Analytic solutions of the vector Burgers’ equation
Authors:
Steven Nerney, Edward J. Schmahl and Z. E. Musielak
Journal:
Quart. Appl. Math. 54 (1996), 63-71
MSC:
Primary 35Q53; Secondary 35C99, 35K55
DOI:
https://doi.org/10.1090/qam/1373838
MathSciNet review:
MR1373838
Full-text PDF Free Access
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Abstract: The well-known analytical solution of Burgers’ equation is extended to curvilinear coordinate systems in three dimensions by a method that is much simpler and more suitable to practical applications than that previously used [22], The results obtained are applied to incompressible flow with cylindrical symmetry, and also to the decay of an initially linearly increasing wind.
H. Bateman, Some recent researches on the motion of fluids, Monthly Weather Rev. 43, 163 (1915)
E. R. Benton, Solutions illustrating the decay of dissipation layers in Burgers’ nonlinear diffusion equation, Phys. Fluids 10, 2113 (1967)
E. R. Benton, A table of solutions of the one-dimensional Burgers equation, Quart. Appl. Math. 30, 195–212 (1972)
J. M. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. Roy. Neth. Acad. Sci. Amsterdam 43, 2–12 (1940)
J. M. Burgers, The Non-Linear Diffusion Equation, D. Reidel Pub. Co., Dordrecht-Holland, 1974
J. I. Castor, D. C. Abbott, and R. I. Klein, Radiatively-driven winds in Of stars, Astrophys. J. 195, 157 (1975)
J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9, 225–236 (1951)
D. B. Friend and D. C. Abbott, The theory of radiatively-driven stellar winds. III. Wind models with finite disk correction and rotation, Astrophys. J. 311, 701 (1986)
Z. A. Gol’berg, Finite amplitude waves in magnetohydrodynamics, JETP 15, 179 (1962)
S. N. Gurbatov, A. I. Saichev, and S. F. Shandarin, The large-scale structure of the universe in the frame of the model of non-linear diffusion, Monthly Notices Roy. Astronom. Soc. 236, 385 (1989)
S. N. Gurbatov, A. N. Malakhov, and A. I. Saichev, Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays, and Particles, Manchester University Press, distributed by St. Martin’s Press, New York, 1991
E. Hopf, The partial differential equation ${u_t} + u{u_x} = \mu {u_{xx}}$, Comm. Pure and Appl. Math. 3, 201–230 (1950)
D. T. Jeng, R. Foerster, S. Haaland, and W. C. Meecham, Statistical initial-value problem for Burgers’ model equation of turbulence, Phys. Fluids 7, 2114 (1966)
J. I. Katz and M. L. Green, A Burgers model of interstellar dynamics, Astronom. and Astrophys. 161 (1986)
L. A. Kofman and S. F. Shandarin, Theory of adhesion for the large-scale structure of the universe, Nature 334, 129 (1988)
L. A. Kofman and A. C. Raga, Modeling structures of knots in jet flows with the Burgers equation, Astrophys. J. 390, 359 (1992)
P. Lagerstrom, J. D. Cole, and L. Trilling, Problems in the Theory of Viscous Compressible Fluids, monograph, California Institute of Technology, 1949
M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics (Batchelor and Davies, eds.), Cambridge Univ. Press, Cambridge, 1956
H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, Von Nostrand Co., London, 1956
B. Van Der Pol, On a non-linear partial differential equation satisfied by the logarithm of the Jacobian theta-functions, with arithmetical applications, Proc. Acad. Sci. Amsterdam A13, 261–271, 272–284 (1951)
G. N. Watson, A Treatise on the Theory of Bessel Functions, second ed., Cambridge Univ. Press, Cambridge, 1962, pp. 393–396
K. B. Wolf, L. Hlavaty, and S. Steinberg, Non-linear differential equations as invariants under group action on coset bundles: Burgers and Korteweg-de Vries equation families, J. Math. Anal. Appl. 114, 340 (1986)
H. Bateman, Some recent researches on the motion of fluids, Monthly Weather Rev. 43, 163 (1915)
E. R. Benton, Solutions illustrating the decay of dissipation layers in Burgers’ nonlinear diffusion equation, Phys. Fluids 10, 2113 (1967)
E. R. Benton, A table of solutions of the one-dimensional Burgers equation, Quart. Appl. Math. 30, 195–212 (1972)
J. M. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. Roy. Neth. Acad. Sci. Amsterdam 43, 2–12 (1940)
J. M. Burgers, The Non-Linear Diffusion Equation, D. Reidel Pub. Co., Dordrecht-Holland, 1974
J. I. Castor, D. C. Abbott, and R. I. Klein, Radiatively-driven winds in Of stars, Astrophys. J. 195, 157 (1975)
J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9, 225–236 (1951)
D. B. Friend and D. C. Abbott, The theory of radiatively-driven stellar winds. III. Wind models with finite disk correction and rotation, Astrophys. J. 311, 701 (1986)
Z. A. Gol’berg, Finite amplitude waves in magnetohydrodynamics, JETP 15, 179 (1962)
S. N. Gurbatov, A. I. Saichev, and S. F. Shandarin, The large-scale structure of the universe in the frame of the model of non-linear diffusion, Monthly Notices Roy. Astronom. Soc. 236, 385 (1989)
S. N. Gurbatov, A. N. Malakhov, and A. I. Saichev, Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays, and Particles, Manchester University Press, distributed by St. Martin’s Press, New York, 1991
E. Hopf, The partial differential equation ${u_t} + u{u_x} = \mu {u_{xx}}$, Comm. Pure and Appl. Math. 3, 201–230 (1950)
D. T. Jeng, R. Foerster, S. Haaland, and W. C. Meecham, Statistical initial-value problem for Burgers’ model equation of turbulence, Phys. Fluids 7, 2114 (1966)
J. I. Katz and M. L. Green, A Burgers model of interstellar dynamics, Astronom. and Astrophys. 161 (1986)
L. A. Kofman and S. F. Shandarin, Theory of adhesion for the large-scale structure of the universe, Nature 334, 129 (1988)
L. A. Kofman and A. C. Raga, Modeling structures of knots in jet flows with the Burgers equation, Astrophys. J. 390, 359 (1992)
P. Lagerstrom, J. D. Cole, and L. Trilling, Problems in the Theory of Viscous Compressible Fluids, monograph, California Institute of Technology, 1949
M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics (Batchelor and Davies, eds.), Cambridge Univ. Press, Cambridge, 1956
H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, Von Nostrand Co., London, 1956
B. Van Der Pol, On a non-linear partial differential equation satisfied by the logarithm of the Jacobian theta-functions, with arithmetical applications, Proc. Acad. Sci. Amsterdam A13, 261–271, 272–284 (1951)
G. N. Watson, A Treatise on the Theory of Bessel Functions, second ed., Cambridge Univ. Press, Cambridge, 1962, pp. 393–396
K. B. Wolf, L. Hlavaty, and S. Steinberg, Non-linear differential equations as invariants under group action on coset bundles: Burgers and Korteweg-de Vries equation families, J. Math. Anal. Appl. 114, 340 (1986)
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© Copyright 1996
American Mathematical Society
