Partial Differential Equations

  • G. W. Bluman
  • J. D. Cole
Part of the Applied Mathematical Sciences book series (AMS, volume 13)


The role of Lie theory in constructing solutions to partial differential equations differs essentially from its role for solving ordinary differential equations.(1) Invariance under a one-parameter Lie group of transformations reduces by one the number of variables appearing in a partial differential equation rather than the order as is the case for an ordinary differential equation. The method is not an encompassing as in Part 1 since it leads to particular (similarity) solutions and not to the “general” solution of a given partial differential equation. Thus boundary conditions play an important role in the applications of Lie theory to partial differential equations.


Partial Differential Equation Fundamental Solution Heat Equation Dimensional Analysis Similarity Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Lie, Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen, Arch, for Math., Vol. VI, No. 3, Kristiana, 1881, p. 328.Google Scholar
  2. 2.
    L. V. Ovsjannikov, Gruppovye Svoystva Differentsialny Uravneni, Novosibirsk, 1962. (Group Properties of Differential Equations, translated by G. Bluman, 1967).Google Scholar
  3. 3.
    E. A. Müller and K. Matschat, Über das Auffinden von Ähnlichkeitslosungen partieller Differentialgleichungssysteme unter Benutzung von Transformationsgruppen, mit Anwendungen auf Probleme der Strömungsphysik, Miszellaneen der Angewandten Mechanik, Berlin, 1962, p. 190.Google Scholar
  4. 4.
    P. W. Bridgman, Dimensional Analysis, 2nd Ed., Yale University Press, 1931.Google Scholar
  5. 5.
    L. I. Sedov, Similarity and Dimensional Methods, Moscow, 6th Ed. (in Russian); English translation, 4th Ed., Academic Press, (1959) .zbMATHGoogle Scholar
  6. 6.
    G. Birkhoff, Hydrodynamics, 2nd Ed., Princeton University Press, 1960.zbMATHGoogle Scholar
  7. 7.
    G. I. Barenblatt and Ya. B. Zelfdovich, Self-similar solutions are intermediate asymptotics, Ann. Rev. of Fluid Mech., 1972.Google Scholar
  8. 8.
    G. W. Bluman, Construction of Solutions to Partial Differential Equations by the Use of Transformation Groups, Ph.D. thesis, California Institute of Technology, 1967.Google Scholar
  9. 9.
    G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation, J. of Math., and Mech., Vol. 18, No. 11, May, 1969, pp. 1025–1042.MathSciNetzbMATHGoogle Scholar
  10. 10.
    G. W. Bluman, Applications of the general similarity solution of the heat equation to boundary-value problems, Quart, of Appl. Math., Vol. 31, No. 4, January 1974, pp. 403–415.MathSciNetzbMATHGoogle Scholar
  11. 11.
    G. W. Bluman, Similarity solutions of the one-dimensional Fokker-Planck equation, Int. J. Non-lin. Mech., 6, pp. 143–153, 1971.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    L. L. Ovsjannikov, Gruppovye Svoystva Uravnenya Nelinaynoy Teploprovodnosty, Dok. Akad. Nauk, CCCP, 1959, 125, 3, p. 492.MathSciNetGoogle Scholar
  13. 13.
    W. F. Ames, Nonlinear Partial Differential Equations in Engineering, Academic Press, 1972, Volume II, Chapter 2.zbMATHGoogle Scholar
  14. 14.
    A. J. A. Morgan, The reduction by one of the number of independent variables in some systems of partial differential equations, Quart. J. of Math., 3, Ser. 2, 1952, pp. 250–259.zbMATHCrossRefGoogle Scholar
  15. 15.
    A. G. Hansen, Similarity Analyses of Boundary Value Problems in Engineering, Prentice-Hall, 1964.zbMATHGoogle Scholar
  16. 16.
    G. Rosen and G. W. Ullrich, Invariance group of the equation \(\frac{{\partial u}} {{\partial t}} = - \mathop u\limits_{-} \cdot \nabla \mathop u\limits_{-}\), SIAM J. on A. Ma., May, 1973, Vol. 24, No. 3, pp. 286–288.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 16.
    J. D. Cole and J. Aroesty, Hypersonic similarity solutions for airfoils supporting exponential shock waves, A.I.A.A. Journal, 8, February, 1970, No. 2, pp. 308–315.zbMATHGoogle Scholar
  18. 18.
    P. Germain, Écoulements Transsoniques Homogènes, Progress in Aeronautical Sciences, Vol. 5, Pergamon Press, 1964, pp. 143–273.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1974

Authors and Affiliations

  • G. W. Bluman
    • 1
  • J. D. Cole
    • 2
  1. 1.Department of MathematicsThe University of British ColumbiaVancouverCanada
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

Personalised recommendations