Xu Xue-zi, Chen Huai-yong. Application of One-Parameter Groups of Transformation in Mechanics[J]. Applied Mathematics and Mechanics, 1990, 11(7): 636-642.
Citation: Xu Xue-zi, Chen Huai-yong. Application of One-Parameter Groups of Transformation in Mechanics[J]. Applied Mathematics and Mechanics, 1990, 11(7): 636-642.

Application of One-Parameter Groups of Transformation in Mechanics

  • Received Date: 1989-08-17
  • Publish Date: 1990-07-15
  • In this paper,including some partial differential equations with a number of independent variables,which can he reduced by the infinitesimal form of the group,we obtain the theory of similarity transformation and its application of the second order nonlinear partial differential equations which have two independent variables and two dependent variables in mechanics.
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  • [1]
    Birkhoff,G.,Hydrodynamics,2nd Ed,Princeton University Press(1960).
    Michal,A.D.,Differential invariants and invariant partial differential equations under continuous transformation groups in normed linear spaces,Proc.Nat.Acad.Sci.,37(1951),623-627.
    Morgan,A.J.A.,The reduction by one of the number of independent variables in some systems of partial differential equations.Quart.J.Math.,Oxford,Ser.2(1952),250-259.
    Hansen,A.G.,Similarity Analyses of Boundary Value Problems in Engineering,Prentice-Hall(1964).
    Ames,W.F.,Similarity for the nonlinear diffusion equation,Ind.Eng.Chem.Fundam.,4,1(1965),72-76.
    Lee,S.Y.and W.F.Ames,Similarity solutions for non-Newtonian fluids.A.I.Ch.E.J.,12.44(1966),700-708.
    Na,T.Y.and A.G.Hansen,Possible similarity solutions of the laminar natural convection flow of non-Newtonian fluids,Int.J.Heat Mass Transfer,9(1966),261-626.
    Na.Y.T.and A.G.Hansen,Similarity solutions of a class of laminar three-dimensional boundary layer equations of power law fluids,Int.J.Non-Linear Mech.,2(1967),373-385.
    Gabbert,C.H.,Similarity for unsteady compressible boundary layers,AIAA J.,5,6(1967),1198-1200.
    Moran.M.J.and R.A.Gaggioli,Reduction of the number of variables in systems of partial differential equations,with auxiliary conditions,SIAM J.Appl.Math.,16(1968),202-215.
    Hansen,A.G.and T.Y.Na,Similarity solutions of laminar,incompressible boundary layer equations of non-Newtonian fluids,J.Basic Eng(1968),71-74.
    Moran,M.J.and R.A.Gaggioli,Similarity analyses via group theory,AIAA J.,6,10(1968),2014-2016.
    Ames.W.F.,Nonlinear partial differential equations in Engineering.Vol.Ⅱ(1972).
    Bluman,G.W and J.D.Cole,Similarity methods for differential equations,Springer-Verlag.New York(1974).
    Dresner,L.Sanilarity Solutions Nonlinear Partial Differential Equations.Pitman(1983).
    Frydryehowicv,W.and M.C.Singh,Group theoretic and similarity analysis of hyperbolic partial differential equations,J.Math.Anal.Appl.,114(1986),75-99.
    Iimol,M.G.and N.L.Kalthia,Similarity solutions of three-dimensional boundary layer equations of non-Newtonian fluids,Int.J.Nonlinear Mech.,21,6(1986),475-481.
    Donato,A.,Similarity analysis and non-linear wave propagation,Int.J.Non-Linear Mech.,22.4(1987),307-314.
    Bluman,G.W.and J.D.Cole,The general similarity solution of the heat equation,J.Math.Mech.,18,11(1969),1025-1042.
    Nariboli,G.A.Self-similar solutions of some nonlinear equations,Appl.Sci.Res.,22(1970),449-461.
    Bluman,G.W.,Slimlarity solutions of the one-dimensional Fokker-Planck equation,Int.J.Non-Linear Mech.,6(1971),143-153.
    Bluman,G.W.and J.D.Cole,Similarity Methods for Differential Equations,Springer,Berlin(1974).
    Bluman,G.W.Applications of the general similarity solution of the heat equation to boundary-value problems Quart.Appl.Math.,(1974),403-415.
    Shen,H.and W.F.Ames,On invariant solutions of the Korteweg-de Vries equation,Phys.Lett.,49A(1974),313-314.
    Mayer,Humi,Invariant solutions for a class of diffusion equations,J.Math.Phys.,18,6(1977),1705-1708.
    Lakshmanan,M.and P.Kaliappan,On the invariant solutions of the Korteweg-de Vries-Burgers equations.Phys.Lett.,71A,2/3(1979),166-168.
    Seshadri(edmonton),R.and M.C.Singh(calgary),Similarity analysis of wave propagation problems in nonlinear rods,Arch.Mech.,32,b(1980),933-945.
    Logan,J.D.and J.D,J.Perey,Similarity solutions for reactive shock hydrodynamics,SIAM.J.Appl.Math.,39,3(1980),512-527.
    Thien,N.P.,A method to obtain some similarity solutions to the generalized Newtonian fluid,ZAMP,32(1981).609-615.
    Lioyd,S.P.,The infinitesimal group of the Navier-Stokes equations,Acta Mech.,38(1981),85-98.
    Ames,W.F.and R J.Lohner,Group properties of Ua=[ƒ(u)ux]x,Int.J.Non-linear Mech.,16,5/6(1981),439-447.
    Hill.J.M.,Solution of Differential Equations by Means of One-Parameter Group pitman(1982).
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