偏微分方程的精确解及Taylor级数解
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摘要
本文以计算机代数和导师张鸿庆教授的“AC=BD”理论为工具，以构造机械化算法为目的，以源于物理，力学，光学等领域中的非线性问题所对应的非线性偏微分代数方程(组)为研究对象，研究了它们的一些问题，如精确解(孤子解，周期解)，Riccati方程展开法，微分代数及Taylor级数解。
第一章介绍了孤立子理论，计算机代数，数学机械化等学科的起源和发展，以及国内外学者在这些方面所做的工作和一些所取得的成就。
第二章以“AC=BD”的理论模式为指导，考虑了非线性偏微分方程(组)的精确解的构造，给出了“AC=BD”理论的基本思想，C-D可积理论在微分方程求解中的应用，然后通过具体的变换给出了构造C-D对的算法。
第三章基于非线性发展方程求解，代数化，算法化，机械化的指导思想，运用吴方法和符号计算为工具，考虑了非线性发展方程精确解的构造，提出了射影Riccati方程展开法，并将其应用到求解二维广义Burgers方程及耦合MKdV-KdV方程中。
第四章介绍了微分代数的基础知识，并讨论了偏微分代数方程的Taylor级数解。在特征集的基础上，讨论了其参数导数构成的状况，并给出算法。
In this dissertation, the nonlinear partial differential algebraic equation or equations(PDAE or PDAEs) related to some nonlinear topics which origin from physics, mechanics and optics et al are studied, including exact solutions(soliton solutions, periodic solution), the projective Riccati equation method and Taylor series solutions. The computer algebra and the " AC = BD " model of Professor Zhang Hongqing are employed as the tools to deal with this problem.
Chapter 1 introduces the origin and development of several subjects related to this paper, such as the soliton theory, computer algebra, mathematics mechanization. The main works and achievements that have been obtained are presented.
Chapter 2 considers the construction of exact solutions of partial differential equations(PDEs) under the guidance of the theory of " AC = BD " . The basic theory pf " AC=BD" and the algorithm to construct the C-D pair are illustrated through some concrete transformations.
Based on the ideas of algebraic method, algorithm realization, and mechanization for solving nonlinear evolution equations, Chapter 3 deals with the construction of exact solutions for nonlinear evolution equations by use of Wu-method and symbolic computation. The projective Riccati equation method is generalized to obtain some new exact solutions for two-dimensional generalized Burgers equation and the coupled MKdv-KdV equations.
Chapter 4 is devoted to studying the Taylor series solutions. Based on the characteristic set, the case of infinity parameters is described by using of finite value and functions. The situation of linear PDAEs is extended to the nonlinear PDAEs.
第一章介绍了孤立子理论，计算机代数，数学机械化等学科的起源和发展，以及国内外学者在这些方面所做的工作和一些所取得的成就。
第二章以“AC=BD”的理论模式为指导，考虑了非线性偏微分方程(组)的精确解的构造，给出了“AC=BD”理论的基本思想，C-D可积理论在微分方程求解中的应用，然后通过具体的变换给出了构造C-D对的算法。
第三章基于非线性发展方程求解，代数化，算法化，机械化的指导思想，运用吴方法和符号计算为工具，考虑了非线性发展方程精确解的构造，提出了射影Riccati方程展开法，并将其应用到求解二维广义Burgers方程及耦合MKdV-KdV方程中。
第四章介绍了微分代数的基础知识，并讨论了偏微分代数方程的Taylor级数解。在特征集的基础上，讨论了其参数导数构成的状况，并给出算法。
In this dissertation, the nonlinear partial differential algebraic equation or equations(PDAE or PDAEs) related to some nonlinear topics which origin from physics, mechanics and optics et al are studied, including exact solutions(soliton solutions, periodic solution), the projective Riccati equation method and Taylor series solutions. The computer algebra and the " AC = BD " model of Professor Zhang Hongqing are employed as the tools to deal with this problem.
Chapter 1 introduces the origin and development of several subjects related to this paper, such as the soliton theory, computer algebra, mathematics mechanization. The main works and achievements that have been obtained are presented.
