On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation

Communications in Nonlinear Science and Numerical Simulation

Volumn 16, Issue 11, 2011, Pages 4215-4231

  Vitanov, Nikolay K ^a,b  

^a INSTITUTE OF MECHANICS   (Bulgaria)

^b MAX PLANCK INSTITUT FÜR PHYSIK KOMPLEXER SYSTEME   (Germany)

Author keywords

Elliptic equation; Equation of Bernoulli; Equation of Riccati; Exact traveling wave solutions; Method of simplest equation; Nonlinear partial differential equations

Indexed keywords

NONLINEAR EQUATIONS; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; RICCATI EQUATIONS;

APPROXIMATE SOLUTION; BERNOULLI; ELLIPTIC EQUATIONS; EQUATION OF BERNOULLI; EQUATION OF RICCATI; EXACT SOLUTION; EXACT TRAVELLING WAVE SOLUTIONS; METHOD OF SIMPLE EQUATION; NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS; SIMPLE++;

POLYNOMIALS;

EID: 79957892557     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2011.03.035     Document Type: Article

References (69)

1. Scott A.C. Nonlinear science. Emergence and dynamics of coherent structures 1999, Oxford University Press.
2. Murray J.D. Lectures on nonlinear differential equation models in biology 1977, Oxford University Press, Oxford, England.
3. May R.M. Stability and complexity in model ecosystems 2001, Princeton University Press, New Jersey.
4. Vitanov N.K., Busse F.H. Bounds on the heat transport in a horizontal fluid layer with stress-free boundaries. ZAMP 1997, 48:310-324.
5. Hoffmann N.P., Vitanov N.K. Upper bounds on energy dissipation in Couette-Ekman flow. Phys Lett A 1999, 255:277-286.
6. Tannenhill J.C., Anderson D.A., Pletcher R.H. Computational fluid mechanics and heat transfer 1997, Taylor and Francis, Philadelphia, PA.
7. Kantz H., Holstein D., Ragwitz M., Vitanov N.K. Markov chain model for turbulent wind speed data. Physica A 2004, 342:315-321.
8. Ablowitz M., Clarkson P.A. Solitons, nonlinear evolution equations and inverse scattering 1991, Cambridge University Press, Cambridge, England.
9. Vitanov N.K., Dimitrova Z.I., Ausloos M. Verhulst-Lotka-Volterra (VLV) model of ideological struggle. Physica A 2010, 389:4970-4980.
10. Vitanov N.K., Panchev S. Generalization of the model of conflict between two armed groups. Comput Rend Acad Bulg Sci 2008, 61:1121-1126.
11. Scott A.C. Neuroscience: a mathematical primer 2002, Springer, New York.
12. Perko L. Differential equations and dynamical systems 2001, Springer, New York.
13. Infeld E., Rowlands G. Nonlinear waves, solitons and chaos 1990, Cambridge University Press, Cambridge, UK.
14. Thietart R.A., Forgues B. Chaos theory and organization. Organiz Sci 1995, 6:19-31.
15. Chaos theory in psychology and the life sciences 1995, Lawrence Erlbaum Associates, Inc, Mahwah, New Jersey. R. Robertson, A. Combs (Eds.).
16. Vitanov N.K., Jordanov I.P., Dimitrova Z.I. On nonlinear population waves. Appl Math Comput 2009, 215:2950-2964.
17. Temam R. Navier-Stokes equations: Theory and numerical analysis. Providence, R.I.: AMS Chelsea Publishing; 2001.
18. Vitanov N.K. Upper bounds on the heat transport in a porous layer. Physica D 2000, 136:322-339.
19. Holmes P., Lumley J.L., Berkooz G. Turbulence, coherent structures, dynamical systems and symmetry 1996, Cambridge University Press, Cambridge, UK.
20. Boeck T, Vitanov NK. Low-dimensional chaos in zero-Prandtl-number Benard-Marangoni convection. Phys Rev E 2002; 65 Article number: 037203.
