George Bluman

  • Position: Professor, UBC
  • Background:
    • B.Sc. (Hon. Math/Physics), UBC (1964)
    • Ph.D. (Applied Math), Caltech (1968)
  • Office: Mathematics Annex 1112
  • E-mail: bluman(at)math(dot)ubc(dot)ca
  • Phone: 604-822-3044     Fax: 604-822-6074

 

Research Interests

Symmetries and differential equations, Conservation laws

Research in Progress

Publications

     Papers

  • Bluman, G, Construction of Solutions to Partial Differential Equations by the Use of Transformation Groups, Ph.D. Thesis, Caltech (1967)
    http://etd.caltech.edu/etd/available/etd-04082005-155822/unrestricted/Bluman_gw_1968.pdf
  • Bluman, G & Cole, J, General similarity solution of the heat equation, J. Math. Mech. 18 (1969), 1025-1042.
    http://www.math.ubc.ca/~bluman/jmm article 1969.pdf
  • Bluman, G, Similarity solutions of the one-dimensional Fokker-Planck equation, Internat. J. Nonlinear Mech., 6 (1971), 143-153.
    IntJ.Nomnline.Mech71.pdf
  • Bluman, G, Applications of the general similarity solution of the heat equation to boundary value problems, Quarterly Appl. Math. XXXI (1974), 403-415.
  • Bluman, G, Use of group methods for relating linear and nonlinear partial differential equations, Proceedings of International Conference on Symmetry,
    Similarity and Group Theoretic Methods in Mechanics, Calgary, 1974, 203-218
  • Milinazzo, F & Bluman, G, Numerical similarity solutions to Stefan Problems, ZAMM 55 (1975), 423-429.
  • Bluman, G & Kumei, S, On the remarkable nonlinear diffusion equation , J. Math. Phys. 21 (1980), 1019-1023.
    JMP1980.pdf
  • Bluman, G, On the transformation of diffusion processes into the Wiener process, SIAM J. Appl. Math., 39 (1980), 238-247.
    SIAM1980.pdf
  • Kumei, S & Bluman, G, When nonlinear differential equations are equivalent to linear differential equations, SIAM J. Appl. Math. 42 (1982), 1157-1173.
    SIAM1982.pdf
  • Bluman, G, Dimensional Analysis, Modelling and Symmetry, Int. J. of Math Ed. in Science and Technology, 14 (1983), 259-272.
  • Bluman, G, On mapping linear partial differential equations to constant coefficient equations, SIAM J. Appl. Math., 43 (1983), 1259-1273.
    SIAM1983.pdf
  • Bluman, G & Gregory, RD, On transformations of the biharmonic equation, Mathematika, 32 (1985), 118-130.
  • Bluman, G & Kumei, S, On invariance properties of the wave equation, J. Math. Phys. 28 (1987), 307-318.
    blumankumeijmp1987.pdf
  • Bluman, G & Tuckwell, H, Techniques for obtaining analytical solutions for Rall's model neuron, J. of Neuroscience Methods 20 (1987), 151-166.
    jneuro.pdf
  • Bluman, G & Kumei, S, Exact solutions for wave equations of two-layered media with smooth transition, J. Math Phys. 29 (1988), 86-96.
    JMP1988aTwo-layered.pdf
  • Bluman, G, Kumei, S & Reid, G, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988), 806-811.
    JMP1988aNewclasses.pdf
  • Bluman, G & Reid, G, New symmetries for ordinary differential equations, IMA J. App. Math., 40 (1988), 87-94.
    IMA_JAM1998.pdf
  • Bluman, G & Kumei S, Use of group analysis in solving overdetermined systems of ordinary differential equations, J. Math. Anal. Appl. 138 (1989), 95-105.
    JMAA1989aOverdetermined.pdf
  • Bluman, G & Reid, G, Sequences of related linear PDE's, J. Math. Anal. Appl. 144 (1989), 565-585.
    JMAA1989bSequences.pdf
  • Bluman, G, Simplifying the form of Lie groups admitted by a given differential equation, J. Math. Anal. Appl. 145 (1990), 52-62.
    JMAA 145 1990 52-62.pdf
  • Bluman, G & Kumei, S, Symmetry-based algorithms to relate partial differential equations. I. Local symmetries, EJAM 1 (1990), 189-216.
  • Bluman, G & Kumei, S, Symmetry-based algorithms to relate partial differential equations. II. Linearization by nonlocal symmetries, EJAM 1 (1990), 217-223.
  • Bluman, G, A reduction algorithm for an ordinary differential equations admitting a solvable Lie group. SIAM J. Appl. Math 50 (1990), 1689-1705.
    SIAM1990aReduction.pdf
  • Bluman, G, Invariant solutions for ordinary differential equations, SIAM J. Appl. Math. 50 (1990), 1706-1715.
