Abstract
In this study, with the help of fractional complex transform and new analytical method namely, improved \(\tan (\phi (\xi )/2)\)-expansion method (ITEM), we obtained new solitary wave solution for time-fractional nonlinear Kuramoto–Sivashinsky equation. By using of the fractional complex transform we could convert a nonlinear fractional differential equation into its equivalent ordinary differential equation form. Description of the method is given and the obtained results reveal that the ITEM is a new significant method for exploring nonlinear partial differential models. It is worth mentioning that some of newly obtained solutions are identical to already published results as Tanh method Sahoo and Saha Ray (Phys A 434:240–245, 2015). Therefore, this method can be applied to study many other nonlinear fractional partial differential equations which frequently arise in engineering, mathematical physics and nonlinear optic. Moreover, by using Matlab, some graphical simulations were done to see the behavior of these solutions.
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References
Aghdaei, M.F., Manafian, J.: Optical soliton wave solutions to the resonant Davey–Stewartson system. Opt. Quant. Electron 48, 1–33 (2016)
Aghdaei, M.F., Manafianheris, J.: Exact solutions of the couple Boiti–Leon–Pempinelli system by the generalized \(\rm (\frac{G^{\prime }}{G})\)-expansion method. J. Math. Ext. 5, 91–104 (2011)
Alfaro, C.M., Depassier, M.C.: A five-mode bifurcation analysis of a Kuramoto–Sivashinsky equation with dispersion. Phys. Lett. A 184, 184–189 (1994)
Alfaro, C.M., Benguria, R.D., Depassier, M.C.: Finite mode analysis of the generalized Kuramoto–Sivashinsky equation. Physica D 61, 1–5 (1992)
Aslan, I., Marinakis, V.: Some remarks on Exp-function method and its applications. Commun. Theor. Phys. 56, 397–403 (2011)
Barker, B., Johnson, M.A., Noble, P., Rodrigues, L.M., Zumbruna, K.: Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation. Physica D 258, 11–46 (2013)
Bekir, A.: Application of the Exp-function method for nonlinear differential-difference equations. Appl. Math. Comput. 215(11), 352–362 (2010)
Bulut, H., Baskonus, H.M., Pandir, Y.: The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation, Abstr. Appl. Anal. (2013). Article ID 636802
Chen, Y., Wang, Q.: Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1 + 1)-dimensional dispersive long wave equation. Chaos Solitons Fract. 24, 745–757 (2005)
Conte, R.: Exact solutions of nonlinear partial differential equations by singularity analysis. In: Lecture Notes in Physics, Springer, p. 183 (2003)
Debnath, L.: Nonlinear Partial Differential Equations for Scientists and Engineers, 2nd edn. Birkhäusher, Boston (2005)
Dehghan, M., Manafian, J.: The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method. Z. Naturforsch 64a, 420–430 (2009)
Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Num. Meth. Part. Differ. Eq. J. 26, 448–479 (2010)
Dehghan, M., Manafian, J., Saadatmandi, A.: Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics, Int. J. Num. Methods Heat Fluid Flow 21, 736–753 (2011)
Ekici, M., Zhou, Q., Sonmezoglu, A., Manafian, J., Mirzazadeh, M.: The analytical study of solitons to the nonlinear Schödinger equation with resonant nonlinearity. Optik-Int. J. Light Electron Opt. 130, 378–382 (2017)
El-Ajou, A., Abu Arqub, O., Momani, S.: Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: a new iterative algorithm. J. Comput. Phys. 293, 81–95 (2015)
Eslami, M., Vajargah, Fathi, Mirzazadeh, M., Biswas, A.: Application of first integral method to fractional partial differential equations. Indian J. Phys. 88, 177–184 (2014)
