New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method

Abstract Our concern in the present paper is to generate a few new explicit and exact solutions for the time-fractional Cahn–Allen and Cahn–Hilliard equations in the context of the conformable fractional derivative. A new version of Kudryashov method with the help of the Maple package is utilized to carry out this purpose. It is believed that the modified Kudryashov method is practically well suited; such that it can be adopted to a wide range of fractional differential equations (FDEs).

[1]  R. Ansari,et al.  Exact solutions of some nonlinear systems of partial differential equations by using the first integral method , 2012 .

[2]  Hossein Aminikhah,et al.  Sub-equation method for the fractional regularized long-wave equations with conformable fractional derivatives , 2016 .

[3]  H. Bulut,et al.  Exact Solutions of Time-Fractional KdV Equations by Using Generalized Kudryashov Method , 2014 .

[4]  M. Mirzazadeh,et al.  Optical solitons with Biswas–Milovic equation by extended G′/G-expansion method , 2016 .

[5]  E. Misirli,et al.  The modified Kudryashov method for solving some fractional-order nonlinear equations , 2014 .

[6]  Bin He,et al.  Exact Solutions of Fractional Burgers and Cahn-Hilliard Equations Using Extended Fractional Riccati Expansion Method , 2014 .

[7]  Hadi Rezazadeh,et al.  The first integral method for Wu–Zhang system with conformable time-fractional derivative , 2016 .

[8]  Adem C. Cevikel,et al.  Soliton solutions of (2+1)-dimensional time-fractional Zoomeron equation , 2016 .

[9]  A. Biswas,et al.  Solving nonlinear fractional differential equations using exp-function and (G/G′) -expansion methods , 2015 .

[10]  M. Darvishi,et al.  Fractional version of (1 + 1)-dimensional Biswas–Milovic equation and its solutions , 2016 .

[11]  Mehrdad Lakestani,et al.  Optical soliton solutions for the Gerdjikov-Ivanov model via tan(ϕ/2)-expansion method , 2016 .

[12]  A. Bekir,et al.  Two reliable methods for solving the time fractional Clannish Random Walker's Parabolic equation , 2016 .

[13]  M. Belić,et al.  Optical solitons in nano-fibers with spatio-temporal dispersion by trial solution method , 2016 .

[14]  J. Manafian,et al.  A new analytical approach to solve some of the fractional-order partial differential equations , 2017 .

[15]  H. Bulut.  EXACT SOLUTIONS FOR SOME FRACTIONAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS VIA KUDRYASHOV METHOD , 2013 .

[16]  Dumitru Baleanu,et al.  New exact solutions of Burgers’ type equations with conformable derivative , 2017 .

[17]  Mojtaba Moradi,et al.  Exact solitary wave solutions to the new (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation , 2017 .

[18]  Hasan Bulut,et al.  Symmetrical hyperbolic Fibonacci function solutions of generalized Fisher equation with fractional order , 2013 .

[19]  A. Sheikhani,et al.  Exact solutions for the fractional differential equations by using the first integral method , 2015 .

[20]  D. Baleanu,et al.  Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations , 2014 .

[21]  E. Zayed,et al.  A new fractional sub-equation method for solving the space-time fractional differential equations in mathematical physics , 2014 .

[22]  B. Zheng.  Exp-Function Method for Solving Fractional Partial Differential Equations , 2013, TheScientificWorldJournal.

[23]  M. Belić,et al.  Soliton solutions to resonant nonlinear schrodinger's equation with time-dependent coefficients by modified simple equation method , 2016 .

[24]  M. Sababheh,et al.  A new definition of fractional derivative , 2014, J. Comput. Appl. Math..

[25]  Ahmet Bekir,et al.  A novel analytical method for time-fractional differential equations , 2016 .

[26]  Emine Misirli,et al.  On the solutions of the nonlinear fractional differential equations via the modified trial equation method , 2018 .

[27]  E. Zayed,et al.  The modified Kudryashov method for solving some seventh order nonlinear PDEs in mathematical physics , 2015 .

[28]  Adem C. Cevikel,et al.  A variety of exact solutions for the time fractional Cahn-Allen equation , 2015 .

[29]  M. Mirzazadeh.  Analytical study of solitons to nonlinear time fractional parabolic equations , 2016 .

[30]  E. Yaşar,et al.  The G′/G,1/G$\boldsymbol {\left (G^{\prime }/G,1/G\right )}$-expansion method for solving nonlinear space–time fractional differential equations , 2016 .

[31]  S. Ray.  New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods , 2016 .

[32]  Hasan Bulut,et al.  The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation , 2013 .

[33]  S. Ray,et al.  Two efficient reliable methods for solving fractional fifth order modified Sawada–Kotera equation appearing in mathematical physics , 2016 .

[34]  Adem C. Cevikel,et al.  Exact solutions of nonlinear time fractional partial differential equations by sub‐equation method , 2015 .

[35]  R. Sakthivel,et al.  Some New Electrostatic Potential Functions Used to Analyze the Ion-Acoustic Waves in a Thomas Fermi Plasma with Degenerate Electrons , 2015 .

[36]  K. Hosseini,et al.  Feng’s First Integral Method for Analytic Treatment of Two Higher Dimensional Nonlinear Partial Differential Equations , 2015 .

[37]  ANJAN BISWAS,et al.  Solitons and periodic solutions to a couple of fractional nonlinear evolution equations , 2014 .

[38]  Y. Pandır,et al.  Symmetric Fibonacci Function Solutions of some Nonlinear Partial Differential Equations , 2014 .

[39]  Liquan Mei,et al.  The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics ✩ , 2012 .

[40]  Anjan Biswas,et al.  Optical solitons with spatio-temporal dispersion by first integral approach and functional variable method , 2014 .

[41]  Ahmet Bekir,et al.  Exact solutions of nonlinear fractional differential equations by (G′/G)-expansion method , 2013 .

[42]  Bulent Kilic,et al.  Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation , 2014 .

[43]  H. Bulut,et al.  Modified Trial Equation Method to the Nonlinear Fractional Sharma-Tasso-Olever Equation , 2013 .

[44]  Ozkan Guner,et al.  Soliton solution of fractional-order nonlinear differential equations based on the exp-function method , 2016 .

[45]  Harun-Or Roshid,et al.  Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp(−ϕ(ξ))-expansion method , 2014, SpringerPlus.

[46]  Yusuf Pandır,et al.  Solving nonlinear space-time fractional differential equations using the exp-function method , 2015 .

[47]  M. Najafi,et al.  Soliton solutions of nonlinear evolution equations in mathematical physics , 2016 .

[48]  A. Zerarka,et al.  Exact structures for the KdV–mKdV equation with variable coefficients via the functional variable method , 2016 .

[49]  Zheng Bin,et al.  （G＇/G）-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics , 2012 .

[50]  D. Baleanu,et al.  Improved (G'/G)-Expansion Method for the Time-Fractional Biological Population Model and Cahn–Hilliard Equation , 2015 .