Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A table of solutions of the one-dimensional Burgers equation


Authors: Edward R. Benton and George W. Platzman
Journal: Quart. Appl. Math. 30 (1972), 195-212
MSC: Primary 35Q99
DOI: https://doi.org/10.1090/qam/306736
MathSciNet review: 306736
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Abstract: The literature relating to the one-dimensional Burgers equation is surveyed. About thirty-five distinct solutions of this equation are classified in tabular form. The physically interesting cases are illustrated by means of isochronal graphs.


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  • [1] W. F. Ames, Nonlinear partial differential equations in engineering, Academic Press, New York-London, 1965. MR 0210342
  • [2] Edwin D. Banta, Lossless propagation of one-dimensional, finite amplitude sound waves, J. Math. Anal. Appl. 10 (1965), 116–173. MR 0170575, https://doi.org/10.1016/0022-247X(65)90153-8
  • [3] J. Bass, Les fonctions pseudo-aléatoires, Mémor. Sci. Math., Fasc. CLIII, Gauthier-Villars, Éditeur-Imprimeur-Libraire, Paris, 1962 (French). MR 0147847
  • [4] H. Bateman, Some recent researches on the motion of fluids, Mon. Weather Rev. 43, 163-170 (1915)
  • [5] R. Bellman, S. P. Azen, and J. M. Richardson, On new and direct computational approaches to some mathematical models of turbulence, Quart. Appl. Math. 23 (1965), 55–67. MR 0178672, https://doi.org/10.1090/S0033-569X-1965-0178672-3
  • [6] E. R. Benton, Some new exact, viscous, nonsteady solutions of Burgers' equation, Phys. Fluids 9, 1247-48 (1966)
  • [7] -, Solutions illustrating the decay of dissipation layers in Burgers' nonlinear diffusion equation, Phys. Fluids 10, 2113-19 (1967)
  • [8] D. T. Blackstock, Thermoviscous attenuation of plane, periodic, finite-amplitude sound waves, J. Acoust. Soc. Amer. 36, 534-542 (1964)
  • [9] David T. Blackstock, Connection between the Fay and Fubini solutions for plane sound waves of finite amplitude, J. Acoust. Soc. Amer. 39 (1966), 1019–1026. MR 0198801, https://doi.org/10.1121/1.1909986
  • [10] J. M. Burgers, Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion, Trans. Roy. Neth. Acad. Sci. Amsterdam, 17, 1-53 (1939)
  • [11] J. M. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Nederl. Akad. Wetensch., Proc. 43 (1940), 2–12. MR 0001147
  • [12] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, Academic Press, Inc., New York, N. Y., 1948, pp. 171–199. edited by Richard von Mises and Theodore von Kármán,. MR 0027195
  • [13] J. M. Burgers, The formation of vortex sheets in a simplified type of turbulent motion, Nederl. Akad. Wetensch., Proc. 53 (1950), 122–133. MR 0034671
  • [14] J. M. Burgers, Correlation problems in a one-dimensional model of turbulence. I, Nederl. Akad. Wetensch., Proc. 53 (1950), 247–260. MR 0035581
  • [15] -, Statistical problems connected with the solution of a simple non-linear partial differential equation, Proc. Roy. Neth. Acad. Sci. Amsterdam B57, 45-72, 159-169, 403-433 (1954)
  • [16] -, An approximate equation for the correlation function connected with a non-linear problem, in Proceedings of the eighth international congress for applied mechanics (University of Istanbul, Turkey) 2, 89-103 (1955)
  • [17] -, A model for one-dimensional compressible turbulence with two sets of characteristics, Proc. Roy. Neth. Acad. Sci. Amsterdam B58, 1-18 (1955)