Helal, M.A., Seadawy, A.R.: Variational method for the derivative nonlinear Schrödinger equation with computational applications. Phys. Scr. 80, 035004 (2009)
Jumarie, G.: Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51, 1367–1376 (2006)
Khater, A.H., Temsah, R.S.: Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods. Comput. Math. Appl. 56, 1465–1472 (2008)
Kudryashov, N.: Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3507–3529 (2009)
Kudryashov, N., Loguinova, N.: Be careful with the Exp-function method. Commun. Nonlinear Sci. Numer. Simul. 14, 1881–1990 (2009)
Kurulay, M., Secer, A., Akinlar, M.A.: A new approximate analytical solution of Kuramoto–Sivashinsky equation using homotopy analysis method. Appl. Math. Inf. Sci. 7, 267–271 (2013)
Manafian, J.: On the complex structures of the Biswas-Milovic equation for power, parabolic and dual parabolic law nonlinearities. Eur. Phys. J. Plus 130, 1–20 (2015)
Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolutionequations by the \(tan(\phi /2)\)-expansion method. Optik-Int. J. Electron Opt. 127, 4222–4245 (2016)
Manafian, J., Lakestani, M.: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 1–12 (2015)
Manafian, J., Lakestani, M.: Application of \(tan(\phi /2)\)-expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity. Optik-Int. J. Electron Opt. 127, 2040–2054 (2016a)
Manafian, J., Lakestani, M.: Dispersive dark optical soliton with Tzitzéica type nonlinear evolution equations arising in nonlinear optics. Opt. Quant. Electron 48, 1–32 (2016b)
Manafian, J., Lakestani, M.: Abundant soliton solutions for the KunduEckhaus equation via \(tan(\phi /2)\)-expansion method. Optik-Int. J. Electron Opt. 127, 5543–5551 (2016c)
Manafian, J., Lakestani, M.: A new analytical approach to solve some of the fractional-order partial differential equations. Indian J. Phys. 91(3), 243–258 (2017)
Manafian, J., Lakestani, M., Bekir, A.: Study of the analytical treatment of the (2 + 1)-dimensional Zoomeron, the Duffing and the SRLW equations via a new analytical approach. Int. J. Appl. Comput. Math. 2, 243–268 (2016a)
Manafian, J., Aghdaei, M.F., Zadahmad, M.: Analytic study of sixth-order thin-film equation by \(tan(\phi /2)\)-expansion method. Opt. Quant. Electron 48, 1–16 (2016b)
Marcinkevicius, R., Navickas, Z., Ragulskis, M., Telksnys, T.: Solitary solutions to a relativistic two-body problem. Astrophys. Space Sci. 361, 201–207 (2016)
Mirzazadeh, M.: Analytical study of solitons to nonlinear time fractional parabolic equations. Nonlinear Dyn. 85, 2569–2576 (2016)
Mirzazadeh, M., Eslami, M.: Exact multisoliton solutions of nonlinear Klein–Gordon equation in \(1+2\) dimensions. Eur. Phys. J. Plus 128, 1–9 (2015)
Mirzazadeh, M., Eslami, M., Milovic, D., Biswas, A.: Topological solitons of resonant nonlinear Schödinger’s equation with dual-power law nonlinearity by G′/G-expansion technique. Optik-Int. J. Light Electron Opt. 125, 5480–5489 (2014)
Mirzazadeh, M., Eslami, M., Arnous, A.H.: Dark optical solitons of Biswas–Milovic equation with dual-power law nonlinearity. Eur. Phys. J. Plus 130, 1–7 (2015)
Mirzazadeh, M., Ekici, M., Zhou, Q., Biswas, A.: Exact solitons to generalized resonant dispersive nonlinear Schrödinger’s equation with power law nonlinearity. Optik-Int. J. Light Electron Opt. 130, 178–183 (2017)
Navickas, Z., Ragulskis, M.: How far one can go with the Exp-function method? Appl. Math. Comput. 211, 522–530 (2009)
