• Sachdev* P.L.
  • Srinivasa Rao Ch.
Part of the Springer Monographs in Mathematics book series (SMM)


Nonlinear partial differential equations (PDEs) do not, in general, admit exact solutions; these solutions are even more rare when initial/boundary conditions are imposed. There are exceptional circumstances when the PDEs enjoy certain symmetries: they are invariant to a class of finite or infinitesimal transformations (Sachdev (2000). When this is the case, the PDEs are exactly reducible to ordinary differential equations (ODEs) if they are functions of two independent variables; the ODEs may occasionally be integrated in a closed form. Alternatively, one may study their qualitative properties and obtain the actual solutions numerically with reference to appropriate initial/boundary conditions. These solutions are called and belong to one of the two classes, first kind and second kind (Zel’dovich 1956, Zel’dovich and Raizer 1967, Barenblatt and Zel’dovich 1972, Sachdev 2000), and solve some degenerate problems for which ‘all, or at least some, of the constant parameters in the initial and boundary conditions of the problem, having the dimensionality of independent variables, tend to zero or infinity.’ These solutions describe those properties of the phenomena that do not depend on the details of the initial and boundary conditions; they do involve some nondimensional parameters which, in some integral sense, represent the memory of initial/boundary conditions. Exceptionally, there may not be any nondimensional parameter of the problem in the asymptotic solution (Barenblatt and Zel’dovich 1972). These special solutions do not describe equilibrium states; they describe intermediate stages when the process of evolution of the solution is continuing and yet the details of initial/boundary conditions have already disappeared. These solutions satisfy some singular, delta functions like, initial conditions.