21. Vitanov N.K. Upper bound on the heat transport in a horizontal fluid layer of infinite Prandtl number. Phys Lett A 1998, 248:338-346.
22. Foias C., Manley O., Rosa R., Temam R. Navier-Stokes equations and turbulence 2001, Cambridge University Press, Cambridge, UK.
23. Vitanov N.K. Upper bounds on convective heat transport in a rotating fluid layer of infinite Prandtl number: case of intermediate Taylor numbers. Phys Rev E 2000, 62:3581-3591.
24. Vitanov N.K. Convective heat transport in a fluid layer of infinite Prandtl number: upper bounds for the case rigid lower boundary and stress-free upper boundary. Eur Phys J B 2000, 15:349-355.
25. Gardner C.S., Greene J.M., Kruskal M.D., Miura R.R. Method for solving Korteweg-de Vries equation. Phys Rev Lett 1967, 19:1095-1097.
26. Ablowitz M.J., Kaup D.J., Newell A.C. Nonlinear evolution equations of physical significance. Phys Rev Lett 1973, 31:125-127.
27. Ablowitz M.J., Kaup D.J., Newell A., Segur H. Inverse scattering transform-Fourier analysis for nonlinear problems. Stud Appl Math 1974, 53:249-315.
28. Remoissenet M. Waves called solitons 1993, Springer, Berlin.
29. Hirota R. Exact solution of Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett 1971, 27:1192-1194.
30. Kudryashov N.A. Exact solutions of the generalized Kuramoto-Sivashinsky equation. Phys Lett A 1990, 147:287-291.
31. Vitanov N.K., Dimitrova Z.I., Kantz H. Modified method of simplest equation and its application to nonlinear PDEs. Appl Math Comput 2010, 216:2587-2595.
32. Vitanov N.K. Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs. Commun Nonlinear Sci Numer Simulat 2011, 16:1176-1185.
33. Kudryashov N.A. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Soliton Fract 2005, 24:1217-1231.
34. Ablowitz M.J., Zeppetela A. Explicit solutions of Fisher equation for a specifical wave speed. Bull Math Biol 1979, 41:835-840.
35. Dimitrova Z I., Vitanov N K. Influence of adaptation on the nonlinear dynamics of a system of competing populations. Phys Lett A 2000, 272:368-380.
36. Dimitrova Z.I., Vitanov N.K. Dynamical consequences of adaptation of growth rates in a system of three competing populations. J Phys A: Math Gen 2001, 34:7459-7473.
37. Dimitrova Z.I., Vitanov N.K. Adaptation and its impact on the dynamics of a system of three competing populations. Physica A 2001, 300:91-115.
38. Dimitrova Z.I., Vitanov N.K. Chaotic pairwise competition. Theoretical Population Biology 2004, 66:1-12.
39. Vitanov N.K., Dimitrova Z.I., Kantz H. On the trap of extinction and its elimination. Phys Lett A 2006, 349:350-355.
40. Vitanov N.K., Jordanov I.P., Dimitrova Z.I. On nonlinear dynamics of interacting populations: coupled kink waves in a system of two populations. Commun Nonlinear Sci Numer Simulat 2009, 14:2379-2388.
41. Vitanov N.K., Dimitrova Z.I. Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics. Commun Nonlinear Sci Numer Simulat 2010, 15:2836-2845.
42. Wang X.Y. Exact and explicit wave solutions for the generalized Fisher equation. Phys Lett A 1988, 131:277-279.
43. Kudryashov N.A. Exact solitary waves of the Fisher equation. Phys Lett A 2005, 342:99-106.
44. Vitanov N.K. Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling wave solutions for a class of PDEs with polynomial nonlinearity. Commun Nonlinear Sci Numer Simulat 2010, 15:2050-2060.
45. Lou S. Symmetry analysis and exact solutions of the 2+1-dimensional sine-Gordon system. J Math Phys 2000, 41:6509-6524.