Navickas, Z., Bikulciene, L., Rahula, M., Ragulskis, M.: Algebraic operator method for the construction of solitary solutions to nonlinear differential equations. Commun. Nonlinear Sci. Num. Simul. 18, 1374–1389 (2013)
Navickas, Z., Marcinkevicius, R., Telksnys, T., Ragulskis, M.: Existence of second order solitary solutions to Riccati differential equations coupled with a multiplicative term. IMA J. Appl. Math. 81, 1163–1190 (2016)
Navickas, Z., Ragulskis, M., Marcinkevicius, R., Telksnys, T.: Kink solitary solutions to generalized Riccati equations with polynomial coefficients. J. Math. Anal. Appl. 448, 156–170 (2017)
Rademacher, J.D.M., Wattenberg, R.W.: Viscous shocks in the destabilized Kuramoto–Sivashinsky equation. J. Comput. Nonlinear Dyn. 1, 336–347 (2006)
Sahoo, S., Saha Ray, S.: New approach to find exact solutions of time-fractional Kuramoto–Sivashinsky equation. Phys. A 434, 240–245 (2015)
Seadawy, A.R.: Approximation solutions of derivative nonlinear Schrodinger equation with computational applications by variational method. Eur. Phys. J. Plus 130, 182 (2015)
Taghizadeh, N., Mirzazadeh, M., Rahimian, M., Akbari, M.: Application of the simplest equation method to some time-fractional partial differential equations. Ain Shams Eng. J. 4(4), 897–902 (2013)
Tariq, H., Akram, G.: New approach for exact solutions of time fractional Cahn–Allen equation and time fractional Phi-4 equation. Phys. A 473, 352–362 (2017)
Tonti, E.: Variational formulation for every nonlinear problem. Int. J. Eng. Sci. 22, 1343–1371 (1984)
Wazwaz, A.M.: New solitary wave solutions to the Kuramoto–Sivashinsky and the Kawahara equations. Appl. Math. Comput. 182, 1642–1650 (2006)
Xu, F., Gao, Y., Yang, X., Zhang, H.: Construction of fractional power series solutions to fractional Boussinesq equations using residual power series method. Math. Probl. Eng. 15 (2016). Article ID 5492535
Yin, W., Xu, F., Zhang, W., Gao, Y.: Asymptotic expansion of the solutions to time-space fractional Kuramoto–Sivashinsky equations. Adv. Math. Phys., 9 (2016). Article ID 4632163
Zhao, X., Wang, L., Sun, W.: The repeated homogeneous balance method and its applications to nonlinear partial differential equations. Chaos Solitons Fract. 28, 448–453 (2006)
Zhou, Q., Mirzazadeh, M.: Analytical solitons for Langmuir waves in plasma physics with cubic nonlinearity and perturbations. Z. Naturforsch. (2016). doi:10.1515/zna-2016-0201
Zhou, Q., Yao, D., Chen, F.: Analytical study of optical solitons in media with Kerr and parabolic-law nonlinearities. J. Mod. Opt. 60, 1652–1657 (2013)
Zhou, Q., Zhu, Q., Bhrawy, A.H., Moraru, L., Biswas, A.: Optical solitons with spatially-dependent coefficients by Lie symmetry. Optoelectron. Adv. Mater.-Rapid Commun. 8, 800–803 (2014)
Zhou, Q., Zhu, Q., Liu, Y., Yu, H., Yao, P., Biswas, A.: Thirring optical solitons in birefringent fibers with spatio-temporal dispersion and Kerr law nonlinearity. Laser Phys. 25, 015402 (2015)
Zhou, Q., Ekici, M., Sonmezoglu, A., Manafian, J., Khaleghizadeh, S., Mirzazadeh, M.: Exact solitary wave solutions to the generalized Fisher equation. Optik 127, 12085–12092 (2016)
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Manafian, J., Foroutan, M. Application of \(\tan (\phi (\xi )/2)\)-expansion method for the time-fractional Kuramoto–Sivashinsky equation. Opt Quant Electron 49, 272 (2017). https://doi.org/10.1007/s11082-017-1107-3
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DOI: https://doi.org/10.1007/s11082-017-1107-3
Keywords
- Improved \(\tan (\phi /2)\)-expansion method
- Time-fractional nonlinear Kuramoto–Sivashinsky equation
- Fractional complex transform
- Soliton wave solutions