Journal of Engineering, Computers & Applied Sciences (JEC&AS) ISSN No: 2319‐5606 Volume 2, No.2, ,February 2013 _________________________________________________________________________________ www.borjournals.com Blue Ocean Research Journals 30 A Coupling Technique for Analytical Solution of Time Fractional Biological Population Model R. N. Prajapati, Department of Mathematics, Dehradun Institute of Technology, Dehradun, Uttrakhand, India R. Mohan, Department of Mathematics, Dehradun Institute of Technology, Dehradun, Uttrakhand, India ABSTRACT In this study, homotopy perturbation transform method (HPTM) is used to obtain the approximate analytical solutions of time fractional biological population model. The solution procedure obtained by proposed method indicate that the approach is easy to implement and accurate. Some numerical examples are given in the support of the validity of the method. These results reveal that the proposed method is very effective and easy to use. The comparisons between exact solution and approximate solution are shown through graphs. Keywords: Homotopy perturbation transform method; Laplace transform; Biological population model; Fractional derivative; Mittag-Leffler function. MSC (2010): 26A33, 34A08, 34A34. 1. Introduction The idea of fractional-order derivatives initially arose from a letter by Leibnitz to L’ Hospital in 1695. Fractional calculus has gained considerable popularity and importance during the past three decades, mainly due to its applications in numerous fields of science and engineering. One of the main advantages of using fractional-order differential equations in mathematical modelling is their non- local property. It is a well- known fact that the integer-order differential operator is a local operator whereas the fractional-order differential operator is non-local in the sense that the next state of the system depends not only upon its current state but also upon all of its proceeding states. In the last few decades, many authors have made notable contributions to both theory and application of fractional differential equations in areas as diverse as finance [1-2], physics [3-6], control theory [7] and hydrology [8-10]. The degenerate parabolic nonlinear partial differential equations arising in the spatial diffusion of biological populations are given as , , , 0 , ) , , , ( )) ( ( )) ( ( R y x t u y x t f u G u G u yy xx t      (1.1) with initial condition , ) 0 , , ( ) 0 , , ( 0 y x u y x u  where u denotes the population density and f represents the population supply due to birth and death. Our model considered as for example in the population of animals. The movements are made generally either by mature animals driven out by invaders or by young animals just reaching maturity moving out of their parental territory to establish breeding territory of their own. In both cases, it is much more plausible to suppose that they will be directed towards nearby vacant territory. In this model, therefore, movement will take place almost exclusively “down” the population density gradient, and will be much more rapid at high population densities than at low ones. In an attempt to model this situation, they considered a walk through a rectangular grid, in which at each step an animal may either stay at its present location or may move in the direction of lowest population density. In this article, we will do it in a practical case , ) ( 2 u u G  besides the theory of the spread of biological populations, the case 2 ) ( u u G  occurs in a variety of different setting. After this assumption, equation (1.1) will be , , , 0 , ) ( ) ( ) ( 2 2 R y x t u f u u u yy xx t      (1.2) with initial condition ). 0 , , ( ) 0 , , ( 0 y x u y x u  The homotopy perturbation method was first proposed by J.H. He [11-15]. Considerable research works have been conducted recently in applying the homotopy perturbation method to a class of linear and non-linear equations by many researchers [12-20]. In this article homotopy perturbation transform method has been used. This method was given by Khan and Wu [21] which is coupling of the Laplace transformation, the homotopy perturbation method and He’s polynomials [22-23]. In recent years, many authors have paid attention to studying the solutions of linear and nonlinear partial differential equations by using various methods with combination of the Laplace transform. Among these are the Laplace