Travel Wave Solution Blast Wave Burger Equation Nonlinear Partial Differential Equation Nonlinear PDEs 
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  1. 1.
    Barenblatt, G. I., Zel’dovich, Ya. B. (1971) Intermediate asymptotics in mathematical physics, Russian Math. Surveys 26, 45–61.CrossRefGoogle Scholar
  2. 2.
    Barenblatt, G. I., Zel’dovich, Ya. B. (1972) Self-similar solutions as intermediate asymptotics, Ann. Rev. Fluid Mech. 4, 285–312.CrossRefGoogle Scholar
  3. 3.
    Bluman, G. W., Kumei, S. (1989) Symmetries and Differential Equations, Springer-Verlag, New York.MATHGoogle Scholar
  4. 4.
    Clarkson, P. A., Kruskal, M. D. (1989) New similarity reductions of the Boussinesq equation, J. Math. Phys. 30, 2201–2213.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Grundy, R. E. (1988) Large time solution of the Cauchy problem for the generalized Burgers equation, Preprint, University of St. Andrews, UK.Google Scholar
  6. 6.
    Grundy, R. E., Sachdev, P. L., Dawson, C. N. (1994a) Large time solution of an initial value problem for a generalised Burgers equation, in Nonlinear Diffusion Phenomenon, P. L. Sachdev and R. E. Grundy (Eds.), 68–83, Narosa, New Delhi.Google Scholar
  7. 7.
    Grundy, R. E., Van Duijn, C. J., Dawson, C. N. (1994b) Asymptotic profiles with finite mass in one-dimensional contaminant transport through porous media: the fast reaction case, Quart. J. Mech. Appl. Math. 47, 69–106.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kanel’, Ya. I. (1962) On the stability of solutions of the Cauchy problem for equations occuring in the theory of combustion, Mat. Sb. 59(101), 245–288.MathSciNetGoogle Scholar
  9. 9.
    Kolmogorov, A. N., Petrovskii, I. G., Piskunov, N. S. (1937) Investigation of the diffusion equation connected with an increasing amount of matter and its application to a biological problem, Bull. MGU A1(6), 1–26.Google Scholar
  10. 10.
    Oleinik, O. A. (1966) Stability of solutions of a system of boundary layer equations for a nonsteady flow of incompressible fluid, Prikl. Mat. Mekh. 30, 417–423.MathSciNetGoogle Scholar
  11. 11.
    Peletier, L. A. (1970) Asymptotic behaviour of temperature profiles of a class of non-linear heat conduction problems, Quart. J. Mech. Appl. Math. 23, 441–447.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Peletier, L. A. (1971) Asymptotic behaviour of solutions of the porous media equation, SIAM J. Appl. Math. 21, 542–551.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Peletier, L. A. (1972) On the asymptotic behaviour of velocity profiles in laminar boundary layers, Arch. Rat. Mech. Anal. 45, 110–119.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Philip, J. R. (1957) The theory of infiltration: 2. The profile of infinity, Soil Sci. 83, 435–448.CrossRefGoogle Scholar
  15. 15.
    Philip, J. R. (1974) Recent progress in the solution of nonlinear diffusion equations, Soil Sci. 117, 257–264.CrossRefGoogle Scholar
  16. 16.
    Sachdev, P. L. (1987) Nonlinear Diffusive Waves, Cambridge University Press, Cambridge, UK.MATHCrossRefGoogle Scholar
  17. 17.
    Sachdev, P. L. (2000) Self-Similarity and Beyond–Exact Solutions of Nonlinear Problems, Chapman & Hall/ CRC Press, New York.MATHCrossRefGoogle Scholar
  18. 18.
    Sachdev, P. L. (2004) Shock Waves and Explosions, Chapman & Hall/ CRC Press, New York.MATHCrossRefGoogle Scholar
  19. 19.
    Sachdev, P. L., Joseph, K. T. (1994) Exact representations of N-wave solutions of generalized Burgers equations, in Nonlinear Diffusion Phenomenon, P. L. Sachdev and R. E. Grundy (Eds.), 197–219, Narosa, New Delhi.Google Scholar
  20. 20.
    Sachdev, P. L., Srinivasa Rao, Ch. (2000) N-wave solution of modified Burgers equation, Appl. Math. Lett. 13, 1–6.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sachdev, P. L., Joseph, K. T., Nair, K. R. C. (1994) Exact N-wave solutions for the nonplanar Burgers equation, Proc. Roy. Soc. London Ser. A 445, 501–517.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Sachdev, P. L., Joseph, K. T., Mayil Vaganan, B. (1996) Exact N-wave solutions of generalized Burgers equations, Stud. Appl. Math. 97, 349–367.MATHMathSciNetGoogle Scholar
  23. 23.
    Sachdev, P. L., Srinivasa Rao, Ch., Enflo, B. O. (2005) Large-time asymptotics for periodic solutions of the modified Burgers equation, Stud. Appl. Math. 114, 307–323.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sachdev, P. L., Srinivasa Rao, Ch., Joseph, K. T. (1999) Analytic and numerical study of N-waves governed by the nonplanar Burgers equation, Stud. Appl. Math. 103, 89–120.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Sedov, L. I. (1946) Propagation of intense blast waves, Prikl. Mat. Mekh. 10, 241–250.Google Scholar
  26. 26.
    Serrin, J. (1967) Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory, Proc. Royal. Soc. London Ser. A 299, 491–507.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Taylor, G. I. (1950) The formation of a blast wave by a very intense explosion I, Proc. Roy. Soc. London Ser. A 201, 159–174.CrossRefGoogle Scholar
  28. 28.
    Zel’dovich, Ya. B. (1956) The motion of a gas under the action of a short term pressure shock, Akust. Zh. 2, 28–38, (Sov. Phys. Acoustics 2, 25–35).Google Scholar
  29. 29.
    Zel’dovich, Ya. B., Frank-Kamenetskii, D. A. (1938) Theory of uniform propagation of flames, Doklady USSR Ac. Sci. 19, 693–697.Google Scholar
  30. 30.
    Zel’dovich, Ya. B., Raizer, Yu. P. (1967) Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. 2, Academic Press, New York.Google Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathmeticsIndian Institute of Technology MadrasChennaiIndia

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