Method of the logistic function for finding analytical solutions of nonlinear differential equations

N. A. Kudryashov

Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 1, pp. 23-37.

Résumé

The method of the logistic function is presented for finding exact solutions of nonlinear differential equations. The application of the method is illustrated by using the nonlinear ordinary differential equation of the fourth order. Analytical solutions obtained by this method are presented. These solutions are expressed via exponential functions.

Détail

Keywords: logistic function, nonlinear wave, nonlinear ordinary differential equation, Painlevé test
Mots-clés : exact solution.

Bibliographie

[1] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, “Method for solving the Korteweg–de Vries equation”, Phys. Rev. Lett., 19 (1967), 1095–1097 | DOI

[2] V. E. Zakharov, A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, Funkts”, Anal. Prilozh., 8:3 (1974), 43–53

[3] V. E. Zakharov, A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, II”, Funkts. Anal. Prilozh., 13:3 (1979), 13–22

[4] R. Hirota, “Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27 (1971), 1192–1194 | DOI

[5] P. J. Olver, Applications of Lie groups to differential equations, Springer-Verlag, New York, 1993

[6] N. A. Kudryashov, “Exact soliton solutions of the generalized evolution equation of wave dynamics”, PMM-J. Appl. Math. Mech., 52 (1988), 361–365 | DOI

[7] N. A. Kudryashov, “Exact solutions of the generalized Kuramoto–Sivashinsky equation”, Phys. Lett. A, 147 (1990), 287–291 | DOI

[8] W. Malfliet, W. Hereman, “The tanh method. I: Exact solutions of nonlinear evolution and wave equations”, Phys. Scripta, 54 (1996), 563–568 | DOI

[9] E. J. Parkes, B. R. Duffy, “An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations”, Comput. Phys. Commun., 98 (1996), 288–300 | DOI

[10] W. Malfliet, W. Hereman, “The tanh method. II: Perturbation technique for conservative systems”, Phys. Scripta, 54 (1996), 569–575 | DOI

[11] N. A. Kudryashov, “Simplest equation method to look for exact solutions of nonlinear differential equations”, Chaos Soliton Frac., 24 (2005), 1217–1231 | DOI

[12] N. A. Kudryashov, N. B. Loguinova, “Extended simplest equation method for nonlinear differential equations”, Appl. Math. Comp., 205 (2008), 396–402 | DOI

[13] N. K. Vitanov, “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., 15 (2010), 2050–2060 | DOI

[14] N. K. Vitanov, “Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs”, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1176–1185 | DOI

[15] A. J. M. Jaward, M. D. Petkovic, A. Biswas, “Modified simple equation method for nonlinear evolution equations”, Appl. Math. Comput., 217 (2010), 869–877 | DOI

[16] M. L. Wang, X. Li, J. Zhang, “The G'/G–expansion method and evolution equation in mathematical physics”, Phys. Lett. A, 372 (2008), 417–421 | DOI

[17] N. A. Kudryashov, “A note on the G'/G–expansion method”, Appl. Math. Comput., 217 (2010), 1755–1758 | DOI

[18] A. Biswas, “Solitary wave solution for the generalized Kawahara equation”, Appl. Math. Lett., 22 (2009), 208–210 | DOI

[19] W. W. Li, Y. Tian, Z. Zhang, “F-expansion method and its application for finding new exact solutions to the sine-Gordon and sinh-Gordon equations”, Applied mathematics and computation, 219:3 (2012), 1135–1143 | DOI

[20] Y. M. Zhao, “F-expansion method and its application for finding new exact solutions to the Kudryashov–Sinelshchikov equation”, Journal of Applied Mathematics, 2013, 895760

[21] Y. He, S. Li, Y. Long, “A improved F-expansion method and its application to Kudryashov–Sinelshchikov equation”, Mathematical Methods in the Applied Sciences, 37:12 (2014), 1717–1722 | DOI

[22] N. A. Kudryashov, “Seven common errors in finding exact solutions of nonlinear differential equations”, Communications in Nonlinear Science and Numerical Simulation, 2009

[23] N. A. Kudryashov, “One method for finding exact solutions of nonlinear differential equations”, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2248–2253 | DOI

[24] N. A. Gershenfield, The Nature of mathematical modeling, Cambridge University Press, Cambridge, UK, 1999

[25] G. C. Wu, D. Baleanu, “Discrete fractional logistic map and its chaos”, Nonlinear Dynamics, 75 (2014), 283–287 | DOI

[26] F. J. Richards, “A flexible growth function for empirical use”, J. Exp. Bot., 10 (1959), 290–300 | DOI

[27] N. A. Kudryashov, “Polynomials in logistic function and solitary waves of nonlinear differential equations”, Appl. Math. Comput., 219 (2013), 9245–9253 | DOI

