New exact wave solutions to the space-time fractional-coupled Burgers equations and the space-time fractional foam drainage equation

References

• Akgul, A., Inc, M., Karatas, E., & Baleanu, D. (2015). Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique. Advances in Difference Equations, 2015(220), 1. doi:10.1186/s13662-0015-0558-8
   Google Scholar
• Akgul, A., Kilicman, A., & Inc, M. (2013). Improve (G′/G)-expansion method for space and time fractional foam drainage and KdV equations. Abstract and Applied Analysis, 2013, 7, Article ID 414353.
   Google Scholar
• Akgül, A., Khan, Y., Akgül, E. K., Baleanu, D., & Al Qurashi, M. M. (2017). Solutions of nonlinear systems by reproducing kernel method. The Journal of Nonlinear Sciences and Applications, 10, 4408–4417.10.22436/jnsa
   Web of Science ®Google Scholar
• Alam, M. N., & Akbar, M. A. (2014a). Application of the new approach of generalized (G′/G)-expansion method to find exact solutions of nonlinear PDEs in mathematical physics. BIBECHANA, 10, 58–70.
   Google Scholar
• Alam, M. N., & Akbar, M. A. (2014b). The new approach of the generalized (G'/G)-expansion method for nonlinear evolution equations. Ain Shams Engineering Journal, 5, 595–603.10.1016/j.asej.2013.12.008
   Google Scholar
• Ali, A. H. A. (2007). The modified extended tanh-function method for solving coupled MKdV and coupled Hirota-Satsuma coupled KdV equations. Physics Letters A, 363(5–6), 420–425.10.1016/j.physleta.2006.11.076
   Web of Science ®Google Scholar
• Ali, A., Inc, M., & Karatas, E. (2015). Useful reproducing kernel functions for solving difference equations. Discrete & Continuous Dynamical Systems-Series S, 8(1), 1055–1064.
   Google Scholar
• Alzaidy, J. F. (2013). The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs. British Journal of Mathematics & Computer Science, 3, 153–163.10.9734/BJMCS
   Google Scholar
• Bekir, A., & Guner, O. (2013). Exact solutions of nonlinear fractional differential equation by (G'/G)-expansion method. Chinese Physics B, 22(11), 1–6.
   Web of Science ®Google Scholar
• Bekir, A., & Güner, Ö. (2014). The (G′/G)-expansion method using modified Riemann–Liouville derivative for some space-time fractional differential equations. Ain Shams Engineering Journal, 5, 959–965.10.1016/j.asej.2014.03.006
   Google Scholar
• Bulut, H., Baskonus, H. M., & Pandir, Y. (2013). The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation. Abstract and Applied Analysis, 2013, 1, Article ID 636802.
   Web of Science ®Google Scholar
• Caputo, M., & Fabrizio, M. A. (2015). A new definition of fractional derivatives without singular kernel. Mathematical and Computer Modelling, 1, 73–85.
   Google Scholar
• Dahmani, Z., Mesmoudi, M. M., & Bebbouchi, R. (2008). The foam-drainage equation with time and space fractional derivative solved by the ADM method. Electronic Journal of Qualitative Theory of Differential Equations, 30, 1–10.
   Google Scholar
• Deng, W. (2009). Finite element method for the space and time fractional Fokker–Planck equation. SIAM Journal on Numerical Analysis, 47(1), 204–226.10.1137/080714130
   Web of Science ®Google Scholar
• Ege, S. M., & Misirli, E. (2014). Solutions of space-time fractional foam drainage equation and the fractional Klein–Gordon equation by use of modified Kudryashov method. International Journal of Research in Advent Technology, 2(3), 384–388.
   Google Scholar
• El-Borai, M. M., El-Sayed, W. G., & Al-Masroub, R. M. (2015). Exact solutions for time fractional coupled Whitham–Broer–Kaup equations via exp-function method. International Research Journal of Engineering and Technology, 2(6), 307–315.
   Google Scholar
• El-Sayed, A. M. A., Behiry, S. H., & Raslan, W. E. (2010). Adomian’s decomposition method for solving an intermediate fractional advection–dispersion equation. Computers & Mathematics with Applications, 59(5), 1759–1765.10.1016/j.camwa.2009.08.065
   Web of Science ®Google Scholar
• Elzaki, S. M. (2015). Exact solutions of coupled burgers equation with time-and space-fractional derivative. International Journal of Applied Mathematical Research, 4(1), 99–105.10.14419/ijamr.v4i1
   Google Scholar
• Fereidoon, A., Yaghoobi, H., & Davoudabadi, M. (2011). Application of the homotopy perturbation method for solving the foam drainage equation. International Journal of Differential Equations, 2011, 13, Article ID 864023.
   Google Scholar
• Guner, O., & Bekir, A. (2017). A novel method for nonlinear fractional differential equations using symbolic computation. Waves in Random and Complex Media, 27(1), 163–170.10.1080/17455030.2016.1213462
   Web of Science ®Google Scholar
• Guo, S., Mei, L., Li, Y., & Sun, Y. (2012). The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics. Physics Letters A, 376, 407–411.
