Email this article FINITE VELOCITY OF PROPAGATION OF PERTURBATIONS FOR A ONE-DIMENSIONAL WAVE INTEGRO-DIFFERENTIAL EQUATION WITH A FRACTIONAL-EXPONENTIAL MEMORY FUNCTION
- Authors: Georgievskii D.V.1,2, Rautian N.A.1,2
-
Affiliations:
- Lomonosov Moscow State University
- Moscow Center of Fundamental and Applied Mathematics
- Issue: Vol 61, No 4 (2025)
- Pages: 570-576
- Section: INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
- URL: https://ogarev-online.ru/0374-0641/article/view/296242
- DOI: https://doi.org/10.31857/S0374064125040105
- EDN: https://elibrary.ru/HKPIER
- ID: 296242
Cite item
Open Access
Access granted
Subscription Access
Abstract
The paper studies a Volterra integro-differential equation, the main part of which is a one-dimensional wave equation perturbed by an integral operator of the Volterra convolution type (wave equation with memory). The kernel function of the integral operator is a sum of fractional exponential functions (Rabotnov functions) with positive coefficients. The issue of the influence of the integral operator on the velocity of propagation of disturbances in the initial value problem for the wave equation with memory is studied. The Volterra integro-differential equation under study describes oscillations of a one-dimensional viscoelastic rod, as well as the process of heat propagation in media with memory (Gurtin–Pipkin equation).
About the authors
D. V. Georgievskii
Lomonosov Moscow State University; Moscow Center of Fundamental and Applied Mathematics
Email: georgiev@mech.math.msu.su
Russia; Russia
N. A. Rautian
Lomonosov Moscow State University; Moscow Center of Fundamental and Applied Mathematics
Email: nadezhda.rautian@math.msu.ru
Russia; Russia
References
- Amendola, G. Thermodynamics of Materials with Memory. Theory and Applications / G. Amendola, M. Fabrizio, J.M. Golden. — New-York ; Dordrecht ; Heidelberg ; London : Springer, 2012. — 576 p.
- Ильюшин, А.А. Основы математической теории термовязкоупругости / А.А. Ильюшин, Б.Е. Победря. — М. : Наука, 1970. — 280 c.
- Il’yushin, A.A. and Pobedrya, B.E., Osnovy matematicheskoi teorii termovyazkouprugosti (Mathematical Theory of Thermoviscoelasticity), Moscow: Nauka, 1970.
- Работнов, Ю.Н. Элементы наследственной механики твердых тел / Ю.Н. Работнов. — М. : Наука, 1977. — 384 c.
- Rabotnov, Yu.N., Elementy nasledstvennoi mekhaniki tverdykh tel (Elements of Hereditary Mechanics of Solids), Moscow: Nauka, 1977.
- Георгиевский, Д.В. Модели теории вязкоупругости / Д.В. Георгиевский. — М. : Ленанд, 2023. — 144 c.
- Georgievskii, D.V., Modeli teorii vyazkouprugosti (Models of Viscoelasticity Theory). Moscow: Lenand, 2023.
- Gurtin, M.E. General theory of heat conduction with finite wave speed / M.E. Gurtin, A.C. Pipkin // Arch. Rat. Mech. Anal. — 1968. — V. 31. — P. 113–126.
- Власов, В.В. Спектральный анализ функционально-дифференциальных уравнений / В.В. Власов, Н.А. Раутиан. — М. : МАКС Пресс, 2016. — 488 с.
- Vlasov, V.V. and Rautian, N.A., Spektral’nyi analiz funktsional’no-differentsial’nykh uravnenii (Spectral Analysis of Functional Differential Equations), Moscow: MAKS Press, 2016.
- Vlasov, V.V. Investigation of integro-differential equations by methods of spectral theory / V.V. Vlasov, N.A. Rautian // J. Math. Sci. — 2024. — V. 278, № 1. — P. 55–81.
- Раутиан, Н.А. Полугруппы, порождаемые вольтерровыми интегро-дифференциальными уравнениями / Н.А. Раутиан // Дифференц. уравнения. — 2020. — Т. 56, № 9. — С. 1226–1244.
- Rautian, N.A., Semigroups generated by Volterra integro-differential equations, Differ. Equat., 2020, vol. 56, no. 9, pp. 1193–1211.
- Раутиан, Н.А. О свойствах полугрупп, порождаемых вольтерровыми интегро-дифференциальными уравнениями с ядрами, представимыми интегралами Стилтьеса / Н.А. Раутиан // Дифференц. уравнения. — 2021. — Т. 57, № 9. — С. 1255–1272.
- Rautian, N.A., On the properties of semigroups generated by Volterra integro-differential equations with kernels representable by Stieltjes integrals, Differ. Equat., 2021, vol. 57, no. 9, pp. 1231–1248.
- Владимиров, В.С. Уравнения математической физики / В.С. Владимиров. — М. : Наука, 1988. — 512 c.
- Vladimirov, V.S., Uravneniya matematicheskoy fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1988.
- Владимиров, В.С. Обобщенные функции в математической физике / В.С. Владимиров. — М. : Наука, 1979. — 320 c.
- Vladimirov, V.S., Oboshchenniye funkcii v matematicheskoy fizike (Generalized Functions in Mathematical Physics), Moscow: Nauka, 1979.
Supplementary files
