Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity
Yıl: 2021 Cilt: 11 Sayı: 2 Sayfa Aralığı: 123 - 127 Metin Dili: İngilizce DOI: 10.11121/ijocta.01.2021.00973 İndeks Tarihi: 17-02-2022
Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity
Öz: This paper proposes a new dispersion-convection-reaction model, which is called the gKdV-Fisher equation, to obtain the travelling wave solutions by using the Riccati equation method. The proposed equation is a third-order dispersive partialdifferential equation combining the purely nonlinear convective term with thepurely nonlinear reactive term. The obtained global and blow-up solutions, whichmight be used in the further numerical and analytical analyses of such models, areillustrated with suitable parameters.
Anahtar Kelime: Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
- [1] Korteweg, D. J., & de Vries, G. (1895). On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39(240), 422-443.
- [2] Tsutsumi, M., Mukasa, T., & Iino, R. (1970). On the generalized Korteweg-de Vries Equation. Proceedings of the Japan Academy, 46(9), 921-925.
- [3] Gardner, C. S., Greene, J. M., Kruskal, M. D., & Miura, R. M. (1974). Korteweg‐devries equation and generalizations. VI. methods for exact solution. Communications on Pure and Applied Mathematics, 27(1), 97-133.
- [4] Tagare, S. G., & Chakrabarti, A. (1974). Solution of a generalized Korteweg-de Vries equation. Physics of Fluids, 17, 1331-1332.
- [5] Rosenau, P., & Hyman, M. (1993). Compactons: Solitons with Finite Wavelength. Physical Review Letters, 70, 564-567.
- [6] Colliander, J. E., & Kenig, C. E. (2002). The generalized Korteweg-de Vries equation on the half line. Communications in Partial Differential Equations, 27, 2187-2266.
- [7] Wazwaz, A. M. (2006). Kinks and solitons for the generalized KdV equation with two power nonlinearities. Applied Mathematics and Computation, 183, 1181-1189.
- [8] Tao, T. (2007). Two remarks on the generalised Korteweg de-Vries equation. Discrete & Continuous Dynamical Systems - A, 18, 1-14.
- [9] Tao, T. (2008). Global behaviour of nonlinear dispersive and wave equations. Current Developments in Mathematics, 2006, 255-340.
- [10] Pinar, Z., & Koçak, H. (2018). Exact solutions for the third-order dispersive-Fisher equations. Nonlinear Dynamics, 91(1), 421-426.
- [11] Rosenau, P. (1998). On a class of nonlinear dispersive-dissipative interactions. Physica D: Nonlinear Phenomena, 123(1-4), 525-546.
- [12] Wazwaz, A. M. (2006). The tanh method for compact and noncompact solutions for variants of the KdV-Burger and the K(n,n)-Burger equations. Physica D: Nonlinear Phenomena, 213, 147-151.
- [13] Galaktionov, V. A., Miditieri, E. L., & Pohozaev, S. I. (2014). Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations. Monographs and Research Notes in Mathematics, Chapman and Hall/CRC, Boca Raton.
- [14] Koçak, H. (2017). Similarity solutions of nonlinear third-order dispersive PDEs: The first critical exponent. Applied Mathematics Letters, 74, 108- 113.
- [15] Koçak, H., & Pinar Z. (2018). On solutions of the fifth-order dispersive equations with porous medium type non-linearity. Waves in Random and Complex Media, 28(3), 516-522.
- [16] Fisher, R. A. (1937). The wave of advance of advantageous genes. Annals of Eugenics, 7(4), 355- 369.
- [17] Kolmogorov, A. N., Petrovskii, I. G., & Piskunov, N. S. (1937). Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem. Byull. Moskov. Gos. Univ., Sect. A, 1, 1–26. (English. transl. In: Dynamics of Curved Fronts, P. Pelce, Ed., Acad. Press, Inc., New York, 1988, 105–130.)
- [18] Ablowitz, M. J., & Zeppetella, A. (1979). Explicit solutions of Fisher's equation for a special wave speed. Bulletin of Mathematical Biology, 41, 835– 840.
- [19] Wazwaz, A. M. (2008). Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations. Applied Mathematics and Computation, 195(2), 754-761.
- [20] Gilding, B. H., & Kersner, R. (2012). Travelling waves in nonlinear diffusion-convection-reaction. Vol 60, Birkhäuser Basel.
- [21] Galaktionov, V. A. (2012). Towards the KPPProblem and log t-Front Shift for Higher-Order Nonlinear PDES II. Quasilinear Bi-and TriHarmonic Equations. arXiv:1210.5063.