[28] N. A. Kudryashov, “Quasi-exact solutions of the dissipative Kuramoto-Sivashinsky equation”, Appl. Math. Comput., 219 (2013), 9213–9218 | DOI

[29] N. A. Kudryashov, “Meromorphic solutions of nonlinear ordinary differential equations”, Commun Nonlinear Sci Numer Simulat., 15 (2010), 2778–2790 | DOI

[30] Y. Kuramoto, T. Tsuzuki, “Persistent propagation of concentration waves in dissipative media far from thermal equilibrium”, Progress of theoretical physics, 55:2 (1976), 356–369 | DOI

[31] G. I. Sivashinsky, “Instabilities, pattern formation, and turbulence in flames”, Ann. Rev. Fluid Mech., 15 (1983), 179–199 | DOI

[32] D. J. Benney, “Long waves on liquid films”, Journal of Mathematics and Physics, 45 (1966), 150–155

[33] J. Topper, T. Kawahara, “Approximate equations for long nonlinear waves on a viscous fluid”, Journal of the Physical society of Japan, 44:2 (1978), 663–666 | DOI

[34] V. Ya. Shkadov, “Solitary waves in a layer of viscous liquid”, Fluid Dynamics, 12:1 (1977), 52–55 | DOI

[35] B. I. Cohen, W. M. Tang, J. A. Krommes, M. N. Rosenbluth, “Non-linear saturation of the dissipative trapped-ion mode by mode coupling”, Nuclear fusion, 16 (1976), 971–992 | DOI

[36] D. Michelson, “Elementary particles as solutions of the Sivashinsky equation”, Physica D, 44 (1990), 502–556 | DOI

[37] N. A. Kudryashov, D. I. Sinelschikov, I. L. Chernyavsky, “Nonlinear evolution equations for description of perturbation in tube”, Nonlinear dynamics, 4:1, 69–86

[38] N. A. Kudryashov, E. D. Zargaryan, “Solitary waves in active-dissipative dispersive media”, J. Phys. A.: Math. Gen., 29 (1996), 8067–8077 | DOI

[39] Z. N. Zhu, “Exact Solutions to the two-dimensional generalized fifth order Kuramoto–Sivashinsky-type equation”, Chinese Journal of Physics, 34:2 (1996), 85–90

[40] N. G. Berloff, L. N. Howard, “Solitary and Periodic Solutions of nonlinear nonintegrable equations”, Studies in Applied Mathematics, 99, 1–24

[41] Z. Fu, S. Liu, S. Liu, “New exact solutions to the KdV–Burgers–Kuramoto equation”, Chaos, Solitons and Fractals, 23 (2005), 609–616 | DOI

[42] N. A. Kudryashov, “Simplest equation method to look for exact solutions of nonlinear differential equations”, Chaos, Solitons and Fractals, 24 (2005), 1217–1231 | DOI

[43] S. Zhang, “New exact solutions of the KdV–Burgers–Kuramoto equation”, Phys Lett. A., 358, 414–420

[44] S. A. Khuri, “Traveling wave solutions for nonlinear differential equations: a unified ansätze approach”, Chaos, Solitons and Fractals, 32 (2007), 252–258 | DOI

[45] J. Nickel, “Travelling wave solutions to the Kuramoto–Sivashinsky equation”, Chaos, Solitons and Fractals, 33 (2007), 1376–1382 | DOI

[46] N. A. Kudryashov, M. V. Demina, “Polygons of differential equations for finding exact solutions”, Chaos, Solitons and Fractals, 33 (2007), 1480–1496 | DOI

[47] H. Aspe, M. C. Depassier, “Evolution equation of survace waves in a convecting fluids”, Physical Review A, 41:6, 19900, 3125–3128 | DOI

[48] A. V. Porubov, “Exact travelling wave solutions of nonlinear evolution of survace waves in a convecting fluids”, Journal of Physics A. Mathematical and General, 26:7 (1993), L797–L800 | DOI

[49] N. A. Kudryashov, D. I. Sinelshchikov, “Exiended models of non-linear waves in liquid with gas bubbles”, International Journal of Non-Linear Mechanics, 63 (2014), 31–38 | DOI

[50] M. V. Demina, N. A. Kudryashov, “Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations”, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1127–1134 | DOI

[51] M. V. Demina, N. A. Kudryashov, “From Laurent series to exact meromorphic solutions: The Kawahara equation”, Phys. Lett. A, 374 (2010), 4023–4029 | DOI

[52] M. V. Demina, N. A. Kudryashov, “On elliptic solutions of nonlinear ordinary differential equations”, Applied Mathematics and Computation, 217:23 (2011), 9849–9853 | DOI