    SIAM1990bInvariant.pdf
  • Bluman, G, Potential symmetries, Proceedings of the Annual Seminar of CMS on Lie Theory Differential Equations and Representation Theory, CRM, Montreal (1990), 85-100.
  • Bluman, G, Linearization of PDEs, Springer Lecture Notes in Physics, Vol. 382 (1991) 285-288.
  • Bluman, G, Use and construction of potential symmetries, Math. Comput. Modelling, 8 (1993), 1-14.
    MathComputModel1993.pdf
  • Bluman, G, Potential symmetries and linearization, in "Applications of Analytic and Geometric Methods to Nonlinear Differential Equations", P.A. Clarkson (ed.) Kluwer, (1993), 363-373.
  • Bluman, G, An overview of potential symmetries, in "Exploiting Symmetry in Applied and Numerical Analysis", E.L. Allgower et al. (eds.), AMS Lectures in Applied Mathematics, vol. 29, (1993), 97-109.
  • Bluman, G, Potential symmetries and equivalent conservation laws, in "Modern Group Analysis: advanced analytical and computational methods in mathematical physics," N. H. Ibragimov et al (eds.), Kluwer, (1993), 71-84.
  • Bluman, G & Doran-Wu, P, The use of factors to discover potential systems or linearizations, Acta Applicandae Mathematicae 41 (1995), 21-43.
  • Anco, S & Bluman, G, Derivation of conservation laws from nonlocal symmetries of differential equations, J. Math. Phys. 37 (1996), 2361-2375.
    JMP1996.pdf
  • Bluman, G & Shtelen, V, New classes of Schrodinger equations equivalent to the free particle equation through non-local transformations, J. Phys. A 29 (1996), 4473-4480.
    jphysA96.pdf
  • Bluman, G, Developments in similarity methods related to pioneering work of Julian Cole, in Mathematics is for Solving Problems, S L. P. Cook, V. Roytburd, M. Tulin (eds.), SIAM (1996) 105-118.
  • Anco, S & Bluman, G, Direct construction of conservation laws from field equations, Phys. Rev. Lett. 78 (1997), 2869-2873.
    PhysRevLett97.pdf
  • Anco, S & Bluman, G, Nonlocal symmetries and nonlocal conservation laws of Maxwell's equations, J. Math. Phys. 38 (1997), 3508-3532.
    JMP1997.pdf
  • Anco, S & Bluman, G, Integrating factors and first integrals for ordinary differential equations, EJAM 9 (1998), 245-259.
    EJAM1998.pdf
  • Bluman, G, Cook, L. Pamela, Flaherty, Joe, Kevorkian, J, Malmuth, Norman, O'Malley, Robert, Schwendeman, Donald W & Tulin, Marshall, Julian D. Cole (1925-1999) Notices of the AMS 47 (2000), 466-473.
  • Anco, S & Bluman, G, Direct construction method for conservation laws of partial differential equations.
    Part I: Examples of conservation law classifications, EJAM 13, (2002) 545-566.
    EJAM2002(1).pdf
  • Anco, S & Bluman, G, Direct construction method for conservation laws of partial differential equations.
    Part II: General treatment, EJAM 13 (2002), 567-585.
    EJAM2002(2).pdf
  • Bluman, G & Shtelen,V, Nonlocal transformations of Kolmogorov equations into the Backward heat equation, J. Math. Anal. Appl. 291 (2004), 419-437.
    blumanshtelenjmaa.pdf
  • Bluman, G & Yan, Z, Nonclassical potential solutions of partial differential equations, EJAM 16 (2005), 239-261.
    http://www.math.ubc.ca/~bluman/bluman yan ejam.pdf
  • Bluman, G, Temuerchaolu & Sahadevan, R, Local and nonlocal symmetries for nonlinear telegraph equations, J. Math. Phys 46 (2005), 023505 (12 journal pages)
    BTSJMP2005.pdf
  • Bluman, G & Temeuerchaolu, Conservation laws for nonlinear telegraph equations, J. Math. Anal Appl. 310 (2005), 459-476.
    BTjmaa2005.pdf
  • Bluman, G & Temuerchaolu, Comparing symmetries and conservation laws of nonlinear telegraph equations, J. Math. Phys. 46 (2005), 073513 (14 journal pages)
    BTjmp2005.pdf
  • Bluman, G, Connections between symmetries and conservation laws., SIGMA, 1 (2005), Paper 011, 16 pages.
    http://www.emis.de/journals/SIGMA/2005/Paper011/sigma05-011.pdf
  • Bluman, G & Cheviakov,A, Framework for potential systems and nonlocal symmetries: Algorithmic approach, J. Math. Phys. 46(2005), 123506 (19 journal pages)
    bcjmp2005.pdf