  • [18] J. M. Burgers, Statistical problems connected with the solution of a nonlinear partial differential equation, Nonlinear Problems of Engineering, Academic Press, New York, 1964, pp. 123–137. MR 0175430
  • [19] -, Functions and integrals connected with solutions of the diffusion or heat flow equation, Tech. Note BN-398, The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, 96 pages (1965)
  • [20] Chong-wei Chu, A class of reducible systems of quasi-linear partial differential equations, Quart. Appl. Math. 23 (1965), 275–278. MR 0186937, https://doi.org/10.1090/S0033-569X-1965-0186937-X
  • [21] Julian D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225–236. MR 0042889, https://doi.org/10.1090/S0033-569X-1951-42889-X
  • [22] D. H. Cooper, Integrated treatment of tracing and tracking error, J. Audio Eng. Soc. 12, 2-7 (1964)
  • [23] S. Crow and G. Canavan, Relationship between a Wiener-Hermite expansion and an energy cascade, J. Fluid Mech. 41, 387-403 (1970)
  • [24] R. D. Fay, Plane sound waves of finite amplitude, J. Acoust. Soc. Amer. 3,222-241 (1931)
  • [25] E. Fubini-Ghiron, Anomalie nella propagazione di onde acustiche di grande ampiezza, Alta Frequenza 4, 530-581 (1935)
  • [26] A. Giorgini, A numerical experiment on a turbulence model, in Developments in mechanics, Johnson Publishing Co., 1968, pp. 1379-1408
  • [27] Z. A. Goldberg, Finite-amplitude waves in magnetohydrodynamics, Soviet Physics JETP 15, 179-181 (1962)
  • [28] L. E. Hargrove, Fourier series for the finite amplitude sound waveform in a dissipationless medium, J. Acoust. Soc. Amer. 32, 511-512 (1960)
  • [29] H. S. Tsien, The equations of gas dynamics. Fundamentals of gas dynamics. Vol. 3. High speed aerodynamics and jet propulsion, Princeton University Press, Princeton, N.J., 1958. MR 0097212
    L. Crocco, One-dimensional treatment of steady gas dynamics. Fundamentals of gas dynamics. Vol 3, Princeton University Press, Princeton, N.J., 1958. MR 0097213
    A. Kantrowitz, One-dimensional treatment of non-steady dynamics. Fundamentals of gas dynamics. Vol 3, Princeton University Press, Princeton, N.J., 1958. MR 0097214
    Wallace D. Hayes, The basic theory of gasdynamic discontinuities. Fundamentals of gas dynamics. Vol 3, Princeton University Press, Princeton, N.J., 1958. MR 0097215
    H. Polachek and R. J. Seeger, Shock wave interactions. Fundamentals of gas dynamics. Vol 3, Princeton University Press, Princeton, N.J., 1958. MR 0097216
    H. Guyford Stever, Condensation phenomena in high speed flows. Fundamentals of gas dynamics. Vol 3, Princeton University Press, Princeton, N.J., 1958. MR 0097217
    Th. von Kármán, H. W. Emmons, Geoffrey Taylor, and R. S. Tankin, Gas dynamics of combustion and detonation. Fundamentals of gas dynamics. Vol 3, Princeton University Press, Princeton, N.J., 1958. MR 0097218
    S. A. Schaaf and P. L. Chambré, Flow of rarefied gases. Fundamentals of gas dynamics. Vol 3, Princeton University Press, Princeton, N.J., 1958. MR 0097219
  • [30] Eberhard Hopf, The partial differential equation 𝑢_{𝑡}+𝑢𝑢ₓ=𝜇𝑢ₓₓ, Comm. Pure Appl. Math. 3 (1950), 201–230. MR 0047234, https://doi.org/10.1002/cpa.3160030302
  • [31] I. Hosokawa and K. Yamamoto, Numerical study of the Burgers model of turbulence based on the characteristic functional formalism, Phys. Fluids 13, 1683-1692 (1970)