Chapter 2 considers the construction of exact solutions of partial differential equations(PDEs) under the guidance of the theory of " AC = BD " . The basic theory pf " AC=BD" and the algorithm to construct the C-D pair are illustrated through some concrete transformations.
Based on the ideas of algebraic method, algorithm realization, and mechanization for solving nonlinear evolution equations, Chapter 3 deals with the construction of exact solutions for nonlinear evolution equations by use of Wu-method and symbolic computation. The projective Riccati equation method is generalized to obtain some new exact solutions for two-dimensional generalized Burgers equation and the coupled MKdv-KdV equations.
Chapter 4 is devoted to studying the Taylor series solutions. Based on the characteristic set, the case of infinity parameters is described by using of finite value and functions. The situation of linear PDAEs is extended to the nonlinear PDAEs.
引文
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[2] J. Boussinesq, J. Math. Pure Appl., 1872(17):55.
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[16] G. Darbous, Compets Rendus Hebdomaaires des Seances de I'Academie des Sciences, Pair, 1882(94):1456.
[17] M. Wadati et al, Prog. Theo. Phys., 1975(53):418.
[18] C. H. Gu, Lett. Math. Phys., 1986,(11)31:325.
[19] C. H. Cu and Z. X. Zhou, Lett. Math. Phys., 1987(13):179; 1994(32):1.
[20] C. S. Gardner, et al, Phys. Rev. Lett. 1967(19):1095.
[21] P. D. Lax, Commun. Pure Appl. Math., 1968(21):467.
[22] R .M.. Miura, J. Math. Phys., 1968(9):1202.
[23] R. Hirota, Phys. Rev. Lett., 1971(27):1192; Prog.. Theor. Phys. Suppl. 1976(59):64
[24] 张鸿庆，大连理工大学学报，1978(18)：23：力学特刊，1981：152。
[25] 张鸿庆，王震宇，科学通报，1985(10)：667。
[26] 张鸿庆，吴方向，力学学报，1992(24)：700：科学通报，1993(13)：671。
[27] 张鸿庆，冯红，应用数学和力学，1995(16)：335。
[28] 张鸿庆 现代数学和力学，Ⅷ(MMM-Ⅶ)，1997：20。
[29] H. Q. Zhang and Y. E Chen, Proceeding of the 3rd ACM, Lanzhou University Press, 1998:147.
[30] 陈玉福，大连理工大学博士论文，1999。
[31] 阎振亚，大连理工大学博士论文，2002。
[32] 张玉峰，大连理工大学博士论文，2002。
[33] 谢福鼎，大连理工大学博士论文，2003。
[34] 夏铁成，大连理工大学博士论文，2003。
[35] Z. Y. Yan, Generalized method and its application in the higher-order nonlinear equation in nonlinear optical fibres. Chaos, Solitons and Fractals. 2003, 16:759.
[36] G J. Reid, Euro. J. Appl. Math., 1991(2):293
[37] 张鸿庆，大连理工大学学报，20(1981)11
[38] 闫振亚，张鸿庆，物理学报，48(1999)1962．
[39] 范恩贵，大连理工大学博士论文，1998
[40] H. Q. Zhang, Proceedings of the Fifth Asia Symposium Computer. Mathematics, World Scientific, Singapore, 2001, pp. 210.
[41] 张鸿庆，冯红，大连理工大学学报，34(1994)249．
[42] 张鸿庆，大连理工大学学报，18(1978)23；力学学报，特刊，1981，152．
[43] 闫振亚，张鸿庆，物理学报，48(1999)1．
[44] G W. Bluman and J. D. Cole, Similarity method for differentials equation, New York: Soringer-Verlag, 1974.
[45] G. W. Bluman and J. D. Cole,, J. Math. Mech., 18(1969)1025.
[46] P. A. Clarkson and M. D. Kruskal, J. Math. Phys., 30(1989)2001.
[2] J. Boussinesq, J. Math. Pure Appl., 1872(17):55.
[3] D.J.Kortewreg and G.deVries, Phil Mag., 1895(39):422.
[4] A. Fermi, J.Pasta and Ulam, I. Los Alamos Rep. LA., 1940:1955.
