## THERMAL SCIENCE

International Scientific Journal

### A SHORT REVIEW ON ANALYTICAL METHODS FOR FRACTIONAL EQUATIONS WITH HE'S FRACTIONAL DERIVATIVE

**ABSTRACT**

He's fractional derivative is adopted in this paper, and analytical methods for fractional differential equations are briefly reviewed, two modifications of the exp-function method, the generalized Kudryashov method and generalized exponential rational function method, are emphasized, and fractional Benjamin-Bona-Mahony equation with He's fractional derivative is used an example to elucidate the solution process.

**KEYWORDS**

PAPER SUBMITTED: 2016-05-13

PAPER REVISED: 2016-06-30

PAPER ACCEPTED: 2016-08-26

PUBLISHED ONLINE: 2017-09-09

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**4**, PAGES [1567 - 1574]

- Yang, X. J., et al., Local Fractional Integral Transforms and Their Applications, Academic Press, New York, USA, 2015
- He, J.-H., Liu, F. J. Local Fractional Variational Iteration Method for Fractal Heat Transfer in Silk Co-coon Hierarchy, Non-Linear Science Letters A, 4 (2013), 1, pp. 15-20
- Liu, F. J., et al., He's Fractional Derivative for Heat Conduction in a Fractal Medium Arising in Silk-worm Cocoon Hierarchy, Thermal Science, 19 (2015), 4, pp. 1155-1159
- Yang, X. J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Itera-tion Method, Thermal Science, 17 (2013), 2, pp. 625-628
- Kolebaje, O., Popoola, O., Assessment of the Exact Solutions of the Space and Time Fractional Benja-min-Bona-Mahony Equation via the G′/G - Expansion Method, Modified Simple Equation Method, and Liu's Theorem, ISRN Mathematical Physics, 2014 (2014), ID 217784
- Mirzazadeh, M., et al., Soliton Solutions to a Few Fractional Nonlinear Evolution Equations in Shallow Water Wave Dynamics, European Physical Journal Plus, 131 (2016), 5, pp. 1-11
- He, J.-H., Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation, de-Verlag im Inter-net GmbH, Berlin, Germany, 2006
- Guner, O., et al., Different Methods for (3+1)-Dimensional Space-Time Fractional Modified KdV-Zak-harov-Kuznetsov Equation, Computers and Mathematics with Applications, 71 (2016), 6, pp. 1259-1269
- Aksoy, E. et al., Exponential Rational Function Method for Space-Time Fractional Differential Equa-tions, Waves in Random and Complex Media, 26 (2016), 2, pp. 142-151
- Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012
- Yang, X. J., et al., Local Fractional Homotopy Perturbation Method for Solving Fractal Partial Differen-tial Equations Arising in Mathematical Physics, Romanian Reports in Physics, 67 (2015), 3, pp. 752-761
- Yang, X. J., et al., Fractal Boundary Value Problems for Integral and Differential Equations with Local Fractional Operators, Thermal Science, 19 (2015), 3, pp. 959-966
- Yang, X. J., et al., On Local Factional Operators View of Computational Complexity: Diffusion and Re-laxation Defined on Cantor Sets, Thermal Science, 20 (2016), Suppl., 3, pp. S755-S767
- He, J. H., Asymptotic Methods for Solitary Solutions and Compactons, Abstract and Applied Analysis, 2012 (2012), ID 916793
- Yang, X. J., et al., A New Fractional Derivative without Singular Kernel: Application to the Modelling of the Steady Heat Flow, Thermal Science, 20 (2016), 2, pp. 753-756
- Yang, A. M., et al., On Steady Heat Flow Problem Involving Yang-Srivastava-Machado Fractional De-rivative without Singular Kernel, Thermal Science, 20 (2016), Suppl. 3, pp. S717-S721
- He, J.-H., et al., A New Fractional Derivative and its Application to Explanation of Polar Bear Hairs, Journal of King Saud University Science, 28 (2016), 2, pp. 190-192
- He, J.-H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
- Wang, K. L., Liu, S. Y., A New Solution Procedure for Nonlinear Fractional Porous Media Equation Based on a New Fractional Derivative, Nonlinear Science Letters A, 7 (2016), 4, pp. 135-140
- Wang, K. L., Liu, S. Y., He's Fractional Derivative for Nonlinear Fractional Heat Transfer Equation, Thermal Science, 20 (2016), 3, pp. 793-796
- Sayevand, K., Pichaghchi, K., Analysis of Nonlinear Fractional KdV Equation Based on He's Fractional Derivative, Nonlinear Science Letters A, 7 (2016), 3, pp. 77-85
- Zhou, X. W., Exp-Function Method for Solving Huxley Equation, Mathematical Problems in Engineer-ing, 2008 (2008), ID 538489
- Li, Z. B., He, J.-H., Fractional Complex Transform for Fractional Differential Equations, Mathematical & Computational Applications, 15 (2010), 5, pp. 970-973
- He, J.-H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
- Li, Z. B., et al., Exact Solutions of Time-Fractional Heat Conduction Equation by the Fractional Com-plex Transform, Thermal Science, 16 (2012), 2, pp. 335-338