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.
J. D. Murray, Journal of Mathematics and Physics 47, 111–133 (1968)
- D. G. Crighton and J. F. Scott, Asymptotic solutions of model equations in nonlinear acoustics, Philos. Trans. Roy. Soc. London Ser. A 292 (1979), no. 1389, 101–134. MR 547939, DOI https://doi.org/10.1098/rsta.1979.0046
- R. W. Lardner and J. C. Arya, Two generalisations of Burgers’ equation, Acta Mech. 37 (1980), no. 3-4, 179–190 (English, with German summary). MR 586056, DOI https://doi.org/10.1007/BF01202942
- J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. MR 608029
- Robin W. Lardner, Higher order shock structure for a class of generalized Burgers’ equations, Arabian J. Sci. Engrg. 9 (1984), no. 2, 109–117 (English, with Arabic summary). MR 765097
- Eberhard Hopf, The partial differential equation $u_t+uu_x=\mu u_{xx}$, Comm. Pure Appl. Math. 3 (1950), 201–230. MR 47234, DOI https://doi.org/10.1002/cpa.3160030302
- Julian D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225–236. MR 42889, DOI https://doi.org/10.1090/S0033-569X-1951-42889-X
D. T. Blackstock, J. Acoust. Soc. Amer. 26, 534–542 (1964)
- 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
- A. Jeffrey and T. Taniuti, Non-linear wave propagation. With applications to physics and magnetohydrodynamics, Academic Press, New York-London, 1964. MR 0167137
- G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
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