## THERMAL SCIENCE

International Scientific Journal

### NEW EXACT SOLUTIONS OF THE SPACE-TIME FRACTIONAL KDV-BURGERS AND NONLINEAR FRACTIONAL FOAM DRAINAGE EQUATION

**ABSTRACT**

The fractional differential equations have been studied by many authors and some effective methods for fractional calculus were appeared in literature, such as the fractional sub-equation method and the first integral method. The fractional complex transform approach is to convert the fractional differential equations into ordinary differential equations, making the solution procedure simple. Recently, the fractional complex transform has been suggested to convert fractional order differential equations with modified Riemann-Liouville derivatives into integer order differential equations, and the reduced equations can be solved by symbolic computation. The present paper investigates for the applicability and efficiency of the exp-function method on some fractional non-linear differential equations.

**KEYWORDS**

PAPER SUBMITTED: 2017-06-15

PAPER REVISED: 2017-11-15

PAPER ACCEPTED: 2017-11-17

PUBLISHED ONLINE: 2018-01-07

**THERMAL SCIENCE** YEAR

**2018**, VOLUME

**22**, ISSUE

**Supplement 1**, PAGES [S15 - S24]

- Guner, O., et al., Different methods for (3+1)-dimensional space-time fractional modified KdVZakharov- Kuznetsov equation, Computers & Mathematics with Applications, 71 (2016), 6, pp. 1259-1269.
- Bekir, A., et al., Exact Solutions for Fractional Differential-Difference Equations by (G '/G)- Expansion Method with Modified Riemann-Liouville Derivative, Advances in Applied Mathematics and Mechanics, 8 (2016), 2, pp. 293-305.
- Guner, O., et al., A variety of exact solutions for the time fractional Cahn-Allen equation, European Physical Journal Plus, 130 (2015), 7, 146.
- Inc, M., et al., Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation, Waves Random Complex Media 24 (2014), 4, pp. 393-403.
- Bekir, A., et al., Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations, Abstract and Applied Analysis, 2013 (2013) Article Number: 426462.
- Guner, O., and Cevikel, A.C., A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations, Scientific World Journal, 2014, Article Number: 489495.
- Lu, B., The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 395 (2012) pp. 684-693.
- Cenesiz, Y., et al., New exact solutions of Burgers' type equations with conformable derivative, Waves in Random and Complex Media 27 (2017) 1103-116.
- Aksoy, E., et al., Soliton solutions of (2+1)-dimensional time-fractional Zoomeron equation, OPTIK, 127 (2016), 17, pp. 6933-6942.
- Bekir, A., et al., Exact solutions of nonlinear time fractional partial differential equations by sub-equation method, Mathematical Methods in The Applied Sciences, 38 (2015), 13, pp. 2779- 2784.
- Taghizadeh N., et al., Application of the simplest equation method to some time-fractional partial differential equations, Ain Shams Engineering Journal, 4 (2013), pp. 897-902.
- Jumarie, G., Table of some basic fractional calculus formulae derived from a modified Riemann-Liouvillie derivative for nondifferentiable functions. Appl. Maths. Lett., 22 (2009), pp. 378-385.
- Li, Z.B., and He, J.H., Fractional complex transform for fractional differential equations, Math. Comput. Appl., 15 (2010), pp. 970-973.
- He, J., et al., Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376 (2012), pp. 257-259
- Saad, M., et al., Using a complex transformation to get an exact solutions for fractional generalized coupled MKDV and KDV equations, Int. Journal of Basic & Applied Sciences, 13 (2013), 1, pp. 23-25.
- Elghareb, T., et al., An Exact Solutions for the Generalized Fractional Kolmogorov-Petrovskii Piskunov Equation by Using the Generalized (G' /G) -expansion Method, Int. Journal of Basic & Applied Sciences, 13 (2013), 1, pp. 19-22.
- He, J.H., and Wu, X.H., Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006), pp. 700-708.
- Bekir, A., and Cevikel, A.C., New solitons and periodic solutions for nonlinear physical models in mathematical physics, Nonlinear Analysis-Real World Applications, 11 (2010), 4, pp. 3275-3285.
- Cevikel, A.C., et al., A procedure to construct exact solutions of nonlinear evolution equations, Pramana - J. Phys., 79 (2012), 3, pp. 337-344.
- Cevikel, A.C., and Bekir, A., New Solitons and Periodic Solutions for (2+1)-dimensional Davey-Stewartson Equations, Chinese Journal of Physics, 51 (2013), 1, pp. 1-13.
- Zhang, S., Application of Exp-function method to high-dimensional nonlinear evolution equation, Chaos Solitons Fract. 38 (2008), pp. 270-276.
- El-Wakil, S.A., et al., Application of Exp-function method for nonlinear evolution equations with variable coefficients, Phys. Lett. A 369 (2007), pp. 62-69.
- Bekir, A., Application of the Exp-function method for nonlinear differential-difference equations, Appl. Math. Comput. 215 (2010), pp. 4049-4053.
- Dai, C.Q., and Chen, J.L., New analytic solutions of stochastic coupled KdV equations, Chaos Solitons Fract. 42 (2008), pp. 2200-2207.
- Younis, M., Soliton Solutions of Fractional order KdV-Burger's Equation, Journal of Advanced Physics, 3, 2014, pp. 325-328(4) Physics (in press)
- Weaire, D., and Hutzler, S., The Physic of Foams, Oxford University Press, Oxford, UK 2000
- Weaire, D., et al., The fluid dynmaics of foams, Journal of Physics: Condensed Matter, 15 (2003) pp. 65-72
- Bekir, A., and Cevikel, A.C., Solitary wave solutions of two nonlinear physical models by tanh-coth method, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 5, pp. 1804-1809
- Verbist, G., et al., The foam drainage equation, Journal of Physics Condensed Matter, 8 (1996), 21, pp. 3715-3731
- Zhang, Y., and Feng, Q., Fractional Riccati Equation Rational Expansion Method For Fractional Differential Equations, Appl. Math. Inf. Sci., 7 (2013), 4, pp. 1575-1584.