基于吴方法的偏微分方程对称计算、判定和分类新算法（英文）
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摘要
综述了作者提出的基于吴方法的偏微分方程(PDE)对称计算、判定和分类新算法的主要进展,并以具体算例说明了给出理论和算法的有效性.算法的主要思想是把PDE对称计算、判定和分类问题转化为微分多项式组的特征列集零点分解问题,从而在吴方法框架内统一处理这些问题.这是吴方法在微分领域中的新应用.
A recent development of the algorithms proposed by the authors for symmetry computation,decision and classification of partial differential equations(PDEs)is reported.The algorithms are designed based on Wu′s method(differential characteristic set algorithm),which provide a systematic and unified mechanical algorithm for these problems.This is a new application of Wu′s method in differential fields.
A recent development of the algorithms proposed by the authors for symmetry computation,decision and classification of partial differential equations(PDEs)is reported.The algorithms are designed based on Wu′s method(differential characteristic set algorithm),which provide a systematic and unified mechanical algorithm for these problems.This is a new application of Wu′s method in differential fields.
引文
[1]Olver P J.Applications of Lie Groups to Differential Equations(2nd ed.)[M].New York:Springer-Verlag,1993.
[2]Ovsiannikov L V.Group Analysis of Differential Equations(Translation edited by W F Ames)[M].New York London:Academic Press,1982.
[3]Ibragimov N H.CRC Handbook of Lie Group Analysis of Differential Equations.Vol 3:New trends in theoretical developments and computational methods[M].Boca Raton,Ann Arbor,London,Tokyo:CRC Press,1994.
[4]Temuer Chaolu.An algorithm for the complete symmetry classification of differential equations based on Wu′s method[J].J Eng Math,2010,66:181-199.
[5]Temuer Chaolu.Wu dchar-set algorithm of symmetry vectors for partial differential equations[J].ACTA Mathematica Scientia,1999,19(3):326-332.
[6]Temuer Chaolu.Wu-differential characteristic set method and its applications to symmetries of PDEs and mechanics[D].DaLian:Dalian University of Technology(DUT),1997.
[7]Temuer Chaolu.New algorithm for classical and nonclassical symmetry of a PDE based on Wu′s method[J].Scientia Sinica:Mathematics,2010,40(4):331-348.
[8]Temuer Chaolu.An algorithm for the complete symmetry classification of differential equations based on Wu′s method[J].J Eng Math,2010,66:181-199.
[9]Temuer Chaolu,Bai Y S.A new algorithmic theory for determining and classifying classical and non-classical symmetries of partial differential equations[J].Sci Sin Math,2010,40(4):331-348.
[10]Temuer Chaolu,Bai Yu shan.Differential characteristic set algorithm for complete symmetry classification of partial differential equations[J].Appl.Math.Mech,2009,30(5):556-566.
[11]Temuer Chaolu,Bluman G.An algorithmic method for showing existence of nontrivial nonclassical symmetries of partial differential equations without solving determining equations[J].Math Anal Appl,2014,411:281-296.
[12]Temuer Chaolu.New algorithm for classical and nonclassical symmetry of a PDE based on Wu′s method[J].Sci Sin Math,2010,40(4):331-348.
[13]Clarkson P A,Mansfield E L.Open problems in symmetry analysis,in Geometrical Study of Differential Equations[J].Editors J A Leslie and T Robart Contemporary Mathematics Series,2001,285:195-205.
[14]Bluman G W,Temuer Chaolu.Conservation laws for nonlinear telegraph equations[J].J Math Anal Appl,2005,310:459-476.
[15]Bluman G,Temuer Chaolu.Local and nonlocal symmetries for nonlinear telegraph equations[J].J Math Phys,2005(a),46(023505):1-9.
[16]Bluman G W,Temuer Chaolu.Comparing symmetries and conservation laws of nonlinear telegraph equations[J].J Math Phys,2005,46(073513):1-14.
[17]Bluman G W,Temuer Chaolu,Anco S C.New conservation laws obtained directly from symmetry action on a known conservation law[J].J Math Anal Appl,2006,322(1):233-250.
[18]Temuer Chaolu,EerdunBuhe,XIA Tiecheng.Nonclassical Symmetry Classification of the Wave Equation with a Nonlinear Source Term[J].Chinese Journal of Contemporary Mathematics,2012,33(2):1-10.
[19]Liu Lihua,Temuer Chaolu.Symmetry analysis of modified 2DBurgers vortex equation for unsteady case Appl[J].Math Mech-English edition,2017,38(3):453-468.
[20]Temuer Chaolu,Bai yushan.An algorithm for approximate symmetry of partial differential equations based on the Wu′s method[J].Chinese J Eng Math,2011,28(5):617-622.
[21]Yin shan,Temuer Chaolu.Classical and non-classical symmetry classifications of nonlinear wave equation with dissipation[J].Appl Math Mech-Engl Ed,2015,36(3):365-378.
[22]Bluman G W,Cole J D.The general similarity solution of the heat equation[J].J Math Mech,1969,18:1025-1042.
[23]Wu W T.Mathematics Mechanization,Mathematics and its applications[M].Beijing:Science Press;Dordrecht/Boston/London:Kluwer Academic Publishers,2000.
