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New traveling wave exact and approximate solutions for the nonlinear Cahn–Allen equation: evolution of a nonconserved quantity

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Abstract

In the present article, we have developed the new exact and approximate solutions of a nonlinear evolution equation that appear in mathematical physics, specifically Cahn–Allen equation by Tanh method via new conformable derivative and a recent analytic iterative technique, named as residual power series method. The obtained results reveal that the proposed methods are the significant addition for exploring nonlinear fractional models in fractional theory and its computations.

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References

  1. Prüss, J., Wilke, M.: Maximal \(L_p\)-regularity and long-time behaviour of the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions. In: Partial Differential Equations and Functional Analysis. Operator Theory: Advances and Applications, vol. 168, pp. 209–236. Birkhäuser, Basel (2006)

  2. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27(6), 1085–1095 (1979)

    Article  Google Scholar 

  3. Jafari, H., Tajadodi, H., Baleanu, D.: Application of a homogeneous balance method to exact solutions of nonlinear fractional evolution equations. J. Comput. Nonlinear Dyn. 9(2), 021019 (2014)

    Article  Google Scholar 

  4. Yasar, E., Giresunlu, I.B.: The \((G^{\prime }/G, 1/G)\)-expansion method for solving nonlinear space-time fractional differential equations. Pramana J. Phys. 87, 17 (2016)

    Article  Google Scholar 

  5. Esen, A., Yagmurlu, M.N., Tasbozan, O.: Approximate analytical solution to time-fractional damped Burger and Cahn–Allen equations. Appl. Math. Inf. Sci. 7, 1951 (2013)

    Article  MathSciNet  Google Scholar 

  6. Hariharan, G.: Haar wavelet method for solving Cahn–Allen equation. Appl. Math. Sci. 3, 2523–2533 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Tascan, F., Bekir, A.: Travelling wave solutions of the Cahn–Allen equation by using first integral method. Appl. Math. Comput. 207, 279–282 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Taghizadeh, N., Mirzazadeh, M., Samiei, A.P., Vahidi, J.: Exact solutions of nonlinear evolution equations by using the modified simple equation method. Ain Shams Eng. J. 3, 321–325 (2012)

    Article  Google Scholar 

  9. Bekir, A.: Multisoliton solutions to Cahn–Allen equation using double exp-function method. Phys. Wave Phenom. 20(2), 118–121 (2012)

    Article  Google Scholar 

  10. Güner, O., Bekir, A., Cevikel, A.C.: A variety of exact solutions for the time fractional Cahn–Allen equation. Eur. Phys. J. Plus 130, 146 (2015). doi:10.1140/epjp/i2015-15146-9

    Article  Google Scholar 

  11. Dai, C.Q., Wang, Y.Y.: Spatiotemporal localizations in (3 + 1)-dimensional PT-symmetric and strongly nonlocal nonlinear media. Nonlinear Dyn. 83, 2453–2459 (2016)

    Article  MathSciNet  Google Scholar 

  12. Dai, C.Q., Fan, Y., Zhou, G.Q., Zheng, J., Chen, L.: Vector spatiotemporal localized structures in (3 + 1)-dimensional strongly nonlocal nonlinear media. Nonlinear Dyn. 86, 999–1005 (2016)

    Article  MathSciNet  Google Scholar 

  13. Zhou, Q., Zhong, Y., Mirzazadeh, M., Bhrawy, A.H., Zerrad, E., Biswas, A.: Thirring combo-solitons with cubic nonlinearity and spatio-temporal dispersion. Waves Random Complex Media 26(2), 204–210 (2016)

    Article  MathSciNet  Google Scholar 

  14. Mirzazadeh, M., Ekici, M., Sonmezoglu, A., Eslami, M., Zhou, Q., Kara, A.H., Milovic, D., Majid, F.B., Biswas, A., Belic, M.: Optical solitons with complex Ginzburg–Landau equation. Nonlinear Dyn. 85(3), 1979–2016 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhou, Q., Mirzazadeh, M.: Analytical solitons for Langmuir waves in plasma physics with cubic nonlinearity and perturbations. Zeitschrift fr Naturforschung A 71(9), 807–815 (2016)

