New Travelling Wave Solutions of Conformable Cahn-Hilliard Equation

Yıl 2022, Cilt: 5 Sayı: 2, 57 - 62, 31.08.2022

Esin Aksoy , Adem Çevikel

https://doi.org/10.33187/jmsm.1149614

Öz

In this article, two methods are proposed to solve the fractional Cahn-Hilliard equation. This model describes the process of phase separation with nonlocal memory effects. Cahn-Hilliard equations have numerous applications in real-world scenarios, e.g., material sciences, cell biology, and image processing. Different types of solutions have been obtained. For this, the fractional complex transformation has been used to convert fractional differential equation to ordinary differential equation of integer order. As a result, these solutions are new solutions that do not exist in the literature.

Anahtar Kelimeler

Conformable derivatives, Exact solution, Nonlinear water waves, Travelling wave solutions

Kaynakça

• [1] G. I. Dolgikha, D. P. Kovalevb, P. D. Kovalevb, Excitation of Under-ice seiches of a sea port of the sea of Okhotsk, Doklady Earth Sciences, 486(2) (2019), 651–653.
• [2] I. Kovacic, M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and Their Behavior, John Wiley and Sons, London, 2011.
• [3] K.S. Miller, B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
• [4] I. Podlubny, Fractional Differential Equations, Academic Press, California, 1999.
• [5] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ, USA; 2000.
• [6] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
• [7] A. Bekir, O. G¨uner, B. Ayhan, A. C. Çevikel, Exact solutions for fractional differential-difference equations by (G0=G)-expansion method with modified Riemann-Liouville derivative, Advances in Applied Mathematics and Mechanics, 8(2) (2016), 293–305.
• [8] O. Güner, E. Aksoy, A. Bekir, A. C. Çevikel, Different methods for (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation, Comput. Math. with Appl., 71 (2016), 1259–1269.
• [9] O. Güner, A. Bekir, A. C. Çevikel, A variety of exact solutions for the time fractional Cahn-Allen equation, European Physical Journal Plus, 130(2015), 146.
• [10] E. Aksoy, A. C. Çevikel, A. Bekir, Soliton solutions of (2+1)-dimensional time-fractional Zoomeron equation, Optik, 127(17) (2016), 6933–6942.
• [11] A. Bekir, E. Aksoy, A. C. Çevikel, Exact solutions of nonlinear time fractional partial differential equations by sub-equation method, Math. Methods Appl. Sci., 38(13) (2015), 2779–2784.
• [12] A. Bekir, O. Güner, A. C. Cevikel, Fractional complex transform and exp-function methods for fractional differential equations, Abstr. Appl. Anal., 2013 (2013), Article ID: 426462, 8 pages.
• [13] O. Güner, A. C. Çevikel, A procedure to construct exact solutions of nonlinear fractional differential equations, Sci. World J., 2014 (2014), 489495.
• [14] A. C. Çevikel, New exact solutions of the space-time fractional KdV-Burgers and non-linear fractional Foam Drainage equation, Therm. Sci., 22 (2018), 15–24.
• [15] A. Bekir, O. G¨uner, A. C. Çevikel, The Exp-function method for some time-fractional differential equations, IEEE/CAA J. Autom. Sin., 4 (2017), 315–321.
• [16] E. Aksoy, A. Bekir, A. C. Çevikel, Study on fractional differential equations with modified Riemann-Liouville derivative via Kudryashov method, Int. J. Nonlinear Sci. Numer. Simul., 20(5) (2019), 511–516.
• [17] H. Rezazadeh, J. Vahidi, A. Zafar, A. Bekir, The functional variable method to find new exact solutions of the nonlinear evolution equations with dual-power-law nonlinearity, Int. J. Nonlinear Sci. Numer. Simul., 21(3-4) (2020), 249-257.
• [18] A. C. Çevikel, A. Bekir, M. Akar, S. San, Procedure to construct exact solutions of nonlinear evolution equations, Pramana Journal of Physics, 79(3) (2012), 337–344.
• [19] N. Taghizadeh, M. Mirzazadeh, M. Rahimian, M. Akbari, Application of the simplest equation method to some time-fractional partial differential equations, Ain Shams Eng., 4 (2013), 897–902.
• [20] N. Savaissou, B. Gambo, H. Rezazadeh, A. Bekir, S. Y. Doka, Exact optical solitons to the perturbed nonlinear Schrodinger equation with dual-power law of nonlinearity, Opt. Quantum Electron., 52 (2020), 318.
• [21] A. C. Çevikel, E. Aksoy, Soliton solutions of nonlinear fractional differential equations with their applications in mathematical physics, Revista Mexicana de Fisica, 67, (3) (2021), 422–428.
• [22] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70.
• [23] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66.
• [24] M. Eslami, H. Rezazadeh, The first integral method for Wu–Zhang system with conformable time-fractional derivative, Calcolo, 53 (2016), 475–485.
• [25] A. C. Çevikel, A. Bekir, New solitons and periodic solutions for (2+1)-dimensional Davey-Stewartson equations, Chinese J. Phys., 51(1) (2013), 1–13.
• [26] A. Bekir, A. C. Çevikel, Solitary wave solutions of two nonlinear physical models by tanh-coth method, Communications in Commun. Nonlinear Sci. Numer. Simul., 14(5) (2009), 1804–1809.
• [27] A. Bekir, A. C. Çevikel, New solitons and periodic solutions for nonlinear physical models in mathematical physics, Nonlinear Anal. Real World Appl., 11(4) (2010), 3275–3285.
• [28] H. Jafari, H. Tajadodi, D. Baleanu, A. A. Al-Zahrani, Y. A. Alhamed, A. H. Zahid, Fractional sub-equation method for the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation, Cent. Eur. J. Phys., 11(10) (2013), 1482–1486.