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A note on solutions of the generalized Fisher equation. (English) Zbl 1327.35165
Summary: The generalized Fisher equation is considered. Possible exact solutions of this equation are found by \(Q\)-function method. The velocities of traveling waves are determined and analyzed.

MSC:
35K55 Nonlinear parabolic equations
35C05 Solutions to PDEs in closed form
35C07 Traveling wave solutions
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[1] Fisher, R. A., The wave of advance of advantageous genes, Ann. Eugenics, 7, 335-369, (1937) · JFM 63.1111.04
[2] Kolmogorov, A. N.; Petrovsky, I. G.; Piscunov, N. S., Investigation of a diffusion equation connected to the growth of materials, and application to a problem in biology, Bull. Univ. Moscow, Ser. Int. Sec. A, 1, 1-26, (1937), (in Russian)
[3] Ablowitz Mark, J.; Zeppetella, Anthony, Explicit solutions of fisher’s equation for a special wave speed, Bull. Math. Biol., 41, 6, 835-840, (1979) · Zbl 0423.35079
[4] Korpusov, M. O.; Ovchinnikov, A. V.; Sveshnikov, A. G., On blow up of generalized Kolmogorov-Petrovskii-piskunov equation, Nonlinear Anal. TMA, 71, 11, 5724-5732, (2009) · Zbl 1180.35134
[5] Kudryashov, N. A., Exact solitary waves of the Fisher equation, Phys. Lett. A, 342, 1-2, 99-106, (2005) · Zbl 1222.35054
[6] Vitanov, N. K.; Jordanov, I. P.; Dimitrova, Z. I., On nonlinear population waves, Appl. Math. Comput., 215, 8, 2950-2964, (2009) · Zbl 1181.92083
[7] Murray, J. D., Mathematical biology. I. an introduction, 556, (2001), Springer-Verlag
[8] Aronson, D. G.; Weinberger, H. F., Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differ. Eq. Relat. Top., 446, 5-49, (1975) · Zbl 0325.35050
[9] Kudryashov, N. A., Exact solutions of the Burgers-Huxley equation, J. Appl. Math. Mech., 68, 3, 413-420, (2004) · Zbl 1092.35084
[10] Feng, Z. A.; Tian, J. A.; Zheng, S. B.; Lu, H. B., Travelling wave solutions of the Burgers-Huxley equation, IMA J. Appl. Math. (Institute of Mathematics and its Applications), 77, 3, 316-325, (2012) · Zbl 1250.35152
[11] Malfliet, W.; Hereman, W., The tanh method: I. exact solutions of nonlinear evolution and wave equations, Phys. Scr., 54, 563-568, (1996) · Zbl 0942.35034
[12] Parkes, E. J.; Duffy, B. R., An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Comm., 98, 288-300, (1996) · Zbl 0948.76595
[13] Biswas, A., Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett., 22, 208-210, (2009) · Zbl 1163.35468
[14] Kudryashov, N. A., Exact solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A, 147, 287-291, (1990)
[15] Kudryashov, N. A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Solitons Fractals, 24, 1217-1231, (2005) · Zbl 1069.35018
[16] 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. Simul., 15, 2050-2060, (2010) · Zbl 1222.35062
[17] Wang, M. L.; Li, X.; Zhang, J., The \(G^\prime / G\)-expansion method and evolution equation in mathematical physics, Phys. Lett. A, 372, 417-421, (2008)
[18] Kudryashov, N. A., A note on the \(G^\prime / G\)-expansion method, Appl. Math. Comput., 217, 1755-1758, (2010) · Zbl 1203.35228
[19] Kudryashov, N. A., One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul., 17, 2248-2253, (2012) · Zbl 1250.35055
[20] Kudryashov, N. A., Polynomials in logistic function and solitary waves of nonlinear differential equations, Appl. Math. Comput., 219, 9245-9253, (2013) · Zbl 1297.35076
[21] Polyanin, A. D.; Zaitsev, V. F., Handbook of nonlinear partial differential equations, (2011), Chapman and Hall/CRC Boca Raton · Zbl 1024.35001
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