Abstract
Hyperbolic models to describe the wave motion in continuous media are considered. Degenerated hyperbolic models as mathematical approximations are constructed. Some hyperbolic models as extensions of the corresponding parabolic models, as well as some new hyperbolic models are presented.
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References
L. Bento, Transverse waves in a relativistic rigid body, Int.J. Theor. Phys., 24 (6) (1985), 653–657.
L. Bers, J. Fritz and M. Schechter, Partial differential equations, Interscience, New York — London — Sydney, 1964.
C. Cattaneo, Sur one forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, Compt. Rend. S. Acad. Sci., 247 (4) (1958), 431–433.
R. Courant, Methods of mathematical physics, I (1935), II (1962), Interscience, New York.
R.W. Davies, The connection between the Smoluchowsky equation and the KramersChandrasekhar equation, Phys. Review, 93 (6) (1954), 1169–1171.
B.I. Davydov, Diffusion equation with taking into account of molecular velocity,(in Russian), Doklady Akad. Nauk. USSR, 1935, 2 (7), (1935), 474–478.
K.S. Eckhoff, On dispersion for linear waves in nonuniform media, SIAM J. Appl. Math., 44 (6) (1984), 1092–1105.
J. Engelbrecht, On the finite velocity of wave motion modelled by nonlinear evolution equations, Proc. of the 2nd Int. Conf. on Nonlinear Hyperbolic Problems. In: Notes on Numerical Fluid Mechanics, Eds J.Ballmann and R.Jeltsch, Vieweg, Braunschweig, 24 (1989), 115–127.
D. Fusco and N. Manganaro, Nonlinear wave features of a hyperbolic model describing dissipative magnetofluid dynamics, J. of Theor. and Appl. Mech., 6 (6), (1987), 761–770.
J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Paris, 1932 (Russian translation).
R. Hersh, Boundary conditions for equations of evolution. Archive Ration. Mech. and Analysis, 16 (4) (1964), 243–264.
D.D. JosephPreziosi L. Heat waves. Rewievs of Modern Physics„ 61 (1) (1989), 41–73.
B.G. Kuznetsov, Hyperbolic modification of Navier-Stokes equations, (in Russian, translated in USA), Applied Mechanics and Engineering Physics, 6 (1993), 133–141.
S.V. Korsunsky and I.T. Selezov, Some wave models of dynamics of electrically conducting fluid, (in Russian). Hydroaeromechanics and the Theory of Elasticity. Dnepropetrovsk, Dneprop. University, 1991,34–40.
P.K. Kythe, Fundamental solutions for differential operators and applications, Birkhäuser Boston, 1996.
J.L. Leander, On the relation between the wavefront speed and the group velocity concept, J. Acoust. Soc. Amer., 100 (6) (1996), 3503–3507.
J.C. Maxwell, On the dynamical theory of gases,Phil. Trans. Roy. Soc., 157 (1967), 49–89.
C. Misokhata, The theory of partial differential equations, 1965, Russian translated edition, 1977.
Ph.M. Morse and H. Feshbach, Methods of theoretical physics, Part 1., McGraw-Hill Book Company, Inc., New York, Toronto, London, 1953.
P. Perzyna and A. Drabik, Analysis of the fundamental equations describing thermoplastic flow process in solid body, Arch. Mech., 43 (2–3) (1991), 287–296.
I.T. Selezov, Conception of hyperbolicity in the theory of control dynamic systems, (in Russian). In: Cybernetics and Computational Engineering. Kiev, Institute of Cybernetics, Ukr. Acad. Sci., 1 (1969), 131–137.
I.T. Selezov, Wave hydraulic models as mathematical approximations, Proc. the 22nd Congress of IAHR. Techn. Session B., 1987, 301–306.
I.T. Selezov, Modelling wave and diffraction processes in continuous media, Kiev, Naukova Dumka, 1989, 204 pp.
I.T. Selezov, Hyperbolic models of wave propagation in bars, plates and shells. Mechanics of Solids, Translated from Izvestia Akademii Nauk Rossii. Meckanika tverdogo tela., 2 (1994), 64–77.
H.E. Wilhelm and S.H. Hong, Stress relaxation waves in fluids, Physical Rewiev, A, 22 (3) (1980), 1266–1271.
M. Zlamal, The parabolic equation as a limiting case of hyperbolic and elliptic equations, Proc. Conf. Prague: Publishing House of the Czhechoslovak Acad. Sci., 1963, 243–247.
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Selezov, I. (1999). Some Hyperbolic Models for Wave Propagation. In: Jeltsch, R., Fey, M. (eds) Hyperbolic Problems: Theory, Numerics, Applications. International Series of Numerical Mathematics, vol 130. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8724-3_34
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DOI: https://doi.org/10.1007/978-3-0348-8724-3_34
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