- [22] Galaktionov, V. A. (2012). Towards the KPP-- Problem and log t-Front Shift for Higher-Order Nonlinear PDEs III. Dispersion and Hyperbolic Equations. arXiv:1210.5084.
- [23] Galaktionov, V. A. (2013). The KPP-Problem and log t-Front Shift for Higher-Order Semilinear Parabolic Equations. Proceedings of the Steklov Institute of Mathematics, 283, 44–74.
- [24] Koçak, H. (2020). Travelling Waves in Nonlinear Media with Dispersion, Dissipation and Reaction. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(9), 093143.
- [25] Chen, H., & Zhang, H. (2004). New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramato-Sivashinsky equation. Chaos, Soltions and Fractals, 19, 71-76.
- [26] Wazwaz, A. M. (2005). The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations. Applied Mathematics and Computation, 169, 321-338.
- [27] Huibin, L., & Kelin, W. (1990). Exact solutions for nonlinear equations. I. Journal of Physics A: Mathematical and General, 23(17), 3923-3928.
- [28] Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7), 650-654.
- [29] Malfliet, W., & Hereman, W. (1996). The Tanh Method: I. Exact Solutions of Nonlinear Evolution and Wave Equations. Physica Scripta, 54, 563-568.
- [30] Fan, E. G. (2002). Traveling Wave Solutions for Nonlinear Equations Using Symbolic Computation. Computers and Mathematics with Applications, 43, 671-680.
- [31] Kudryashov, N. A. (2005). Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons & Fractals, 24 (5), 1217-1231.
- [32] Kudryashov, N. A. (2009). Seven common errors in finding exact solutions of nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 14, 3507-3529.
- [33] Wazwaz, A. M. (2009). Partial Differential Equations and Solitary Wave Theory. Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg.
- [34] Griffiths, G. W., & Schiesser, W. E. (2011). Traveling Wave analysis of Partial Differential Equations: Numerical and Analytical Methods with MATLAB and MAPLE. Academic Press, New York.
- [35] Polyanin, A. D., & Zaitsev, V. F. (2012). Handbook of Nonlinear Partial Differential Equations. Chapman and Hall/CRC, Boca Raton.
- [36] Manafian, J., & Lakestani, M. (2017). A new analytical approach to solve some of the fractionalorder partial differential equations. Indian Journal of Physics, 91, 243-258.
| APA | Kocak H (2021). Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. , 123 - 127. 10.11121/ijocta.01.2021.00973 |
| Chicago | Kocak Huseyin Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. (2021): 123 - 127. 10.11121/ijocta.01.2021.00973 |
| MLA | Kocak Huseyin Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. , 2021, ss.123 - 127. 10.11121/ijocta.01.2021.00973 |
| AMA | Kocak H Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. . 2021; 123 - 127. 10.11121/ijocta.01.2021.00973 |
| Vancouver | Kocak H Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. . 2021; 123 - 127. 10.11121/ijocta.01.2021.00973 |
| IEEE | Kocak H "Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity." , ss.123 - 127, 2021. 10.11121/ijocta.01.2021.00973 |
| ISNAD | Kocak, Huseyin. "Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity". (2021), 123-127. https://doi.org/10.11121/ijocta.01.2021.00973 |
| APA | Kocak H (2021). Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11(2), 123 - 127. 10.11121/ijocta.01.2021.00973 |
| Chicago | Kocak Huseyin Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 11, no.2 (2021): 123 - 127. 10.11121/ijocta.01.2021.00973 |
| MLA | Kocak Huseyin Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), vol.11, no.2, 2021, ss.123 - 127. 10.11121/ijocta.01.2021.00973 |
| AMA | Kocak H Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. An International Journal of Optimization and Control: Theories & Applications (IJOCTA). 2021; 11(2): 123 - 127. 10.11121/ijocta.01.2021.00973 |
| Vancouver | Kocak H Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity. An International Journal of Optimization and Control: Theories & Applications (IJOCTA). 2021; 11(2): 123 - 127. 10.11121/ijocta.01.2021.00973 |
| IEEE | Kocak H "Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity." An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11, ss.123 - 127, 2021. 10.11121/ijocta.01.2021.00973 |
| ISNAD | Kocak, Huseyin. "Kink and anti-kink wave solutions for the generalized KdV equation withFisher-type nonlinearity". An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 11/2 (2021), 123-127. https://doi.org/10.11121/ijocta.01.2021.00973 |