46. Martinov N., Vitanov N. On some solutions of the two-dimensional sine-Gordon equation. J Phys A: Math Gen 1992, 25:L419-L426.
47. Martinov N., Vitanov N. New class of running-wave solutions of the 2+1-dimensional sine-Gordon equation. J Phys A: Math Gen 1994, 27:4611-4618.
48. Vitanov N.K. On traveling waves and double-periodic structures in two-dimensional sine-Gordon systems. J Phys A: Math Gen 1996, 29:5195-5207.
49. Clarkson P.A., Mansfield E.L., Milne A.E. Symmetries and exact solutions of a (2+1)-dimensional sine-Gordon system. Phil Trans Roy Soc London A 1996, 354:1807-1835.
50. Vitanov N.K., Martinov N.K. On the solitary waves in the sine-Gordon model of the two-dimensional Josephson junction. Z Phys B 1996, 100:129-135.
51. Vitanov N.K. Breather and soliton wave families for the sine-Gordon equation. Proc Roy Soc London A 1998, 454:2409-2423.
52. Radha R., Lakshamanan M. The (2+1)-dimensional sine-Gordon equation; integrability and localized solutions. J Phys A: Math Gen 1996, 29:1551-1562.
53. Nakamura A. Exact cylindrical soliton solutions of the sine-Gordon equation, the sinh-Gordon equation and the periodic Toda equation. J Phys Society Japan 1988, 57:3309-3322.
54. Martinov N., Vitanov N. On the correspondence between the self-consistent 2D Poisson-Boltzmann structures and the sine-Gordon waves. J Phys A: Math Gen 1992, 25:L51-L56.
55. Martinov N.K., Vitanov N.K. On the self-consistent thermal equilibrium structures in two-dimensional negative temperature systems. Canadian J Phys 1994, 72:618-624.
56. Panchev S., Spassova T., Vitanov N.K. Analytical and numerical investigation of two families of Lorenz-like dynamical systems. Chaos Soliton Fract 2007, 33:1658-1671.
57. Thaker W.C. Some exact solutions to the nonlinear shallow-water equations. J Fluid Mech 1981, 107:499-508.
58. Malfliet W., Hereman W. The tanh-method: I. Exact solutions of nonlinear evolution and water wave equations. Physica Scripta 1996, 54:563-568.
59. Debnath L. Nonlinear water waves 1994, Academic Press, New York.
60. Ivanov R.I. Water waves and integrability. Phil Trans R Soc A 2007, 365:2267-2280.
61. Johnson R.S. The classical problem of water waves: a reservoir of integrable and nearly-integrable equations. J Nonlinear Math Phys 2003, 10(Supplement 1):72-92.
62. Kudryashov N.A., Loguinova N.B. Extended simplest equation method for nonlinear differential equations. Appl Math Comput 2008, 205:396-402.
63. Kudryashov N.A., Demina M.V. Polygons of differential equations for finding exact solutions. Chaos Soliton Fract 2007, 33:480-496.
64. Kudryashov N.A., Loguinova N.B. Be careful with the exp-function method. Commun Nonlinear Sci Numer Simulat 2009, 14:1881-1890.
65. Kudryashov N.A. Seven common errors in finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simulat 2009, 14:3507-3529.
66. Hone A.N.W. Painleve tests, singularity structure and integrability. Lect Notes Phys 2009, 767:245-277.
67. Vitanov N.K., Dimitrova Z.I., Vitanov K.N. On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis-Processi equation and b-equation. Commun Nonlinear Sci Numer Simulat 2011, 16:3033-3044.
68. Kudryashov N.A. Nonlinear differential equations with exact solutions via the Weierstrass function. Zeitschrift für Naturforschung A 2004, 59:443-454.
69. Janke E., Emde F. Tables of functions with formulae and curves 1945, Dover, New York.