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Research Papers

# Improved $(G'/G)$-Expansion Method for the Time-Fractional Biological Population Model and Cahn–Hilliard Equation

[+] Author and Article Information
Dumitru Baleanu

Department of Mathematics,
Çankaya University,
Balgat 06530, Ankara, Turkey
Institute of Space Sciences,
Magurele-Bucharest, Romania
e-mail: dumitru@cankaya.edu.tr

Yavuz Uğurlu

Department of Mathematics,
Science Faculty,
Fırat University,
Elazığ 23119, Turkey
e-mail: matematikci_23@yahoo.com.tr

Mustafa Inc

Department of Mathematics,
Science Faculty,
Fırat University,
Elazığ 23119, Turkey
e-mail: minc@firat.edu.tr

Bulent Kilic

Department of Mathematics,
Science Faculty,
Fırat University,
Elazığ 23119, Turkey
e-mail: bulentkilic@firat.edu.tr

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 24, 2014; final manuscript received November 22, 2014; published online April 16, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(5), 051016 (Sep 01, 2015) (8 pages) Paper No: CND-14-1220; doi: 10.1115/1.4029254 History: Received September 24, 2014; Revised November 22, 2014; Online April 16, 2015

## Abstract

In this paper, we used improved $(G'/G)$-expansion method to reach the solutions for some nonlinear time-fractional partial differential equations (fPDE). The fPDE is reduced to an ordinary differential equation (ODE) by means of Riemann–Liouille derivative and a basic variable transformation. Various types of functions are obtained for the time-fractional biological population model (fBPM) and Cahn–Hilliard (fCH) equation.

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## Figures

Fig. 1

Profiles of Eq. (25) with α = 0.5,A = 1,B = 2,C = 0.5,h = 0.1,r = 0.3,c = 0.5,a0 = 0.6,v = 0.2,y = 0.5,c1 = 2,c2 = 3

Fig. 2

Profiles of Eq. (27) with α = 0.9,A = 1.5,C = 2,h = 0.1,r = 0.3,c = 0.5,a0 = 0.6,v = 0.2,y = 0.5,c1 = c2 = 3,t = 0.5

Fig. 3

Profiles of Eq. (29) with α = 0.9,A = 1,C = 0.5,h = 0.1,r = 0.3,c = 0.5,a0 = 0.6,v = 0.2,y = 0.5,c1 = c2 = 3,t = 0.5

Fig. 4

Profiles of Eq. (30) with α = 0.1,A = 1,B = 2,C = 0.5,h = 0.1,c = 0.5,v = 0.2,y = 0.5,t = 0.5

Fig. 5

Profiles of Eq. (41) with α = 0.01,A = 1,B = 2,C = 0.5,h = 0.1,c = 1.5,y = 0.5,t = 0.5

Fig. 6

Profiles of Eq. (44) with α = 0.2,A = 5,B = 0.5,C = 0.1,r = 2,c = 1.5,y = 1,c1 = 3,c2 = 2,t = 0.5

Fig. 7

Profiles of Eq. (45) with α = 0.9,A = 3,B = 0.5,C = 0.1,r = 4,c = 1.5,y = 1,t = 0.5

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