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An algorithmic method for showing existence of nontrivial non-classical symmetries of partial differential equations without solving determining equations. (English) Zbl 1308.35013

Summary: In this paper, based on differential characteristic set theory and the associated algorithm (also called Wu’s method), an algorithmic method is presented to decide on the existence of a nontrivial non-classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system. The theory and algorithm give a partial answer for the open problem posed by P. A. Clarkson and E. L. Mansfield [Contemp. Math. 285, 195–205 (2001; Zbl 1172.35306)] on non-classical symmetries of partial differential equations. As applications of our algorithm, non-classical symmetries and corresponding invariant solutions are found for several evolution equations.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs

Citations:

Zbl 1172.35306

Software:

diffgrob2; SYMMAN; SADE; GeM
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arrigo, D. J.; Hill, J. M.; Broadbridge, P., Nonclassical symmetry reductions of the linear diffusion equation with a nonlinear source, IMA J. Appl. Math., 52, 1-24 (1994) · Zbl 0791.35060
[2] Barenblatt, G. I., Scaling, Self-similarity, and Intermediate Asymptotics (1996), Cambridge University Press · Zbl 0907.76002
[3] Baumann, G.; Haager, G.; Nonnenmacher, T. F., Applications of nonclassical symmetries, J. Phys. A, 27, 6479-6493 (1994) · Zbl 0849.35126
[4] Bluman, G. W., Construction of solutions to partial differential equations by the use of transformation groups (1967), California Institute of Technology, PhD thesis
[5] Bluman, G. W.; Chaolu, Temuer, Conservation laws for nonlinear telegraph equations, J. Math. Anal. Appl., 310, 459-476 (2005) · Zbl 1077.35087
[6] Bluman, G. W.; Cole, J. D., The general similarity solution of the heat equation, J. Math. Mech., 18, 1025-1042 (1969) · Zbl 0187.03502
[7] Bluman, G. W.; Kumei, S., Symmetries and Differential Equations (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0698.35001
[8] Bluman, G. W.; Cheviakov, A. F.; Anco, S. C., Applications of Symmetry Methods to Partial Differential Equations (2010), Springer: Springer New York · Zbl 1223.35001
[9] Boulier, F.; Lazard, D.; Ollivier, F.; Petitot, M., Representation for the radical of a finitely generated differential ideal, (Proc. ISSAC ʼ95 (1995), ACM: ACM New York), 158-166 · Zbl 0911.13011
[10] Carminati, J.; Vu, K., Symbolic computation and differential equations: Lie symmetries, J. Symbolic Comput., 29, 95-116 (2000) · Zbl 0958.68543
[11] Chaolu, Temuer, Wu-differential characteristic set method and its applications to symmetries of PDEs and mechanics (1997), Dalian University of Technology (DUT), (in Chinese)
[12] Chaolu, Temuer, Wuwen-tsun-differential characteristic algorithm of symmetry vectors for partial differential equations, Acta Math. Sci., 19, 3, 326-332 (1999), (in Chinese) · Zbl 0938.35007
[13] Chaolu, Temuer, An algorithmic theory of reduction of a differential polynomial system, Adv. Math. (China), 32, 2, 208-220 (2003)
[14] Chaolu, Temuer, An algorithm for the complete symmetry classification of differential equations based on Wuʼs method, J. Engrg. Math., 66, 181-199 (2010) · Zbl 1204.35021
[15] Chaolu, Temuer, New algorithm for classical and nonclassical symmetry of a PDE based on Wuʼs method, Sci. Sin. Math., 40, 4, 331-348 (2010), (in Chinese) · Zbl 1488.35016
[16] Chaolu, Temuer; Gao, X. S., Nearly differential characteristic set for a differential polynomial system, Acta Math. Sinica, 45, 6, 1041-1050 (2002), (in Chinese) · Zbl 1033.68049
[17] Cheviakov, A. F., GeM software package for computation of symmetries and conservation laws of differential equations, Comput. Phys. Comm., 176, 48-61 (2007) · Zbl 1196.34045
[18] Clarkson, P. A., Nonclassical symmetry reductions of the Boussinesq equation, Chaos Solitons Fractals, 12, 5, 2261-2301 (1995) · Zbl 0952.37019
[19] Clarkson, P. A.; Mansfield, E. L., Symmetry reductions and exact solutions of a class of nonlinear heat equations, Phys. D, 70, 250-288 (1994) · Zbl 0812.35017
[20] Clarkson, P. A.; Mansfield, E. L., Algorithms for the non-classical method of symmetry reductions, SIAM J. Appl. Math., 54, 1693-1719 (1994) · Zbl 0823.58036
[21] Clarkson, P. A.; Mansfield, E. L., Open problems in symmetry analysis, (Leslie, J. A.; Robart, T., Geometrical Study of Differential Equations. Geometrical Study of Differential Equations, Contemp. Math., vol. 285 (2001), American Mathematical Society: American Mathematical Society Providence, RI, USA), 195-205 · Zbl 1172.35306
[22] Craddock, M.; Lennox, K. A., Lie group symmetries as integral transforms of fundamental solutions, J. Differential Equations, 232, 652-674 (2007) · Zbl 1147.35009
