIAM J. Math. Anal. 13 (1982), no. 3, 459–483. MR 653467, DOI https://doi.org/10.1137/0513032
- Kenneth B. Hannsgen, Indirect abelian theorems and a linear Volterra equation, Trans. Amer. Math. Soc. 142 (1969), 539–555. MR 246058, DOI https://doi.org/10.1090/S0002-9947-1969-0246058-1
- Kenneth B. Hannsgen, Uniform $L^{1}$ behavior for an integrodifferential equation with parameter, SIAM J. Math. Anal. 8 (1977), no. 4, 626–639. MR 463848, DOI https://doi.org/10.1137/0508050
- J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations, Advances in Math. 22 (1976), no. 3, 278–304. MR 500024, DOI https://doi.org/10.1016/0001-8708%2876%2990096-7
A. C. Pipkin, Lectures on viscoelasticity theory, Springer-Verlag, Heidelberg, 1972
- Daniel F. Shea and Stephen Wainger, Variants of the Wiener-Lévy theorem, with applications to stability problems for some Volterra integral equations, Amer. J. Math. 97 (1975), 312–343. MR 372521, DOI https://doi.org/10.2307/2373715
D. R. Bland, The theory of linear viscoelasticity, Pergamon Press, New York, 1960
R. W. Carr and K. B. Hannsgen, A nonhomogeneous integrodifferential equation in Hilbert space, SIAM J. Math. Anal. 10, 961–984 (1979)
R. W. Carr and K. B. Hannsgen, Resolvent formulas for a Volterra equation in Hilbert space, SIAM J. Math. Anal., 13, 459–483 (1982).
K. B. Hannsgen, Indirect abelian theorems and a linear Volterra equation, Trans. Amer. Math. Soc. 142, 539–555 (1969)
K. B. Hannsgen, Uniform L$^{1}$ behavior for an integrodifferential equation with parameter, SIAM J. Math. Anal. 8, 626–639 (1977)
J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations, Advances in Math. 22, 278–304 (1976)
A. C. Pipkin, Lectures on viscoelasticity theory, Springer-Verlag, Heidelberg, 1972
D. F. Shea and S. Wainger, Variants of the Wiener-Lévy theorem, with applications to stability problems for some Volterra integral equations, Amer. J. Math. 97, 312–343 (1975)
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Article copyright:
© Copyright 1983
American Mathematical Society