    Google Scholar 

  16. Dai, C.Q., Xu, Y.J.: Exact solutions for a Wick-type stochastic reaction Duffing equation. Appl. Math. Model. 39, 7420–7426 (2015)

    Article  MathSciNet  Google Scholar 

  17. Mirzazadeh, M., Eslami, M., Biswas, A.: Soliton solutions of the generalized Klein–Gordon equation by using G’/G-expansion method. Comput. Appl. Math. 33(3), 831–839 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mirzazadeh, M., Eslami, M., Bhrawy, A.H., Biswas, A.: Integration of complex-valued Klein–Gordon equation in Q-4 field theory. Rom. J. Phys. 60(3–4), 293–310 (2015)

    Google Scholar 

  19. Dai, C.Q., Wang, Y.Y.: Controllable combined Peregrine soliton and Kuznetsov-Ma soliton in PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 80, 715–721 (2015)

    Article  MathSciNet  Google Scholar 

  20. Dai, C.Q., Wang, Y., Liu, J.: Spatiotemporal Hermite–Gaussian solitons of a (3 + 1)-dimensional partially nonlocal nonlinear Schrodinger equation. Nonlinear Dyn. 84, 1157–1161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zeng, Z.-F., Liu, J.-G., Nie, B.: Multiple-soliton solutions, soliton-type solutions and rational solutions for the (3 + 1)-dimensional generalized shallow water equation in oceans, estuaries and impoundments. Nonlinear Dyn. 86, 667–675 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Biswas, A., Moosaei, H., Eslami, M., Mirzazadeh, M., Zhou, Q., Bhrawy, A.H.: Optical soliton perturbation with extended tanh function method. Optoelectron. Adv. Mater. Rapid Commun. 8((11–12)), 1029–1034 (2014)

    Google Scholar 

  23. Mirzazadeh, M., Biswas, A.: Optical solitons with spatio-temporal dispersion by first integral approach and functional variable method. Optik Int. J. Light Electron Opt. 125(19), 5467–5475 (2014)

    Article  Google Scholar 

  24. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amesterdam (2006)

    MATH  Google Scholar 

  25. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  26. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  27. He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30(3), 700708 (2006)

    Article  MathSciNet  Google Scholar 

  28. Bekir, A.: Application of the exp-function method for nonlinear differential-difference equations. Appl. Math. Comput. 215(11), 40494053 (2010)

    MathSciNet  Google Scholar 

  29. Gepreel, K.A., Omran, S.: Exact solutions for nonlinear partial fractional differential equations. Chin. Phys. B 21(11), 110–204 (2012)

    Article  Google Scholar 

  30. Zheng, B., Wen, C.: Exact solutions for fractional partial differential equations by a new fractional sub-equation method. Adv. Differ. Equ. 2013, 199 (2013)

    Article  MathSciNet  Google Scholar 

  31. Lu, B.: The first integral method for some time fractional differential equations. J. Math. Anal. Appl. 395(2), 684693 (2012)

    Article  MathSciNet  Google Scholar 

  32. Liu, W., Chen, K.: The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations. Pramana J. Phys. 81(3), 377–384 (2013)

    Article  Google Scholar 

  33. 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)

    Article  Google Scholar 

  34. 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, 8 (2013)

    MathSciNet  Google Scholar 

  35. Darvishi, M.T., Kavitha, L., Najafi, M., Senthil Kumar, V.: Elastic collision of mobile solitons of a (3 + 1)-dimensional soliton equation. Nonlinear Dyn. 86, 765–778 (2016)

    Article  Google Scholar 

  36. Sahoo, S., Saha Ray, S.: New solitary wave solutions of time-fractional coupled Jaulent–Miodek equation by using two reliable methods. Nonlinear Dyn. 85, 1167–1176 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wazwaz, A.M.: The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput. 188, 1467–1475 (2007)