[23] Fushchich, W. I.; Serov, N. I., Conditional invariance and reduction of nonlinear heat equations, Dokl. Akad. Nauk Ukrain. SSR Ser. A, 7, 24-27 (1990) · Zbl 0727.35066
[24] Fushchich, W. I.; Shtelen, W. M.; Serov, N. I., Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0838.58043
[25] Gao, X. S.; Chou, S. C., A zero structure theorem for a differential parametric system, J. Symbolic Comput., 16, 585-595 (1993) · Zbl 0805.12003
[26] Gao, X. S.; Wang, D. K.; Liao, Q.; Yang, H., Equation Solving and Machine Proving - Problem Solving with MMP (2006), Science Press: Science Press Beijing, (in Chinese). The latest version of MMP can be downloaded from
[27] Gerdt, V. P., On decomposition of algebraic PDE systems into simple subsystems, Acta Appl. Math., 101, 39-51 (2008) · Zbl 1154.68111
[28] Hereman, W., Symbolic software for Lie symmetry analysis, (Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3 (1996), CRC Press: CRC Press Boca Raton, FL), 367-413
[29] (Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3: New Trends in Theoretical Developments and Computational Methods (1996), CRC Press: CRC Press Boca Raton, Ann Arbor, London, Tokyo) · Zbl 0864.35003
[30] Kolchin, E. R., Differential Algebra and Algebraic Groups (1973), Academic Press: Academic Press New York, London · Zbl 0264.12102
[31] Kudryashov, N. A.; Loguinova, N. B., Extended simplest equation method for nonlinear differential equations, Appl. Math. Comput., 205, 396-402 (2008) · Zbl 1168.34003
[32] Kunzinger, M.; Popovych, R. O., Singular reduction operators in two dimensions, J. Phys. A, 41, 505201 (2008), 24 pp. · Zbl 1152.35303
[33] Mansfield, E. L., Applications of the differential algebra package diffgrob2 to classical symmetries of differential equations, J. Symbolic Comput., 5-6, 23, 517-533 (1993)
[34] Miller, W., Symmetry and Separation of Variables, Encyclopedia Math. Appl., vol. 4 (1977), Addison-Wesley Publishing Company, Inc.
[35] Olver, P. J., Applications of Lie Groups to Differential Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0785.58003
[36] Olver, P. J.; Vorobʼev, E. M., Nonclassical and conditional symmetries, (Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3 (1996), CRC Press: CRC Press Boca Raton, FL), 291-328
[37] Popovych, R. O., Reduction operators of linear second-order parabolic equations, J. Phys. A, 41, 185202 (2008), 31 pp. · Zbl 1145.35316
[38] Pucci, E.; Saccomandi, G., On the weak symmetry groups of partial differential equations, J. Math. Anal. Appl., 163, 588-598 (1992) · Zbl 0755.35003
[39] Reid, G. J., Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution, European J. Appl. Math., 2, 293-318 (1991) · Zbl 0768.35001
[40] Reid, G. J., Finding abstract Lie symmetry algebras of differential equations without integrating determining equations, European J. Appl. Math., 2, 319-340 (1991) · Zbl 0768.35002
[41] Reid, G. J.; Wittkopf, A. D.; Boulton, A., Reduction of systems of nonlinear partial differential equations to simplified involutive forms, European J. Appl. Math., 7, 605-635 (1996) · Zbl 0892.35041
[42] Ritt, J. F., Differential Algebra, Amer. Math. Soc. Colloq. Publ., vol. 33 (1950), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0037.18501
[43] Rocha Filho, T. M.; Figueiredo, A., [SADE] A Maple package for the symmetry analysis of differential equations, Comput. Phys. Comm., 182, 467-476 (2011) · Zbl 1217.65165
[44] Schwarz, F., An algorithm for determining the size of symmetry groups, Computing, 49, 95-115 (1992) · Zbl 0759.68042
[45] Seiler, W. M., Involution and symmetry reductions, Math. Comput. Modelling, 25, 63-73 (1997) · Zbl 0906.34006
[46] Smyth, N. F., The effect of conductivity on hotspots, J. Aust. Math. Soc. Ser. B, 33, 403-413 (1992) · Zbl 0758.35042
[47] Vorobʼev, E. M., Symmetries of compatibility conditions for overdetermined systems of differential equations, Acta Appl. Math., 26, 61-86 (1992) · Zbl 0763.35013
[48] Vorobʼev, E. M., Symmetry analysis of nonlinear differential equations with the “Mathematica” program SYMMAN, Math. Comput. Modelling, 8/9, 25, 141-152 (1997) · Zbl 0887.35007
[49] Wu, W. T., On the foundation of algebraic differential geometry, J. Syst. Sci. Complex., 2, 289-312 (1989) · Zbl 0739.14001
[50] Wu, W. T., Mathematics Mechanization, Math. Appl., vol. 489 (2000), Science Press, Kluwer Academic Publishers: Science Press, Kluwer Academic Publishers Beijing, Dordrecht/Boston/London
[51] Zhdanov, R. Z.; Lahno, V. I., Conditional symmetry of a porous medium equation, Phys. D, 122, 178-186 (1998) · Zbl 0952.76087
[52] Zhdanov, R. Z.; Tsyfra, I. M.; Popovych, R. O., A precise definition of reduction of partial differential equations, J. Math. Anal. Appl., 238, 1, 101-123 (1999) · Zbl 0936.35012
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