  • Bluman, G, Temuerchaolu & Anco, S, New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl. 322 (2006), 233-250. BTAjmaa2006.pdf
  • Bluman, G, Cheviakov, A & Ivanova, N, Framework for nonlocally related partial differential equations systems and nonlocal symmetries: Extension, simplification, and examples, J. Math. Phys. 47 (2006) 113505 (23 journal pages)
    BCIJMP2006.pdf
  • Bluman, G & Cheviakov, A, Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation. . Math. Anal. Appl. 333 (2007) 93-111.
    Bcjmaa2007.pdf
  • Bluman, G, Cheviakov, A & Senthilvelan, M, Solution and asymptotic/blow-up behaviour of a class of nonlinear dissipative systems J. Math. Anal. Appl. 339 (2008) 1199-1209. BCSjmaa2008.pdf
  • Anco, S, Bluman, G & Wolf, T, Invertible mappings of nonlinear PDEs to linear PDEs through admitted conservation laws, Acta Appl. Math 101 (2008) 21-38.
    ACTA2008.pdf
  • Bluman, G, Nonlocal extensions of similarity methods, J. Nonlin. Math. Phys. 15 (2008) 1-24.
    JNLMP2008.pdf
  • Bluman, G, Cheviakov, A & Ganghoffer, J-F, Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics, J. Eng. Math. 62 (2008) 203-221.
    JEngMath2008.pdf
  • Bluman, G, Cheviakov, A & Ganghoffer, J-F, On the nonlocal symmetries, group invariant solutions and conservation laws of the equations of nonlinear dynamical compressible elasticitiy. Proceedings of IUTAM Symposium on Progress in the Theory and Numerics of Configurational Mechanics, P. Steinmann (ed.), Springer, (2009), 107-120.
    IUTAM2009.pdf
  • Bluman, G, Cheviakov, A & Anco, S, Construction of conservation laws: how the direct method generalizes Noether’s theorem, Proceedings of the Fourth International Workshop “Group Analysis of Differential Equations and Integrable Systems”, (2010) pp. 13-35.
    cyprus proceedings paper.pdf
  • Bluman, G., Broadbridge, P., King, J.R, and Ward, M.J., Similarity: generalizations, applications and open problems, Journal of Engineering Mathematics, 66 (2010), 1-9.
    JEngMath6620101-9.pdf
  • Cheviakov, A & Bluman, G., On locally and nonlocally related potential systems, J. Math. Phys. 51 (2010) 073502 (23 journal pages)
    jmp512010073502.pdf
  • Cheviakov, A and Bluman, G, Multidimensional partial differential equations systems: Generating new systems via conservation laws, potentials, gauges, subsystems. J. Math. Phys. 51 (2010) 103521 (26 journal pages)
    jmp512010103521.pdf
  • Cheviakov, A and Bluman, G, Multidimensional partial differential equations systems: Nonlocal symmetries, nonlocal conservation laws, exact solutions. J. Math. Phys. 51 (2010) 103522 (26 journal pages)
    jmp512010103522.pdf
  • Bluman, G and Ganghoffer, J-F. Connecting Euler and Lagrange systems as nonlocally related system of dynamical nonlinear elasticity. Arch. Mech. 63 (2011), 363-382.
    bluman-ganghoffer3.pdf
  • Bluman, G and Dridi, R. New solutions for ordinary differential equations, Journal of Symbolic Computation 47 (2012), 76-88.
    article30JSC dridi.pdf
  • Adamuti-Trache, M, Bluman, G, and Tiedje, T. Student Performance in first year university Physics and Mathematics courses, International Journal of Science Education 35 (2013), 2905-2927.
    int journal of science education November 2013.pdf
  • Bluman, G, Tian, Shou-fu, Yang, ZZ, Nonclassical analysis of the nonlinear Kompaneets equation, Journal of Engineering Mathematics, online July 3, 2012, 11pp.
    nonlinear kompaneets eqn.pdf
  • Bihlo, Alexander and Bluman, G. Conservative parametrization schemes. J. Math. Phys. 54 (2013) 083101, 24 pp.
    conservarive parametrization schemes.pdf
  • Bluman, G and Yang, ZZ. A symmetry-based method for constructing nonlocally related PDE systems. J. Math. Phys. 54 (2013) 093504, 22 pp.
    symmetry based method to construct nonlocally related systems.pdf
  • Temuer Chaolu and Bluman, G, An algorithmic method for showing existence of nontrivial nonclassical symmetries of partial differential equations without solving determining equations, J. Math. Anal. Appl. 411 (2014), 281-296.
    algorithmic method for existence of nonclassical symmetries.pdf

