Third-order solutions of Burgers’ equation
Author:
R. W. Lardner
Journal:
Quart. Appl. Math. 44 (1986), 293-301
MSC:
Primary 35Q20; Secondary 76L05
DOI:
https://doi.org/10.1090/qam/856182
MathSciNet review:
856182
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Abstract: Burgers’ equation with small dissipation coefficient and general initial conditions is considered. The first three terms are calculated in both the inner (i.e., close to the shock) and outer (away from the shock) expansions. It is shown that these two expansions can be matched and that this third-order matching essentially completes the determination of the secònd-order inner solution but leaves an undetermined function in the third-order solution. It is also shown that the second-order inner solution can be determined completely, without use of the third-order inner solution, by use of an integral conservation property.
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J. D. Murray, Journal of Mathematics and Physics 47, 111–133 (1968)
D. G. Crighton and J. F. Scott, Phil. Trans. Roy. Soc. 292, 103–134 (1979)
J. C. Arya and R. W. Lardner, Acta Mech. 37, 179–190 (1980)
J. Kevorkian and J. D. Cole, Perturbation methods in applied mathematics, Springer-Verlag, Berlin, pp. 356–362 (1981)
R. W. Lardner, Arabian J. Sci. Engng. 9, 109–117 (1984)
E. Hopf, Commun. Pure Appl. Math. 3, 201–230 (1950)
J. D. Cole, Quart. Appl. Math. 9, 225–236 (1951)
D. T. Blackstock, J. Acoust. Soc. Amer. 26, 534–542 (1964)
M. J. Lighthill, Surveys in mechanics (ed. by G. K. Batchelor and R. M. Davies) Cambridge Univ. Press, 250–351 (1956)
A. Jeffrey and T. Taniuti, Nonlinear wave propagation, Academic Press, New York, 1964
G. B. Whitham, Linear and nonlinear waves, John Wiley, New York (1974)
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Article copyright:
© Copyright 1986
American Mathematical Society
