ON THE KOLMOGOROV-PETROVSKII-PISKUNOV EQUATION

Year 2013, Volume: 62 Issue: 1, 1 - 10, 01.02.2013

Arzu Ögün Ünal

https://doi.org/10.1501/Commua1_0000000681

Cited By: 7

Abstract

We prove existence and uniqueness of the solutions of Kolmogorov Petrovskii-Piskunov (KPP) equation. We study asymptotic stability and instability of the equilibrium solution u(x; t) = 0 of KPP equation with subject
to the traveling wave solutions. We show that KPP equation has not got any
periodic traveling wave solution. Also, we obtain some exact traveling wave
solutions of KPP equation by the first integral method.

Keywords

Existence and uniqueness of solutions, asymptotic stability, instability, periodicity, traveling wave solutions

References

• [1] Abbasbandy S., Shirzadi A., The Örst integral method for modiÖed Benjamin-Bona-Mahony equation, Commun. Nonlinear Sci. Numer. Simul., 15 (2010),1759ñ1764.
• [2] Adomian G., The generalized Kolmogorov-Petrovskii-Piskunov equation, Foundation of Pyhsics Letters, 8 (1995), 99-101.
• [3] Ali A. H. A., Raslan K. R., The Örst integral method for solving a system of nonlinear partial di§erential equations, Int. J. Nonlinear Sci., 5 (2008),111ñ119.
• [4] Allen L. J. S., An Introduction to Mathematical Biology, 2007, Pearson.
• [5] Branco J.R., Ferreira J.A., Oliveira P. Numerical methods for the generalized Fisherñ KolmogorovñPetrovskiiñPiskunov equation, Applied Numerical Mathematics, 57 (2007), 89- 102.
• [6] Britton N. F. Reaction-Di§usion Equations and Their Applications to Biology, 1986, Academic Press, New York.
• [7] Deng X., Exact peaked wave solution of CH- equation by the Örst-integral method, Appl. Math. Comput., 206 (2008), 806ñ809.
• [8] Feng J., Li W., Wan Q., Using G0 G -expansion method to seek the traveling wave solution of KolmogorovñPetrovskiiñPiskunov equation, Applied Mathematics and Computation, 217 (2011), 5860-5865.
• [9] Feng Z.S., The Örst-integral method to the BurgersñKdV equation, J. Phys. A, 35 (2002), 343ñ350.
• [10] Hong W. P., Jung Y.D., Auto-b‰clund transformation and analytic solutions for general variable coe¢ cient KdV equation, Phys. Lett. A, 257 (1999), 149ñ152.
• [11] Khater A.H., Maláiet W., Callebaut D.K, Kamel E.S., The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction di§usion equations, Chaos Solitons Fractals, 14(3) (2002), 513 - 522.
• [12] Kolmogorov A. N., Petrovskii I. G., Piskunov N. S., Etude de la di§usion avec croissance de la quantitÈ de matiËre et son application ‡ un problËme biologique, Moscow Univ. Math. Bull., 1 (1937), 1ñ25.
• [13] Liu C., The relation between the kink-type solution and the kink-bell-type solution of nonlinear evolution equations, Physics Letters A, 312 (2003), 41-48.
• [14] Lu B., Zhang H.Q., XIE F.D., Travelling wave solutions of nonlinear partial equations by using the Örst integral method, Appl. Math. Comput., 216 (2010),1329-1336.
• [15] Ma W. X., Fuchssteiner B., Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation, Int. J. Non-Linear Mech., 31 (1996), 329-338.
• [16] Murray J. D., Mathematical Biology I: An Introduction, 2002, Springer, Berlin.
• [17] ÷g¸n A., Kart C., Exact solutions of Fisher and generalized Fisher equations with variable º coe¢ cients, Acta Math. Appl. Sin. Engl Ser., 23 (2007), 563-568.
• [18] Raslan K. R., The Örst integral method for solving some important nonlinear partial di§erential equations, Nonlinear Dynam., 53 (2008), 281ñ286.
• [19] S¨mmons G. F., Di§erential Equations, 1989, McGraw-Hill, New York, pp. 341.
• [20] Taghizadeh N., Mirzazadeh M., Farahrooz F., Exact solutions of the nonlinear Schrˆdinger equation by the Örst integral method, J. Math. Anal. Appl., 374 (2011), 549ñ553