Advertisement

Operator Method for Solution of PDEs Based on Their Symmetries

  • Samuil D. Eidelman
  • Yakov Krasnov
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 157)

Abstract

We touch upon “operator analytic function theory” as the solution of frequent classes of the partial differential equations (PDEs).

Keywords

Linear partial differential equations Second-order constant coefficient PDEs Cauchy problem Explicit solutions Symmetry operator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, II, Comm. Pure Applied Math. 12, (1959), 623–727, 17, (1964), 35-92.Google Scholar
  2. [2]
    A.A. Albert Nonassociative algebras. Ann. of Math. 43, 1942, pp. 685–707Google Scholar
  3. [3]
    G.W. Bluman, J.D. Cole, Similarity methods for differential equations. Applied Mathematical Sciences, 13. Springer-Verlag, New York, 1974, 332 pp.Google Scholar
  4. [4]
    G.W. Bluman, S. Kumei, Symmetries and differential equations. Applied Mathematical Sciences, 81, Springer-Verlag, New York, 1989, 412 pp.Google Scholar
  5. [5]
    F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis. Pitman Research Notes in Math. 76, 1982, 308 pp.Google Scholar
  6. [6]
    J.R. Cannon, The one-dimensional heat equation. Encyclopedia of Math. and its Appl., 23. Addison-Wesley, MA, 1984, 483 pp.Google Scholar
  7. [7]
    A. Coffman, D. Legg, Y. Pan, A Taylor series condition for harmonic extensions. Real Anal. Exchange 28 (2002/03), no. 1, pp. 235–253.Google Scholar
  8. [8]
    M.J. Craddock, A.H. Dooley, Symmetry group methods for heat kernels. J. Math. Phys. 42 (2001), no. 1, pp. 390–418.CrossRefGoogle Scholar
  9. [9]
    L. Ehrenpreis, A fundamental principle for systems of linear equations with constant coefficients. in Proc. Intern. Symp. Linear Spaces, Jerusalem, 1960, pp. 161–174Google Scholar
  10. [10]
    S. Eidelman, Parabolic systems. North-Holland Publishing Company, 1969, 469 pp.Google Scholar
  11. [11]
    R.P. Gilbert, J.L. Buchanan, First Order Elliptic Systems. Mathematics in Science and Engineering; 163, Academic Press, 1983, 281 pp.Google Scholar
  12. [12]
    G.R. Goldstein, J.A. Goldstein, E. Obrecht, Structure of solutions to linear evolution equations: extensions of d’Alembert’s formula. J. Math. Anal. Appl. 201, (1996), no. 2, pp. 461–477.CrossRefGoogle Scholar
  13. [13]
    I.J. Good, A simple generalization of analytic function theory. Exposition. Math, 6, no. 4, 1988, pp. 289–311.Google Scholar
  14. [14]
    G.N. Hile, A. Stanoyevitch, Expansions of solutions of higher order evolution equations in series of generalized heat polynomials. Vol. 2002, No. 64, pp. 1–25.Google Scholar
  15. [15]
    L. Hormander, The analysis of linear partial differential operators II, Springer Verlag, Berlin, 1983.Google Scholar
  16. [16]
    P.W. Ketchum, Analytic functions of hypercomplex variables, Trans. Amer. Mat. Soc., 30, # 4, 1928, pp. 641–667.MathSciNetGoogle Scholar
  17. [17]
    Y. Krasnov, Symmetries of Cauchy-Riemann-Fueter equation, Complex Variables, vol. 41, 2000, pp. 279–292.Google Scholar
  18. [18]
    Y. Krasnov, The structure of monogenic functions. Clifford Algebras and their Applications in Mathematical Physics, vol. 2, Progr. Phys., 19, Birkhäuser, Boston, 2000, pp. 247–272.Google Scholar
  19. [19]
    B. Malgrange, Sur les systèmes différentiels à coefficients constants. (French) 1963 Les Équations aux Dérivées Partielles (Paris, 1962) pp. 113–122Google Scholar
  20. [20]
    W. Miller, Symmetry and Separation of Variables. Encyclopedia of Mathematics and its Applications, Addison-Wesley, 4, 1977, 285 pp.Google Scholar
  21. [21]
    P.J. Olver, Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics 107, Springer, New York, 1993.Google Scholar
  22. [22]
    P.J. Olver, Symmetry and explicit solutions of partial differential equations. Appl. Numer. Math. 10 (1992), no. 3–4, pp. 307–324.CrossRefGoogle Scholar
  23. [23]
    P.J. Olver, V.V. Sokolov, Integrable Evolution Equations on Associative Algebras, Commun. Math. Phys. 193, (1998), pp. 245–268CrossRefGoogle Scholar
  24. [24]
    L.V. Ovsiannikov, Group analysis of differential equations. English translation. Academic Press, Inc., New York-London, 1982. 416 pp.Google Scholar
  25. [25]
    V.P. Palamodov, A remark on the exponential representation of solutions of differential equations with constant coefficients. Mat. Sb. 76(118) 1968, pp. 417–434.Google Scholar
  26. [26]
    P.S. Pedersen, Cauchy’s integral theorem on a finitely generated, real, commutative. and associative algebra. Adv. Math. 131 (1997), no. 2, pp. 344–356.CrossRefGoogle Scholar
  27. [27]
    P.S. Pedersen, Basis for power series solutions to systems of linear, constant coefficient partial differential equations. Adv. Math. 141 (1999), no. 1, pp. 155–166.CrossRefGoogle Scholar
  28. [28]
    I.G. Petrovsky, Partial Differential Equations, CRC Press, Boca Raton, 1996.Google Scholar
  29. [29]
    S.P. Smith, Polynomial solutions to constant coefficient differential equations. Trans. Amer. Math. Soc. 329, (1992), no. 2, pp. 551–569.Google Scholar
  30. [30]
    F. Sommen, N. Van Acker, Monogenic differential operators, Results in Math. Vol. 22, 1992, pp. 781–798.Google Scholar
  31. [31]
    F. Treves, Linear partial differential operators, 1970.Google Scholar
  32. [32]
    I. Vekua, Generalized analytic functions. London. Pergamon, 1962.Google Scholar
  33. [33]
    D.V. Widder Analytic solutions of the heat equation. Duke Math. J. 29, 1962, pp. 497–503.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Samuil D. Eidelman
    • 1
  • Yakov Krasnov
    • 2
  1. 1.Department of MathematicsSolomonov UniversityKievUkraine
  2. 2.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael

Personalised recommendations