Applications of Differential Form Wu’s Method to Determine Symmetries of (Partial) Differential Equations
Abstract
:1. Introduction
2. Wu’s Method (Characteristic Set Method)
 (a)
 ${\mathcal{A}}_{1}\prec {\mathcal{A}}_{2}\prec \cdots {\mathcal{A}}_{s}$;
 (b)
 ${\mathcal{A}}_{j}$ is reduced w.r.t. ${\mathcal{A}}_{i}$ for $i=1,2,\cdots ,j1.$
Algorithm 1: Wu’s algorithmfor determining a dcharset of a DPS 
$$\begin{array}{c}\hfill \begin{array}{ccccccccc}\mathrm{PS}=& {\mathrm{PS}}_{0}& \subset & {\mathrm{PS}}_{1}& \subset & \cdots & \subset & {\mathrm{PS}}_{s}& \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ & {\mathrm{BS}}_{0}& \succ & {\mathrm{BS}}_{1}& \succ & \cdots & \succ & {\mathrm{BS}}_{s}& =\mathrm{CS}\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ & {\mathrm{RIS}}_{0}\uparrow & & {\mathrm{RIS}}_{1}\uparrow & & \cdots \uparrow & & {\mathrm{RIS}}_{s}=\varnothing & \end{array}\end{array}$$
$$\phantom{\rule{56.9055pt}{0ex}}\left\{\begin{array}{cc}\begin{array}{c}{\mathrm{BS}}_{i}{\mathrm{is}\mathrm{a}\mathrm{base}\mathrm{set}\mathrm{of}\mathrm{PS}}_{i}{\mathrm{and}\mathrm{BS}}_{i}\succ {\mathrm{BS}}_{i+1},\hfill \\ {R}_{i}={\mathrm{Prem}\left(\right(\mathrm{PS}}_{i}\setminus {\mathrm{BS}}_{i})/{\mathrm{BS}}_{i})\setminus \left\{0\right\},\hfill \\ {\mathrm{IT}}_{i}={\mathrm{Prem}(\mathrm{IP}/\mathrm{BS}}_{i})\backslash \left\{0\right\},{\mathrm{for}\mathrm{any}\mathrm{IP}\mathrm{of}\mathrm{BS}}_{i},\hfill \\ {\mathrm{RIS}}_{i}={\mathrm{IT}}_{i}\cup {R}_{i},\hfill \\ {\mathrm{PS}}_{i}={\mathrm{PS}}_{0}\cup {\mathrm{BS}}_{i1}\cup {\mathrm{RIS}}_{i1},i=0,1,2,\cdots ,s,\hfill \end{array}& \left(W\right)\end{array}\right.$$

3. DcharSet Algorithm for the Symmetry Computation
3.1. A Differential Algebra Version of the Lie Criterion
3.2. Algorithms
Algorithm 2: Producing determining equations (DTEs)for the classical symmetry of (partial) differential equations (PDEs) $\mathcal{R}=0$ 
Input: A differential polynomial system (DPS) $\mathcal{R}$ with a given differential monomial order. Output: A sequence of DTEs ${\mathrm{DTEs}}_{\mathrm{i}}$ for symmetries of $\mathcal{R}=0$. Begin: Step 1: Compute the irreducible dtriset ${\mathrm{ICS}}_{\mathrm{j}}$ of $\mathcal{R}$ and obtain ${\mathrm{IS}}_{\mathrm{j}}$ such that $\mathrm{Z}\left(\mathcal{R}\right)={\cup}_{\mathrm{j}}{\mathrm{Z}(\mathrm{ICS}}_{\mathrm{j}}/{\mathrm{IS}}_{\mathrm{j}})$ (by Algorithm 1 (Theorem 2)). Step 2: Repeat for j: Apply Theorem 3 to ${\mathrm{ICS}}_{\mathrm{j}}$ and let $\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}DTE{s}_{j}=\{\mathrm{the}\mathrm{coefficients}\mathrm{of}\partial U\mathrm{in}\mathrm{Prem}(\mathrm{Pr}\mathcal{X}\left({\mathrm{ICS}}_{\mathrm{j}}\right)/{\mathrm{ICS}}_{\mathrm{j}})$}. End of repeat for j. Step 3: Return ${\mathrm{DTEs}}_{\mathrm{j}}$. End 
Algorithm 3: Reducingdetermining equations (DTEs): $\mathrm{PS}=0$ 
Input: PS with a given differential monomial order. Output: Sequences of dcharsets ${\mathrm{CS}}_{i}$ or irreducible dcharset ${\mathrm{ICS}}_{j}$ of PS. Begin Step 1: Compute decomposition (by Algorithm 1): $\mathrm{Z}\left(\mathrm{PS}\right)={\cup}_{i}{\mathrm{Z}(\mathrm{CS}}_{i}/{\mathrm{IS}}_{i})={\cup}_{j}{\mathrm{Z}(\mathrm{ICS}}_{j}/{\mathrm{IS}}_{j}).$ Step 2: Return ${\mathrm{CS}}_{i}$ or ${\mathrm{ICS}}_{j}$. End 
