A personal overview on the reduction methods for partial differential equations

Note di Matematica

Volumn 23, Issue 2, 2004, Pages 217-248

  Saccomandi, Giuseppe ^a  

^a UNIVERSITY OF SALENTO   (Italy)

Author keywords

Non classical symmetries; Reduction methods; Semi inverse methods; Weak symmetries

Indexed keywords

EID: 33845227972     PISSN: 11232536     EISSN: 15900932     Source Type: Journal    
DOI: None     Document Type: Article

References (49)

1. W. F. Ames: Nonlinear partial differential equations in engineering. Vol. II, Academic Press, New York-London, 1972.
2. D. J. Arrigo, J. R. Beckham: Nonclassical symmetries of evolutionary partial differential equations and compatibility, J. Math. Anal. Appl., 289 (2004), 55-65.
3. G. Barenblatt: Scaling, Cambridge University Press, Cambridge, 2003.
4. R. Berker: Intégration des équations du mouvement d'un uide visqueux incompressible, in Handbuch der Physik, Bd. VIII/2, Springer, Berlin, 1963.
5. G. Birkhoff: Hydrodynamics. A study in logic, fact, and similitude, Princeton University Press, Princeton, 1960.
6. G. Bluman, J. Cole: The general similarity solution of the heat equation, J. Math. Mech., 18 (1969), 1025-1042.
7. G. Bluman, S. Kumei: Symmetries and Differential Equations, Springer, New York, 1989.
8. R. E. Boisvert, W. F. Ames, U. N. Srivastava: Group properties and new solutions of Navier-Stokes equations, J. Enggr. Math., 17 (1983), 203-221.
9. G. I. Burde: The construction of special explicit solutions of the boundary-layer equations. Steady ows, Quart. J. Mech. Appl. Math. 47 (1994), 247-260.
10. G. I. Burde: The construction of special explicit solutions of the boundary-layer equations. Unsteady ows, Quart. J. Mech. Appl. Math. 48 (1995), 611-633.
11. G. I. Burde: New similarity reductions of the steady-state boundary layer equations, J. Phys. A: Math. Gen., 29 (1996), 1665-1683.
12. K.-S. Chou, C. Qu: Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations, J. Phys. A: Math. Gen., 32 (1999), 6271-6286.
13. F. Cianetti, E. Pucci: Solutions of the Boussinesq equation generated by weak symmetries, Rend. Mat. Appl. 12 (1992), 769-781.
14. G. Cicogna, G. Gaeta: Partial Lie-point symmetries of differential equations, J. Phys. A: Math. Gen., 34 (2001), 491-512.
15. P. A. Clarkson: Nonclassical symmetry reductions of the Boussinesq equation, Solitons & Fractals, 5 (1995), 2261-2301.
16. P. A. Clarkson, M. Kruskal: New similarity reductions of the Boussinesq equation, J. Math. Phys. 30 (1988), 2201-2213.
17. R. Courant, D. Hilbert: Methoden der mathematischen Physik, Vol. 2, Springer, Berlin, 1937.
18. J. L. Ericksen: Special topics in elastostatics, Advances in Applied Mechanics, 17 (1978), 189-244.
19. J. Goard: Generalised conditional symmetries and Nucci's method of iterating the nonclassical symmetries method Applied Mathematics Letters 16 (2003), 481-486.
20. M. E. Gurtin: An introduction to continuum mechanics, Academic Press, New York-London, 1981.
21. S. Hood: On direct, implicit reductions of a nonlinear diffusion equation with an arbitrary function-generalizations of Clarkson's and Kruskal's method, IMA J. Appl. Math. 64 (2000), 223-244.
22. D. Levi, P. Winternitz: Nonclassical symmetry reduction: example of the Boussinesq equation, J. Phys. A: Math. Gen., 22 (1989), 2915-2924.
23. S. Y. Lou: A note on the new similarity reductions of the Boussinesq equation, Phys. Lett. A 151 (1990), 133-135.
