Symbolic methods to construct exact solutions of nonlinear partial differential equations

Mathematics and Computers in Simulation

Volumn 43, Issue 1, 1997, Pages 13-27

  Hereman, Willy ^a   Nuseir, Ameina ^a  

^a Department of Geology and Geological Engineering   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; COMPUTER SOFTWARE; INTEGRATION; NUMERICAL ANALYSIS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; SOLITONS;

FITZHUGH-NAGUMO EQUATION; KAUP-KUPERSHMIDT EQUATION; PAINLEVE EXPANSION METHOD; SINGULAR MANIFOLD METHOD; SOLITARY WAVE; SOLITON SOLUTIONS; SYMBOLIC MANIPULATION PROGRAM;

NONLINEAR EQUATIONS;

EID: 0030829213     PISSN: 03784754     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0378-4754(96)00053-5     Document Type: Article

References (26)

1. M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991).
2. M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981).
3. M.J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher's equation for a special wave speed, Bull. Math. Biology 41 (1979) 835-840.
4. 4, J. Math. Phys. 28 (1987) 538-555.
5. F. Calogero, Why are certain nonlinear PDEs both widely applicable and integrable?, in: V.E. Zakharov ed., What is Integrability (Springer, Berlin, 1991).
6. P.J. Caudrey, R.K. Dodd and J.D. Gibbon, A new hierarchy of Korteweg-de Vries equations, Proc. Roy. Soc. Lond. Ser. A 351 (1976) 407-422.
7. P.A. Clarkson and E.L. Mansfield, On a shallow water wave equation, Nonlinearity 7 (1994) 975-1000.
8. R.K. Dodd and J.D. Gibbon, The prolongation structure of a higher order Korteweg-de Vries equation, Proc. Roy. Soc. Lond. Ser A 358 (1977) 287-296.
9. P.G. Drazin and R.S. Johnson, Solitons: An Introduction (Cambridge University Press, Cambridge, 1989).
10. A. Fordy and J. Gibbons, Some remarkable nonlinear transformation, Phys. Lett. A 75 (1980) 325.
11. W. Hereman, Symbolic software for the study of nonlinear partial differential equations, in: R. Vichnevetsky, D. Knight and G. Richter, eds., Advances in Computer Methods for Partial Differential Equations VII (IMACS, New Brunswick, NJ, 1993) 326-332.
12. W. Hereman and W. Zhuang, Symbolic computation of solitons with Macsyma, in: W.F. Ames and P.J. van der Houwen, eds., Computational and Applied Mathematics II: Differential Equations (North-Holland, Amsterdam, 1992) 287-296.
13. W. Hereman and W. Zhuang, Symbolic software for soliton theory, Acta Applicandae Mathematicae 39 (1995) 361-378.
14. R. Hirota and A. Ramani, The Miura transformation of Kaup's equation and of Mihailov's equation, Phys. Lett. A 76 (1980) 95-96.
15. M. Ito, An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders, J. Phys. Soc. Japan 49 (1980) 771-778.
16. M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. Res. Inst. Math. Sci. (Kyoto University) 19 (1983) 943-1001.
17. x + 6Rψ = λψ, Stud. Appl. Math. 62 (1980) 189-216.
18. T. Kawahara and M. Tanaka, Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation, Phys. Lett. A 97 (1983) 311-314.
19. B.A. Kupershmidt, A super KdV equation: an integrable system, Phys. Lett. A 102 (1984) 213-215.
20. P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968) 467-490.
21. W. Malfliet, Traveling wave solutions of Fisher type, Preprint, Department of Physics, University of Antwerp, Antwerp, Belgium, 1994.
22. W. Mafliet, The tanh method in nonlinear wave theory, Habilitation Thesis, Department of Physics, University of Antwerp, Antwerp, Belgium, 1994.
23. J.D. Murray, Mathematical Biology, Biomathematics, Texts 19 (Springer, Berlin, 1989).
24. A. Nuseir, Symbolic computation of exact solutions of nonlinear partial differential equations using direct method, Ph.D. Thesis, Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO, 1995.
25. J. Satsuma and D.J. Kaup, A Bäcklund transformation for a higher order Korteweg-de Vries equation, J. Phys. Soc. Japan 43 (1977) 692-697.
26. K. Sawada and T. Kotera, A method of finding N-soliton solutions of the KdV and KdV-like equation, Progr. Theoret. Phys. 51 (1974) 1355-1367.