[24]Wu W T.On the foundation of algebraic differential geometry[J].Sys Sci Math Sci,1989,2:289-312.
[25]Gao X S,Chou S C.A zero structure theorem for a differential parametric system[J].J Symbolic Compu,1993,16:585-595.
[26]Temuer Chaolu,Gao X S.Nearly dchar-set for a differential polynomial system[J].Acta Mathematica Science,2002,45(6):1041-1050.
[27]Temuer Chaolu.An algorithmic theory of reduction of a differential polynomial system[J].Adv Math(China),2003,32(2):208-220.
[2]Ovsiannikov L V.Group Analysis of Differential Equations(Translation edited by W F Ames)[M].New York London:Academic Press,1982.
[3]Ibragimov N H.CRC Handbook of Lie Group Analysis of Differential Equations.Vol 3:New trends in theoretical developments and computational methods[M].Boca Raton,Ann Arbor,London,Tokyo:CRC Press,1994.
[4]Temuer Chaolu.An algorithm for the complete symmetry classification of differential equations based on Wu′s method[J].J Eng Math,2010,66:181-199.
[5]Temuer Chaolu.Wu dchar-set algorithm of symmetry vectors for partial differential equations[J].ACTA Mathematica Scientia,1999,19(3):326-332.
[6]Temuer Chaolu.Wu-differential characteristic set method and its applications to symmetries of PDEs and mechanics[D].DaLian:Dalian University of Technology(DUT),1997.
[7]Temuer Chaolu.New algorithm for classical and nonclassical symmetry of a PDE based on Wu′s method[J].Scientia Sinica:Mathematics,2010,40(4):331-348.
[8]Temuer Chaolu.An algorithm for the complete symmetry classification of differential equations based on Wu′s method[J].J Eng Math,2010,66:181-199.
[9]Temuer Chaolu,Bai Y S.A new algorithmic theory for determining and classifying classical and non-classical symmetries of partial differential equations[J].Sci Sin Math,2010,40(4):331-348.
[10]Temuer Chaolu,Bai Yu shan.Differential characteristic set algorithm for complete symmetry classification of partial differential equations[J].Appl.Math.Mech,2009,30(5):556-566.
[11]Temuer Chaolu,Bluman G.An algorithmic method for showing existence of nontrivial nonclassical symmetries of partial differential equations without solving determining equations[J].Math Anal Appl,2014,411:281-296.
[12]Temuer Chaolu.New algorithm for classical and nonclassical symmetry of a PDE based on Wu′s method[J].Sci Sin Math,2010,40(4):331-348.
[13]Clarkson P A,Mansfield E L.Open problems in symmetry analysis,in Geometrical Study of Differential Equations[J].Editors J A Leslie and T Robart Contemporary Mathematics Series,2001,285:195-205.
[14]Bluman G W,Temuer Chaolu.Conservation laws for nonlinear telegraph equations[J].J Math Anal Appl,2005,310:459-476.
[15]Bluman G,Temuer Chaolu.Local and nonlocal symmetries for nonlinear telegraph equations[J].J Math Phys,2005(a),46(023505):1-9.
[16]Bluman G W,Temuer Chaolu.Comparing symmetries and conservation laws of nonlinear telegraph equations[J].J Math Phys,2005,46(073513):1-14.
[17]Bluman G W,Temuer Chaolu,Anco S C.New conservation laws obtained directly from symmetry action on a known conservation law[J].J Math Anal Appl,2006,322(1):233-250.
[18]Temuer Chaolu,EerdunBuhe,XIA Tiecheng.Nonclassical Symmetry Classification of the Wave Equation with a Nonlinear Source Term[J].Chinese Journal of Contemporary Mathematics,2012,33(2):1-10.
[19]Liu Lihua,Temuer Chaolu.Symmetry analysis of modified 2DBurgers vortex equation for unsteady case Appl[J].Math Mech-English edition,2017,38(3):453-468.
[20]Temuer Chaolu,Bai yushan.An algorithm for approximate symmetry of partial differential equations based on the Wu′s method[J].Chinese J Eng Math,2011,28(5):617-622.
[21]Yin shan,Temuer Chaolu.Classical and non-classical symmetry classifications of nonlinear wave equation with dissipation[J].Appl Math Mech-Engl Ed,2015,36(3):365-378.
[22]Bluman G W,Cole J D.The general similarity solution of the heat equation[J].J Math Mech,1969,18:1025-1042.
[23]Wu W T.Mathematics Mechanization,Mathematics and its applications[M].Beijing:Science Press;Dordrecht/Boston/London:Kluwer Academic Publishers,2000.
[24]Wu W T.On the foundation of algebraic differential geometry[J].Sys Sci Math Sci,1989,2:289-312.
[25]Gao X S,Chou S C.A zero structure theorem for a differential parametric system[J].J Symbolic Compu,1993,16:585-595.
[26]Temuer Chaolu,Gao X S.Nearly dchar-set for a differential polynomial system[J].Acta Mathematica Science,2002,45(6):1041-1050.
[27]Temuer Chaolu.An algorithmic theory of reduction of a differential polynomial system[J].Adv Math(China),2003,32(2):208-220.