24. D. K. Ludlow, P. A. Clarkson, A. P. Bassom: Similarity reductions and exact solutions for the two-dimensional incompressible Navier-Stokes equations, Stud. Appl. Math. 103 (1999), 183-240.
25. P. F. Neményi: Recent developments in inverse and semi-inverse methods in the mechanics of continua, Advances in Applied Mechanics, 2 (1951) 123-151.
26. M. C. Nucci: Nonclassical symmetries and Bácklund transformations, J. Math. Anal. and Appl., 178 (1993), 294-300.
27. M. C. Nucci, P. A. Clarkson: The nonclassical method is more general than the direct method for symmetry reductions. An example of the FitzHugh-Nagumo equation, Phys. Lett. A 164 (1992), 49-56.
28. A. D. Polyanin: Exact solutions to the Navier-Stokes equations with generalized separation of variables, Doklady Physics, 46 (2001), 726-731.
29. P. J. Olver: Direct reduction and differential constraints, Proc. Roy. Soc. London Ser. A 444 (1994), 509-523.
30. P. J. Olver: Applications of Lie groups to differential equations, 2nd Edition, Springer, New-York, 1998.
31. P. J. Olver, P. Rosenau: The construction of special solutions to partial differential equations Phys. Lett., 147A (1996), 107-112.
32. P. J. Olver, P. Rosenau, Group-invariant solutions of differential equations SIAM J. of Appl. Math. (1997), 107-112.
33. P. J. Olver, E. M. Vorobev: Nonclassical and conditional symmetries, in CRC handbook of Lie group analysis of differential equations, (N. H. Ibragimov ed.), CRC Press, Boca Raton, 1994.
34. L. V. Ovsiannikov: Group Analysis of Differential Equations, Academic Press, New York, 1982.
35. E. Pucci: Similarity reductions of partial differential equations, J. Phys. A: Math. Gen., 25 (1992), 2631-2640.
36. E. Pucci, G. Saccomandi: On the weak symmetry groups of partial differential equations, J. Math. Anal. Appl., 163 (1992), 588-598.
37. E. Pucci, G. Saccomandi: Quasisolutions as group-invariant solutions for partial differential equations, Studies Appl. Math., 94 (1995), 211-223.
38. E. Pucci, G. Saccomandi: Evolution equations, invariant surface conditions and functional separation of variables, PhysicaD, 139 (2000), 28-47.
39. C. Qu, P. G. Estevez: Extended rotation and scaling groups for nonlinear evolution equations, Nonlinear Analysis, 52 (2003), 1655-1673.
40. J. F. Ritt: Differential algebra, Dover Publications, New York 1966.
41. L. A. Rubel: Closed-form solutions of some partial differential equations via quasisolutions. I, Illinois J. Math., 35 (1991), 690-709.
42. G. Saccomandi: Potential symmetries and direct reduction methods of order two, J. Phys. A: Math. Gen., 30 (1997), 2211-2217.
43. G. Saccomandi: Universal results in finite elasticity in Nonlinear Elasticity (Y. Fu and R. Ogden eds.), Cambridge University Press, Cambridge, 2001.
44. G. Saccomandi: A remarkable class of non classical symmetries of the boundary layers equations, J. of Phys. A: Math. Gen., 37 (2004), 7005-701.
45. H. Schlichting: Boundary-layer theory, McGraw-Hill, New-York, 1979.
46. W. Seiler: Involution and symmetry reductions. Algorithms and software for symbolic analysis of nonlinear systems, Math. Comput. Modelling, 25 (1997), 63-73.
47. A. M. Vinogradov: Local symmetries and conservation laws, Acta Applicandae Mathematicae, 2 (1981), 21-78.
48. C. Y. Wang: Exact solutions of the unsteady Navier-Stokes equations in Mechanics Pan-America 1989 AMR, 42 (1989), S269-S282.
49. C. Y. Wang: Exact solutions of the steady-state Navier-Stokes equations, Annual Review of Fluid Mechanics, 23 (1991), 159-177.