[5] J. K. Perring and T. H. R. Skyrme, Nucl. Phys., 1962(31):550.
[6] N. J. Zabusky and M.D. Kruskal, Phys. Rev. Lett., 1965(15):240
[7] M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transformation, SIAM: Philadelphia, 1981
[8] M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equation and inverse scattering, Cambridge University Press, 1991.
[9] A. C. Newell, Soliton in mathematics and physics, SIAM: Philadelphia, 1985.
[10] L.D. Faddeev and L.A. takhtajan, Hamiltonian method in the theory of solitons, Berlin: Springer-Verlag, 1987.
[11] 谷超豪等，孤立子理论与应用，杭州：浙江科技出版社，1987。
[12] A.V. Backlund, Universitests Arsskrift 10, 1885.
[13] H. D. Wahlquist and F. B. Estabrook, phys. Rev. Lett., 1973(31):1386.
[14] J. Weiss, M. Tabor and G. Carnvale, J. Math. Phys., 1983(24):522.
[15] J. Weiss, J. Math. Phys., 1983(24):1405; 1984(25):13; 1985(26):2174
[16] G. Darbous, Compets Rendus Hebdomaaires des Seances de I'Academie des Sciences, Pair, 1882(94):1456.
[17] M. Wadati et al, Prog. Theo. Phys., 1975(53):418.
[18] C. H. Gu, Lett. Math. Phys., 1986,(11)31:325.
[19] C. H. Cu and Z. X. Zhou, Lett. Math. Phys., 1987(13):179; 1994(32):1.
[20] C. S. Gardner, et al, Phys. Rev. Lett. 1967(19):1095.
[21] P. D. Lax, Commun. Pure Appl. Math., 1968(21):467.
[22] R .M.. Miura, J. Math. Phys., 1968(9):1202.
[23] R. Hirota, Phys. Rev. Lett., 1971(27):1192; Prog.. Theor. Phys. Suppl. 1976(59):64
[24] 张鸿庆，大连理工大学学报，1978(18)：23：力学特刊，1981：152。
[25] 张鸿庆，王震宇，科学通报，1985(10)：667。
[26] 张鸿庆，吴方向，力学学报，1992(24)：700：科学通报，1993(13)：671。
[27] 张鸿庆，冯红，应用数学和力学，1995(16)：335。
[28] 张鸿庆 现代数学和力学，Ⅷ(MMM-Ⅶ)，1997：20。
[29] H. Q. Zhang and Y. E Chen, Proceeding of the 3rd ACM, Lanzhou University Press, 1998:147.
[30] 陈玉福，大连理工大学博士论文，1999。
[31] 阎振亚，大连理工大学博士论文，2002。
[32] 张玉峰，大连理工大学博士论文，2002。
[33] 谢福鼎，大连理工大学博士论文，2003。
[34] 夏铁成，大连理工大学博士论文，2003。
[35] Z. Y. Yan, Generalized method and its application in the higher-order nonlinear equation in nonlinear optical fibres. Chaos, Solitons and Fractals. 2003, 16:759.
[36] G J. Reid, Euro. J. Appl. Math., 1991(2):293
[37] 张鸿庆，大连理工大学学报，20(1981)11
[38] 闫振亚，张鸿庆，物理学报，48(1999)1962．
[39] 范恩贵，大连理工大学博士论文，1998
[40] H. Q. Zhang, Proceedings of the Fifth Asia Symposium Computer. Mathematics, World Scientific, Singapore, 2001, pp. 210.
[41] 张鸿庆，冯红，大连理工大学学报，34(1994)249．
[42] 张鸿庆，大连理工大学学报，18(1978)23；力学学报，特刊，1981，152．
[43] 闫振亚，张鸿庆，物理学报，48(1999)1．
[44] G W. Bluman and J. D. Cole, Similarity method for differentials equation, New York: Soringer-Verlag, 1974.
[45] G. W. Bluman and J. D. Cole,, J. Math. Mech., 18(1969)1025.
[46] P. A. Clarkson and M. D. Kruskal, J. Math. Phys., 30(1989)2001.