Abstract
This chapter is devoted to the study of conditions of hyperbolicity for intergro-differential operators of (1.1.34) and (1.1.35) types, that is, conditions under which the operators in question describe finite speed wave propagation. In Sects. 2.1 and 2.2, we deal with the one-dimensional case; Sects. 2.3, 2.4, and 2.5 are devoted to the case of n spatial dimensions.
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Notes
- 1.
Moreover, σj(λ) prove to be holomorphic [2].
- 2.
The expression
$$ {\Delta}^{(1)}\left(f;{\sigma}_1,{\sigma}_2\right)=\frac{f\left({\sigma}_2\right)-f\left({\sigma}_1\right)}{\sigma_2-{\sigma}_1} $$is called the first-order divided difference for the function f(σ) with respect to points σ1 and σ2; the expression
$$ {\varDelta}^{(2)}\;\left(f;{\sigma}_1,{\sigma}_2,{\sigma}_3\right)=\frac{\varDelta^{(1)}\left(f;{\sigma}_2,{\sigma}_3\right)-{\varDelta}^{(1)}\left(f;{\sigma}_1,{\sigma}_2\right)}{\sigma_3-{\sigma}_1} $$is called the second-order divided difference for the function f(σ)with respect to points σ1, σ2, σ3,etc.
References
Vladimirov, V.S.: Distributions in Mathematical Physics, pp. 1–318. Nauka, Moscow (1979)
Marcushevitc, A.l.: Theory of Analytic Functions, vol. 1, pp. 1–486. Nauka, Moscow (1967)
Lions, J.-L. J. Anal. Math. 2, 369, 1952–1953.
Atiah, M.F., Bott, R., Garding, L.: Acta Math. 24, 109 (1970)
Lax, A.: Comm. Pure Appl. Math. 9, 135 (1956)
Münster, M. Rocky Mountain J. of Math. 3, 443 (1978).
Courant, R., Lax, A.: Comm. Pure Appl. Math. 8, 497 (1955)
Lokshin, A.A., Suvorova, J.V.: Mathematical Theory of Wave Propagation in Media with Memory, pp. 1–151. Moscow University Press, Moscow (1982)
Lokshin, A.A.: Trudy Sem. Petrovskogo, vol. 7, p. 148 (1982)
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Lokshin, A.A. (2020). General Hyperbolic Operators with Memory. In: Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity. Springer, Singapore. https://doi.org/10.1007/978-981-15-8578-4_2
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