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,
Öğretmenler Cad. 14,
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

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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Hereman, W., Banerjee, P. P., Korpel, A., Assanto, G., van Immerzeele, A., and Meerpoel, A., 1986, “Exact Solitary Wave Solutions of Nonlinear Evolution and Wave Equations Using a Direct Algebraic Method,” J. Phys. A, 19(5), pp. 607–628. [CrossRef]
Wang, D.-S., 2010, “Complete Integrability and the Miura Transformation of a Coupled KdV Equation,” Appl. Math. Lett., 23(6), pp. 665–669. [CrossRef]
Geng, X., and He, G., 2010, “Darboux Transformation and Explicit Solutions for the Satsuma–Hirota Coupled Equation,” Appl. Math. Comput., 216(9), pp. 2628–2634. [CrossRef]
Cesar, A., Gómez, S., and Alvaro, H. S., 2008, “The Cole–Hopf Transformation and Improved tanh–coth Method Applied to New Integrable System (KdV6),” Appl. Math. Comput., 204(2), pp. 957–962. [CrossRef]
Lei, Y., Fajiang, Z., and Yinghai, W., 2002, “The Homogeneous Balance Method, Lax Pair, Hirota Transformation and a General Fifth-Order KdV Equation,” Chaos, Solitons Fractals, 13(2), pp. 337–340. [CrossRef]
Wang, M. L., and Wang, Y. M., 2001, “A New Bäcklund Transformation and Multi-Soliton Solutions to the KdV Equation With General Variable Coefficients,” Phys. Lett. A, 287(3–4), pp. 211–216. [CrossRef]
Taşcan, F., and Bekir, A., 2009, “Analytic Solutions of the (2 + 1)-Dimensional Nonlinear Evolution Equations Using the Sine–Cosine Method,” Appl. Math. Comput., 215(8), pp. 3134–3139. [CrossRef]
Wang, M. L., Zhou, Y. B., and Li, Z. B., 1996, “Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Equations in Mathematical Physics,” Phys. Lett. A, 216(1–5), pp. 67–75. [CrossRef]
Fan, E. G., 2002, “Auto-Bäcklund Transformation and Similarity Reductions for General Variable Coefficient KdV Equations,” Phys. Lett. A, 294(1), pp. 26–30. [CrossRef]
Baleanu, D., Hakimeh, M., and Shahram, R., 2013, “On a Nonlinear Fractional Differential Equation on Partially Ordered Metric Spaces,” Adv. Differ. Equations, 2013, p. 83. [CrossRef]
Abdelouahab, K., and Baleanu, D., 2011, “Homotopy Perturbation Method for the Coupled Fractional Lotka–Volterra Equations,” Rom. J. Phys., 56(3–4), pp. 333–338.
Yang, X. J., Baleanu, D., and Zhong, W. P., 2013, “Approximate Solutions for Diffusion Equations on Cantor Space–Time,” Proc. Rom. Acad. Ser. A, 14(2), pp. 127–133.
Biswas, A., Bhrawy, A. H., Abdelkawy, M. A., Alshaery, A. A., and Hilal, E. M., 2014, “Symbolic Computation of Some Nonlinear Fractional Differential Equations,” Rom. J. Phys., 59(5–6), pp. 433–442.
Meng, F., 2013, “A New Approach for Solving Fractional Partial Differential Equations,” J. Appl. Math., 2013, p. 256823.
Jafari, H., Tajadodi, H., Kadkhoda, N., and Baleanu, D., 2013, “Fractional Subequation Method for Cahn–Hilliard and Klein–Gordon Equations,” Abstr. Appl. Anal., 2013, p. 587179. [CrossRef]
Li, Z., Liu, X., and Zhang, W., 2012, “Application of Improved (G′/G)-expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations,” Adv. Appl. Math. Mech., 4(1), pp. 122–131.
Jumarie, G., 2006, “Modified Riemann–Liouville Derivative and Fractional Taylor Series of Non-Differentiable Functions Further Results,” Comput. Math. Appl., 51(9–10), pp. 1367–1376. [CrossRef]
Jumarie, G., 2007, “Fractional Hamilton–Jacobi Equation for the Optimal Control of Nonrandom Fractional Dynamics With Fractional Cost Function,” Appl. Math. Comput., 23(1–2), pp. 215–228.
Jumarie, G., 2009, “Table of Some Basic Fractional Calculus Formulae Derived From a Modified Riemann–Liouville Derivative for Non-Differentiable Functions,” Appl. Math. Lett., 22(3), pp. 378–385. [CrossRef]
Lu, Y. G., 2000, “Hölder Estimates of Solutions of Biological Population Equations,” Appl. Math. Lett., 13(6), pp. 123–126. [CrossRef]
Gurtin, M., and MacCamy, R. C., 1977, “On the Diffusion of Biological Populations,” Math. Biosci., 33(1–2), pp. 35–49. [CrossRef]
Bear, J., 1972, Dynamics of Fluids in Porous Media, American Elsevier, New York.
Li, Z. B., and He, J. H., 2012, “Converting Fractional Differential Equations Into Partial Differential Equations,” Therm. Sci., 16(2), pp. 331–334. [CrossRef]
He, J. H., and Li, Z. B., 2011, “Application of the Fractional Complex Transform to Fractional Differential Equations,” Nonlinear Sci. Lett. A, 2(3), pp. 121–126.
El-Sayed, A. M. A., Rida, S. Z., and Arafa, A. A. M., 2009, “Exact Solutions of Fractional-Order Biological Population Model,” Commun. Theor. Phys., 52(6), pp. 992–996. [CrossRef]
Zhang, S., and Zhang, H. Q., 2011, “Fractional Sub-Equation Method and Its Applications to Nonlinear Fractional PDEs,” Phys. Lett. A, 375(7), pp. 1069–1073. [CrossRef]
Shakeri, F., and Dehghan, M., 2007, “Numerical Solution of a Biological Population Model Using He's Variational Iteration Method,” Comput. Math. Appl., 54(7–8), pp. 1197–1209. [CrossRef]
Devendra, K., and Jagdev, S. S., 2013, “Application of Homotopy Analysis Transform Method to Fractional Biological Population Model,” Rom. Rep. Phys., 65(1), pp. 63–75.
Ugurlu, Y., and Kaya, D., 2008, “Solutions of the Cahn–Hilliard Equation,” Comput. Math. Appl., 56(12), pp. 3038–3045. [CrossRef]
Dahmani, Z., and Benbachir, M., 2009, “Solutions of the Cahn–Hilliard Equation With Time- and Space-Fractional Derivatives,” Int. J. Nonlinear Sci., 8(1), pp. 19–26.


Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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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