  • [32] D. T. Jeng, R. Foerster, S. Haaland and W. C. Meecham, Statistical initial-value problem for Burgers' model equation of turbulence, Phys. Fluids 9, 2114-2120 (1966)
  • [33] W. Kahng and A. Siegel, The Cameron-Martin-Wiener method in turbulence and in Burgers’ model: General formulae, and application to late decay, J. Fluid Mech. 41 (1970), 593–618. MR 0278617, https://doi.org/10.1017/S0022112070000770
  • [34] Winfield Keck and Robert T. Beyer, Frequency spectrum of finite amplitude ultrasonic waves in liquids., Phys. Fluids 3 (1960), 346–352. MR 0121003, https://doi.org/10.1063/1.1706039
  • [35] N. N. Kočina, On periodic solutions of Burgers’ equation, J. Appl. Math. Mech. 25 (1962), 1597–1607. MR 0185286, https://doi.org/10.1016/0021-8928(62)90138-7
  • [36] R. H. Kraichnan, Langrangian-history statistical theory for Burgers' equation, Phys. Fluids 11, 265-277 (1968)
  • [37] Martin D. Kruskal and Norman J. Zabusky, Stroboscopic-perturbation procedure for treating a class of nonlinear wave equations, J. Mathematical Phys. 5 (1964), 231–244. MR 0161029, https://doi.org/10.1063/1.1704113
  • [38] Paco A. Lagerstrom, Julian D. Cole, and Leon Trilling, Problems in the Theory of Viscous Compressible Fluids, California Institute of Technology, Pasadena, California, 1949. MR 0041617
  • [39] W. Lick, The propagation of disturbances on glaciers, J. Geophys. Res. 75, 2189-2197 (1970)
  • [40] M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, Surveys in mechanics, Cambridge, at the University Press, 1956, pp. 250–351 (2 plates). MR 0077346
  • [41] William C. Meecham and Armand Siegel, Wiener-Hermite expansion in model turbulence at large Reynolds numbers, Phys. Fluids 7 (1964), 1178–1190. MR 0167097, https://doi.org/10.1063/1.1711359
  • [42] -, and M.-Y. Su, Prediction of equilibrium properties for nearly normal model turbulence, Phys. Fluids 12, 1582-1591 (1968)
  • [43] J. S. Mendousse, Nonlinear dissipative distortion of progressive sound waves at moderate amplitude, J. Acoust. Soc. Amer. 25, 51-54 (1953)
  • [44] C. N. K. Mooers, Gerstner wave's Fourier decomposition and related identities, J. Geophys. Res. 73, 5843-5847 (1968)
  • [45] Y. Ogura, A note on the energy transfer in Burgers' model of turbulence, 75th Anniversary Volume of the Journal of the Meteorological Society of Japan, 1957, pp. 92-94
  • [46] S. A. Orszag and L. R. Bissonnette, Dynamical properties of truncated Wiener-Hermite expansions, Phys. Fluids 10, 2603-2613 (1967)
  • [47] G. W. Platzman, An exact integral of complete spectral equations for unsteady one-dimensional flow, Tellus 16, 422-431 (1964)
  • [48] L. A. Pospelov, Propagation of finite-amplitude elastic waves, Soviet Physics Acoust. 11, 302-304 (1966)
  • [49] W. H. Reid, On the transfer of energy in Burgers’ model of turbulence, Appl. Sci. Res. A. 6 (1956), 85–91. MR 0083324
  • [50] E. Y. Rodin, Propagation of waves of finite amplitude in thermoviscous media, NASA CR-643, 82 pages(1966)
  • [51] Ervin Y. Rodin, A Riccati solution for Burgers’ equation, Quart. Appl. Math. 27 (1969/1970), 541–545. MR 0259394, https://doi.org/10.1090/S0033-569X-1970-0259394-3
  • [52] Ervin Y. Rodin, On some approximate and exact solutions of boundary value problems for Burgers’ equation, J. Math. Anal. Appl. 30 (1970), 401–414. MR 0257586, https://doi.org/10.1016/0022-247X(70)90171-X