     Books

  • Group Properties of Differential Equations (translation with numerous corrections of book in Russian by LV Ovsiannikov), 1967.
  • Bluman, G & Cole, J, Similarity Methods for Differential Equations, Springer-Verlag New York, Heidelberg, Berlin, 1974, 332 pp (Vol. 13, Appl. Math. Sci.).
  • Bluman, G, Problem Book for First Year Calculus, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984, 384 pp, clothbound; 1985, paperback.
  • Bluman, G & Maskall, D, Problems for Grade 12 Mathematics Students Useful as Preparation for the Euclid Mathematics Contest and/or First Year Calculus. Dept. of Math., UBC, 1985, 113 pp (revised,1986).
  • Bluman, G & Kumei, S, Symmetries and Differential Equations, Springer-Verlag New York, Heidelberg, Berlin, 1989, 412 pp (Vol. 81, Appl. Math. Sci; reprinted with corrections, 1996); Chinese edition, 1991.
  • Bluman, G & Anco, S, Symmetry and Integration Methods for Differential Equations, Springer, New York, 2002, 420 pp (Vol. 154, Appl. Math. Sci; Chinese edition, 2004; Chinese edition in English (see www.wpcbj.com.cn) 2004; Chinese translation (see www.sciencep.com) 2009
  • Bluman, G, Cheviakov, A & Anco, S, Applications of Symmetry Methods to Partial Differential Equations, 417pp. Springer, New York, Vo. 168, Appl. Math. Sci. 2010.

     Book Reviews

  • "Symposium Transsonicum II (IUTAM)", Canadian Aeronautics and Space Journal, 24 (1978), 57-58.
  • "Group Invariance in Engineering Boundary Value Problems", by R. Seshadri and T.Y. Na, SIAM Review, 28 (1986), 248-249.
  • "Application of Lie Groups to Differential Equations", by Peter Olver, Vol. 107 of GTM, AMS Bulletin 18 (1988), 73-78.
  • "Nonlinear Boundary Value Problems in Science and Engineering" by C. Rogers and W.F. Ames, SIAM Review 33 (1991), 294-296.
  • "Differential Equations and Group Methods" by J. M. Hill, SIAM Review 36 (1994), 117.

Courses

Curriculum Vitae