Construction of Exact Solutions to Partial Differential Equations with CRE Method

Year 2019, Volume: 2 Issue: 2, 105 - 113, 27.06.2019

Filiz Taşcan , Arzu Akbulut

https://doi.org/10.33434/cams.486401

Cited By: 5

Abstract

In this article, the consistent Riccati expansion (CRE) method is presented for constructing new exact solutions of (1+1) dimensional nonlinear dispersive modified Benjamin Bona Mahony (DMBBM) and mKdV-Burgers equations. The exact solutions obtained are composed of hyperbolic and exponential functions. The outcomes obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear partial differential equations.

Keywords

Partial differential equations, Exact solution, The consistent Riccati expansion

References

• [1] M. Kaplan, A. Akbulut, A. Bekir, Exact travelling wave solutions of the nonlinear evolution equations by auxiliary equation method, Z. Naturforsch A, 70 (2015), 969–974.
• [2] A. Bekir, A. Akbulut, M. Kaplan, Exact solutions of nonlinear evolution equations by using modified simple equation method, Int. J. Nonlinear Sci., 19 (2015), 159-164.
• [3] F. Tas¸can, A. Yakut, Conservation laws and exact solutions with symmetry reduction of nonlinear reaction diffusion equations, Int. J. Nonlinear Sci. Numer. Simul., 16 (2015), 191–196.
• [4] M. Ekici, M. Mirzazadeh, Q. Zhou, S. P. Moshokoa, A. Biswas, M. Belic, Solitons in optical metamaterials with fractional temporal evolution, Optik, 127 (2016), 10879-10897.
• [5] M. Mirzazadeh, M. Eslami, D. Milovic, A. Biswas, Topological solitons of resonant nonlinear Sch¨odinger’sequation with dual-power law nonlinearity by (G0=G)-expansion technique, Optik, 125 (19), (2014) 5480-5489.
• [6] Q. Feng, F. Meng, Explicit solutions for space-time fractional partial differential equations in mathematical physics by a new generalized fractional Jacobi elliptic equation-based sub-equation method, Optik, 127 (2016), 7450-7458.
• [7] A. Akbulut, M. Kaplan, F. Tas¸can, The investigation of exact solutions of nonlinear partial differential equations by using exp$\left(-\Phi \left( \xi \right) \right) $ method, Optik, 132 (2017), 382-387.
• [8] A. Biswas, M. Mirzazadeh, M. Eslami, D. Milovic, M. Belic, Solitons in optical metamaterials by functional variable method and first integral approach, Frequenz, 68 (11-12) (2014), 525-530.
• [9] B. Lu, The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 395 (2012), 684-693.
• [10] H.M. Bas¸konus¸, H. Bulut , Analytical studies on the (1+1)-dimensional nonlinear Dispersive Modified Benjamin-Bona- Mahony equation defined by seismic sea waves, Waves Random Complex Media, doi:10.1080/17455030.1062577, 2015.
• [11] J. Xiang-Li, L. Sen-Yue, CRE method for solving mKdV equation and new interactions between solitons and cnoidal periodic waves, Commun. Theor. Phys., 63 (2015), 7-9.
• [12] M. Chen, H. Hu, H. Zhu, Consistent Riccati expansion and exact solutions of the Kuramoto-Sivashinsky equation, Appl. Math. Lett., 49 (2015), 147-151.
• [13] K. Khan, M.A. Akbar, S.M. Raynaul Islam , Exact solutions for (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations, SpringerPlus, 3 (2014), 8 pages.
• [14] E.M.E. Zayed, S. Al-Joudi, Applications of an extended (G’/G)-expansion method to find exact solutions of nonlinear PDEs in mathematical physics, Math. Probl. Eng., Article ID 768573, doi:10.1155/2010/768573, (2010), 19 pages.
• [15] M. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A., 213 (1996), 279-287.