   Web of Science ®Google Scholar
• Hashemi, M. S., Inc, M., Karatas, E., & Akgül, A. (2017). A numerical investigation on Burgers equation by the MQL-GPS method. Journal of Advanced Physics, 6(3), 413–417.10.1166/jap.2017.1357
   Web of Science ®Google Scholar
• He, J. H. (2012). Asymptotic methods for solitary solutions and compacts. Abstract and Applied Analysis, 2012. Article ID 916793, 130 p.
   Google Scholar
• He, J. H., Elagan, S. K., & Li, Z. B. (2012). Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physics Letters A, 376(4), 257–259.10.1016/j.physleta.2011.11.030
   Web of Science ®Google Scholar
• Helal, M. A., & Mehanna, M. S. (2007). The tanh method and Adomian decomposition method for solving the foam drainage equation. Applied Mathematics and Computation, 190(1), 599–609.10.1016/j.amc.2007.01.055
   Web of Science ®Google Scholar
• Hutzler, S., Weaire, D., Saugey, A., Cox, S., & Peron, N. (2014). The Physics of foam drainage. S. J. Cox.
   Google Scholar
• Inc, M. (2008). The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. Journal of Mathematical Analysis and Applications, 345(1), 476–484.10.1016/j.jmaa.2008.04.007
   Web of Science ®Google Scholar
• Inc, M., Kilic, B., Karatas, E., Al-Qurashi, M. M., & Tchier, F. (2016). On soliton solutions of the Wu-Zhang system. Open Physics. doi:10.1515/Phys-2016-0004,14,76-80
   Web of Science ®Google Scholar
• Inc, M., Kılıç, B., Karataş, E., & Akgül, A. (2017). Solitary wave solutions for the Sawada–Kotera equation. Journal of Advanced Physics, 6(2), 288–293.10.1166/jap.2017.1318
   Web of Science ®Google Scholar
• Jumarie, G. (2006). Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications, 51(9–10), 1367–1376.10.1016/j.camwa.2006.02.001
   Web of Science ®Google Scholar
• Kaplan, M., Bekir, A., Akbulut, A., & Aksoy, E. (2015). The modified simple equation method for nonlinear fractional differential equations. Romanian Journal of Physics, 60(9–10), 1374–1383.
   Web of Science ®Google Scholar
• Khani, F., Hamedi-Nezhad, S., Darvishi, M. T., & Ryu, S. W. (2009). New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method. Nonlinear Analysis: Real World Applications, 10(3), 1904–1911.10.1016/j.nonrwa.2008.02.030
   Web of Science ®Google Scholar
• Kurudy, M. (2010). The approximate and exact solutions of the space and time fractional Burgers equations. International Journal of Research and Reviews in Applied Sciences, 3(3), 257–263.
   Google Scholar
• Lu, B. (2012a). Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations. Physics Letters A, 376, 2045–2048.10.1016/j.physleta.2012.05.013
   Web of Science ®Google Scholar
• Lu, B. (2012b). The first integral method for some time fractional differential equations. Journal of Mathematical Analysis and Applications, 395, 684–693.
   Web of Science ®Google Scholar
• Momani, S., Odibat, Z., & Erturk, V. S. (2007). Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation. Physics Letters A, 370, 379–387.10.1016/j.physleta.2007.05.083
   Web of Science ®Google Scholar
• Neamaty, A., Agheli, B., & Darzi, R. (2015). Variational iteration method and He’s polynomials for time fractional partial differential equations. Progress in Fractional Differentiation and Applications, 1, 47–55.
   Google Scholar
• Rabtah, A. A., Erturk, R. S., & Momani, S. (2010). Solutions of a fractional oscillator by using differential transform method. Computers & Mathematics with Applications, 59, 1356–1362.10.1016/j.camwa.2009.06.036
   Web of Science ®Google Scholar
• Wang, G. W., & Xu, T. Z. (2014). The modified fractional sub-equation method and its applications to nonlinear fractional partial differential equations. Romanian Journal of Physics, 59(7–8), 636–645.
   Web of Science ®Google Scholar
• Younis, M. (2013). The first integral method for time-space fractional differential equations. Journal of Advanced Physics, 2, 220–223.10.1166/jap.2013.1074
   Google Scholar
• Younis, M., & Zafar, A. (2014). Exact solutions to nonlinear differential equations of fractional order via (G'/G)-expansion method. Applied Mathematics, 5(1), 1–6.10.4236/am.2014.51001
   Google Scholar
• Zayed, E. M. E., Amer, Y. A., & Al-Nowehy, A. G. (2016). The modified simple equation method and the multiple exp-function method for solving nonlinear fractional Sharma–Tasso–Olver equation. Acta Mathematicae Applicatae Sinica, English Series, 32(4), 793–812.10.1007/s10255-016-0590-9
   Web of Science ®Google Scholar
• Zhao, J., Tang, B., Kumar, S., & Hou, Y. (2012). The extended fractional sub-equation method for nonlinear fractional differential equations. Mathematical Problems in Engineering, 2012, 11, Article ID 924956.
   Web of Science ®Google Scholar
• Zheng, B. (2013). Exp-function method for solving fractional partial differential equations. Scientific World Journal. doi:10.1155/2013/465723
   Google Scholar
• Zheng, B., & Feng, Q. (2014). The Jacobi elliptic equation method for solving fractional partial differential equations. Abstract and Applied Analysis, 2014, 9, Article ID 249071.
   Google Scholar