4. Applications
4.1. Computing Classical Symmetry
4.2. Computing Symmetry of NonSolvedForm Equation
4.3. Computing Nonclassical Symmetry
4.4. Computing Symmetry Classification
5. Concluding Remarks
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
 Ovsiannikov, L.V. Group Analysis of Differential Equations; Ames, W.F., Ed.; Academic Press: New York, NY, USA; London, UK, 1982. [Google Scholar]
 Bluman, G.; Kumei, S. Symmetries and Differential Equations; Springer: New York, NY, USA, 1991. [Google Scholar]
 Olver, P.J. Applications of Lie Groups to Differential Equations, 2nd ed.; Springer: New York, NY, USA, 1993. [Google Scholar]
 Lie, S. Über die Integration durch bestimmte Integrale von einer Klass linear partieller Differentialgleichungen. Arch. Math. 1881, VI, 328–368. [Google Scholar]
 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. Eur. J. Appl. Math. 1991, 2, 293–318. [Google Scholar] [CrossRef]
 Reid, G.J. Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. Eur. J. Appl. Math. 1991, 2, 319–340. [Google Scholar] [CrossRef]
 Reid, G.J.; Wittkopf, A.D.; Boulton, A. Reduction of systems of nonlinear partial differential equations to simplified involutive forms. Eur. J. Appl. Math. 1996, 7, 605–635. [Google Scholar] [CrossRef]
 Schwarz, F. An algorithm for determining the size of symmetry groups. Computing 1992, 49, 95–115. [Google Scholar] [CrossRef]
 Schwarz, F. Reduction and completion algorithm for partial differential equations. In Proceedings of the ISSAC ’92 Papers from the International Symposium on Symbolic and Algebraic Computation, Berkeley, CA, USA, 27–29 July 1992; pp. 49–56. [Google Scholar]
 Wolf, T.; Brand, A. Investigating DEs with CRACK and related programs. ACM SIGSAM Bull. 1995, 29, 1–8. [Google Scholar] [CrossRef]
 Mansfield, E.L. Differential Gröbner Bases. Ph.D. Thesis, University of Sydney, Sydney, Australia, 1991; pp. 1–111. [Google Scholar]
 Mansfield, E.L. Applications of the differential algebra package diffgrob2 to classical symmetries of differential equations. J. Symb. Comp. 1993, 5–6, 517–533. [Google Scholar]
 Lisle, I.G.; Reid, G.J. Symmetry classification using noncommutative invariant differential operators. Found. Comput. Math. 2006, 6, 353–386. [Google Scholar] [CrossRef]
 Boulier, F.; Lazard, D.; Ollivier, F.; Petitot, M. Representation for the radical of a finitely generated differential ideal. In Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, Montreal, QC, Canada, 10–12 July 1995; pp. 158–166. [Google Scholar]
 Hubert, E. Essential components of algebraic differential equations. J. Symb. Comp. 1999, 28, 657–680. [Google Scholar] [CrossRef]
 Ibragimov, N.H. CRC handbook of Lie Group Analysis of Differential Equations. In New Trends in the Oretical Developments and Computational Methods; CRC Press: London, UK, 1994. [Google Scholar]
 Hereman, W. Review of Symbolic Software for Lie Symmetry Analysis; CRC Press: Boca Raton, FL, USA, 1996; pp. 367–413. [Google Scholar]
 Topunov, V.L. Reducing systems of linear partial differential equations to a passive form. Acta Appl. Math. 1989, 16, 191–206. [Google Scholar] [CrossRef]
 Yun, Y.; Chaolu TEMUER. Classical and nonclassical symmetry classifications of nonlinear wave equation with dissipation. Appl. Math. Mech. 2015, 36, 365–379. [Google Scholar] [CrossRef]