  • [53] I. Rudnick, On the attenuation of a repeated sawtooth shock wave, J. Acoust. Soc. Amer. 25, 1012-1013 (1953)
  • [54] P. G. Saffman, Lectures on homogeneous turbulence, in Topics in nonlinear physics (N. J. Zabusky, editor), Springer-Verlag New York, 1968, pp. 485-614
  • [55] M. E. Shvets and V. P. Meleshko, Numerical algorithm of a solution of the system of equations of hydrodynamics of the atmosphere, Izv. Acad. Sci. USSR Atmospher. Ocean. Phys. 1, 519-520 (1965)
  • [56] A. Siegel, T. Imamura and W. C. Meecham, Wiener-Hermite functional expansion in turbulence with the Burgers model, Phys. Fluids 6, 1519-1521 (1963)
  • [57] Armand Siegel, Tsutomu Imamura, and William C. Meecham, Wiener-Hermite expansion in model turbulence in the late decay stage, J. Mathematical Phys. 6 (1965), 707–721. MR 0175174, https://doi.org/10.1063/1.1704328
  • [58] S. I. Soluyan and R. V. Khokhlov, Finite amplitude acoustic waves in a relaxing medium, Soviet Physics Acoust. 8 (1962), 170–175. MR 0153253
  • [59] C. H. Su and C. S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Mathematical Phys. 10 (1969), 536–539. MR 0271526, https://doi.org/10.1063/1.1664873
  • [60] T. Tatsumi, Nonlinear wave expansion for turbulence in the Burgers model of a fluid, Phys. Fluids 12 (Part II), II 258-II 264 (1969)
  • [61] G. I. Taylor, The conditions necessary for discontinuous motion in gases, Proc. Roy. Soc. A84, 371-377 (1910)
  • [62] B. van der Pol, On a non-linear partial differential equation satisfied by the logarithm of the Jacobean theta-functions, with arithmetical applications, Proc. Acad. Sci. Amsterdam A13, 261-284 (1951)
  • [63] Robert Aubrey Walsh, INITIAL VALUE PROBLEMS ASSOCIATED WITH BURGERS’ EQUATION, ProQuest LLC, Ann Arbor, MI, 1968. Thesis (S.C.D.C.)–Washington University in St. Louis. MR 2617602
  • [64] J. J. Walton, Integration of the Lagrangian-history approximation to Burgers' equation, Phys. Fluids 13, 1634-1635 (1970)
  • [65] Peter J. Westervelt, The mean pressure and velocity in a plane acoustic wave in a gas, J. Acoust. Soc. Amer. 22 (1950), 319–327. MR 0036644, https://doi.org/10.1121/1.1906606
  • [66] N. J. Zabusky, Phenomena associated with the oscillations of a nonlinear model string: The problem of Fermi, Pasta, and Ulam, in Proceedings of the conference on mathematical models in the physical sciences (S. Drobot, editor), Prentice Hall, New York, 1963, pp. 99-133
  • [67] P. A. Blythe, Non-linear wave propagation in a relaxing gas, J. Fluid Mech. 37, 31-50 (1969)
  • [68] J. G. Jones, On the near-equilibrium and near-frozen regions in an expansion wave in a relaxing gas, J. Fluid Mech. 19 (1964), 81–102. MR 0163556, https://doi.org/10.1017/S0022112064000556
  • [69] W. Lick, Nonlinear wave propagation in fluids, in Annual reviews of fluid mechanics 2, (M. van Dyke, W. G. Vincenti, and J. V. Wehausen, editors), Annual Reviews Inc., Palo Alto, California, 1970, pp. 113-136
  • [70] J. P. Moran and S. F. Shen, On the formation of weak plane shock waves by impulsive motion of a piston, J. Fluid Mech. 25, 705-718 (1966)
  • [71] M. Morduchow and A. J. Paullay, Stability of normal shock waves with viscosity and heat conduction, Phys. Fluids 14, 323-331 (1971)

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DOI: https://doi.org/10.1090/qam/306736
Article copyright: © Copyright 1972 American Mathematical Society


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