    MathSciNet  MATH  Google Scholar 

  38. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Benkhettou, N., Hassani, S., Torres, D.F.M.: A conformable fractional calculus on arbitrary time scales. J. King Saud Univ. Sci. 28(1), 93–98 (2016). arXiv:1505.03134

    Article  Google Scholar 

  41. Bayour, B., Torres, D.F.M.: Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math. (2006). doi:10.1016/j.cam.2016.01.014

    MATH  Google Scholar 

  42. Anderson, D.R., Avery, R.I.: Fractional-order boundary value problem with Sturm–Liouville boundary conditions. Electron. J. Differ. Equ. 2015(29), 1–10 (2015)

    MathSciNet  MATH  Google Scholar 

  43. Eslami, M.: Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations. Appl. Math. Comput. 285, 141–148 (2016)

    MathSciNet  Google Scholar 

  44. Eslami, M., Rezazadeh, H.: The first integral method for Wu-Zhang system with conformable time-fractional derivative. Calcolo (2015). doi:10.1007/s10092-015-0158-8

    MATH  Google Scholar 

  45. Kurt, A., Cenesiz, Y., Tasbozan, O.: On the solution of Burgers equation with the new fractional derivative. Open Phys. 13, 355–360 (2015). doi:10.1515/phys-2015-0045

    Article  Google Scholar 

  46. Chung, W.S.: Fractional Newton mechanics with conformable fractional derivative. J. Comput. Appl. Math. 290, 150158 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Salahshour, S., Ahmadian, A., Ismail, F., Baleanu, D., Senu, N.: A new fractional derivative for differential equation of fractional order under interval uncertainty. Adv. Mech. Eng. 7(12), 1–11 (2015)

    Article  Google Scholar 

  48. Abu Arqub, O.: Series solution of fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 5, 3152 (2013)

    Article  MathSciNet  Google Scholar 

  49. Abu Arqub, O., El-Ajou, A., Bataineh, A.: A representation of the exact solution of generalized Lane–Emden equations using a new analytical method. Abstr. Appl. Anal. 2013, 378593 (2013). doi:10.1155/2013/378593. 10 pp

    Article  MathSciNet  MATH  Google Scholar 

  50. Abu, O., Arqub, A., El-Ajou, S.: Momani, constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. J. Comput. Phys. 293, 385–399 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tariq, H., Akram, G.: Residual power series method for solving time-space-fractional Benney-Lin equation arising in falling film problems. J. Appl. Math. Comput. (2016). doi:10.1007/s12190-016-1056-1

    Google Scholar 

  52. El-Ajou, A., Abu Arqub, O., Momani, S., Baleanu, D., Alsaedi, A.: A novel expansion iterative method for solving linear partial differential equations of fractional order. Appl. Math. Comput. 257, 119–133 (2015)

    MathSciNet  MATH  Google Scholar 

  53. Abu Arqub, O., Abo-Hammour, Z., Al-Badarneh, R., Momani, S.: A reliable analytical method for solving higher-order initial value problems. Discrete Dyn. Nat. Soc. 2013, 673829 (2013). doi:10.1155/2013/673829. 12 pp

    Article  MathSciNet  Google Scholar 

  54. Alquran, M.: Analytical solutions of fractional foam drainage equation by residual power series method. Math. Sci. 8(4), 153–160 (2014)

    Article  MathSciNet  Google Scholar 

  55. Kumar, S., Kumar, A., Baleanu, D.: Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burgers equations arise in propagation of shallow water waves. Nonlinear Dyn. (2016). doi:10.1007/s11071-016-2716-2

    MathSciNet  MATH  Google Scholar 

  56. Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylors formula. Appl. Math. Comput. 186(1), 286–293 (2007)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, 54590, Pakistan

    Hira Tariq & Ghazala Akram

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  1. Hira Tariq
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  2. Ghazala Akram
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Tariq, H., Akram, G. New traveling wave exact and approximate solutions for the nonlinear Cahn–Allen equation: evolution of a nonconserved quantity. Nonlinear Dyn 88, 581–594 (2017). https://doi.org/10.1007/s11071-016-3262-7

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