 LIU, L.; Chaolu TEMUER. Symmetry analysis of modified 2D Burgers vortex equation for unsteady case. Appl. Math. Mech. 2017, 38, 453–468. [Google Scholar] [CrossRef]
 Bluman, G.W.; Cole, J.D. The general similiarity solution of the heat equation. J. Math. Mech. 1969, 18, 1025–1042. [Google Scholar]
 Cherniha, R. Conditional symmetries for systems of PDEs: New definitions and their application for reactiondiffusion systems. J. Phys. A Math. Theor. 2010, 43, 2591–2599. [Google Scholar] [CrossRef]
 Clarkson, P.A.; Mansfield, E.L. Open problems in symmetry analysis. In Geometrical Study of Differential Equations; Contemporary Mathematics Series; Leslie, J.A., Robart, T., Eds.; American Mathematical Society: Providence, RI, USA, 2001; Volume 285, pp. 195–205. [Google Scholar]
 Wu, W.T. Mathematics Mechanization; Science Press Beijing: Beijing, China; Kluwer Academic Publishers: London, UK, 2000. [Google Scholar]
 Wu, W.T. On the foundation of algebraic differential geometry. Syst. Sci. Math. Sci. 1989, 2, 289–312. [Google Scholar]
 Wu, W.T. Basic principles of mechanical theoremproving in elementary geometry. J. Syst. Sci. Math. Sci. 1984, 4, 207–235. [Google Scholar]
 Kolchin, E.R. Differential Algebra and Algebraic Groups; Academic Press: New York, NY, USA; London, UK, 1973. [Google Scholar]
 Ritt, J.F. Differential Algebra; AMS Colloquium Publications: New York, NY, USA, 1950. [Google Scholar]
 Gao, X.S.; Wang, D.K.; Liao, Q.; Yang, H. Equation Solving and Machine Proving Problem Solving with MMP; Science Press: Beijing, China, 2006. (In Chinese) [Google Scholar]
 Chaolu, T.; Gao, X.S. Nearly dcharset for a differential polynomial system. Acta Math. Sci. 2002, 45, 1041–1050. (In Chinese) [Google Scholar]
 Chaolu, T. An algorithmic theory of reduction of a differential polynomial system. Adv. Math. 2003, 32, 208–220. (In Chinese) [Google Scholar]
 Nicoleta, B.; Jitse, N. On a new procedure for finding nonclassical symmetries. J. Symb. Comput. 2004, 38, 1523–1533. [Google Scholar]
 Clarkson, P.A. Nonclassical symmetry reductions for the Boussinesq equation. Chaos Solitons Fract. 1995, 5, 2261–2301. [Google Scholar] [CrossRef]
 Bluman, G.; Chaolu, T. Local and nonlocal symmetries for nonlinear telegraph equations. J. Math. Phys. 2005, 46, 1–9. [Google Scholar] [CrossRef]
 Bluman, G.; Chaolu, T. Conservation laws for nonlinear telegraphtype equations. J. Math. Anal. Appl. 2005, 310, 459–476. [Google Scholar] [CrossRef]
 Seidenberg, A. An elimination theory for differential algebra. Univ. Calif. Publ. Math. 1956, 3, 31–38. [Google Scholar]
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Chaolu, T.; Bilige, S. Applications of Differential Form Wu’s Method to Determine Symmetries of (Partial) Differential Equations. Symmetry 2018, 10, 378. https://doi.org/10.3390/sym10090378
Chaolu T, Bilige S. Applications of Differential Form Wu’s Method to Determine Symmetries of (Partial) Differential Equations. Symmetry. 2018; 10(9):378. https://doi.org/10.3390/sym10090378
Chicago/Turabian StyleChaolu, Temuer, and Sudao Bilige. 2018. "Applications of Differential Form Wu’s Method to Determine Symmetries of (Partial) Differential Equations" Symmetry 10, no. 9: 378. https://doi.org/10.3390/sym10090378