Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

Abstract In this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.


Introduction
The fractional calculus deal with number of problem arising in the eld of uid mechanics, biology, di usion, fractional signal, image processing and many other physical process. Fractional di erential equations are used to model these types of the problems. In the various eld of engineering and science, it is very important to nd the approximate or the exact solution of some nonlinear partial di erential equations [1]. There are several potent methods such as Homotopy perturbation [2,3]; homotopy perturbation transformation method (HPTM) [4] and homotopy perturbation sumudu transformation method (HP-STM) have been proposed to obtain the approximate or the exact solutions of nonlinear equations [5][6][7][8][9].
In 1970, Keller and Segel presented a mathematical formulation of cellular slime mold aggregation pro-cess [10]. Recently many researchers use di erent methods to solve Keller-Segel equation [11][12][13]. Di erent types of numerical methods are used to solve nonlinear partial di erential equations [14][15][16][17][18][19]. The solution of multidimensional linear and nonlinear partial di erential equations are established by using combination of least square approximation and homotopy perturbation approximation [20]. A new semi-analytical method called the homotopy analysis Shehu transform method is used to solve multidimensional fractional di usion equations and this method is combination of the homotopy analysis method and the Laplace-type integral transform transform [21]. Solution of Reaction-Di usion-Convection Problem and nonlinear equation is discussed by homotopy perturbation technique [22,23]. Non-linear Fisher equation is solved with help of homotopy perturbation method then solution is compared with solution from Variational Iteration Method (VIM) and Adomian Decomposition Method (ADM) [24]. In this paper, we propose HPSTM for the solution of fractional attractor one dimensional Keller Siegel equation. The simpli ed form of the Keller Segel equation in one dimension is given as [25]: Subject to the boundary conditions And the initial conditions indicates the sensitivity of the cells, χ(ρ) called the sensitivity function of ρ ∈ ( , ∞). Di erent form of χ(ρ) like ρ, ρ and log ρ have been suggested. But in this paper, we discuss the two cases of the fractional attractor one dimensional Keller-Siegel equations one with chemo tactic sensitivity function χ (ρ) = and other with χ (ρ) = ρ.
De nition: The Riemann-Liouville fractional integral operator of order α > of a function f (t) ∈ Cµ, µ ≥ − is de ned as where D α t is Caputo derivative operator and Γ (α) is the Gamma function.

Sumudu Transformation:
The Sumudu transformation over the set of function is de ned by Watugala (1993) as Some properties of the Sumudu transformations are The Sumudu transformation of the Caputo fractional derivative is de ned as

Homotopy perturbation Sumudu transformation method (HPSTM)
To illustrate the idea of HPSTM technique, we consider the fractional attractor one dimensional Keller-Segel equation Where D µ t and D η t is the Caputo fractional derivative of the function U(x, t) and ρ(x, t) respectively. Apply sumudu transformation on both sides of the equation (3), we have Using di erentiation property of the sumudu transformation and the initial conditions, we get Operating with inverse sumudu transformation on both sides, Now we apply Homotopy perturbation method, where the nonlinear term can be decomposed as for some He's polynomial Hn given by Substituting these values in (5), we have On comparing the like powers of p on both sides, Similarly we can nd all the values of U , U , U , U , . . . . and ρ , ρ , ρ , . . . . The approximate solution of equation (3) can be calculated by setting p → .

Application of HPSTM
In order to understand the solution procedure of the homotopy perturbation sumudu transform method, we consider the following examples: Solution of fractional attractor 1-D Keller Segel equation Example: Consider the following coupled system: Subject to the boundary conditions Case-I: Consider the sensitivity function χ (ρ) = , then the chemo-tactic term i.e. ∂ ∂x v ∂χ(ρ) ∂x = . Hence Keller Segel equation reduces to: By applying HPSTM on Eq. (36), we have On looking at the like terms of p of Eq. (37) & (38) and using (35), we have The approximation solution of Eq. (36) obtained as p → , Now, for the solution of Eq. (39), we apply HPSTM on Eq.
Where ∞ n= p n H n (x, t) = ∂v ∂x An initial couple of terms of He's polynomial i.e. Hn (x, t) are given below: . . .
On looking at the like term of pof Eq. (40) and (41) and using Eq. (35) and He's polynomial we get On using Eq. (42) and as p → , the approximation solu-

Homotopy perturbation transform method (HPTM)
To elucidate the basic idea of this method, we consider coupled attractor for one-dimensional Keller-Segel equation: Subjected to initial condition: Taking Laplace transform on both sides of equation (20) L Applying the di erentiation property of Laplace transform, we have Taking the inverse Laplace transform on both sides of equation (24) and (25) U The Laplace transform and the homotopy perturbation method are coupled here by using He's polynomials. Comparing the coe cients of like powers of p, the following approximations are obtained And so on. Setting, p = results the approximate solution of equation (20) U .

Application of HPTM
In the order to understand solution of the homotopy perturbation transform method, we consider the following example: Example: The simpli ed form of the Keller Segel equation in one dimension in given as By applying HPTM on equation (16) On looking at the coe cients of like powers of p of Eq. (17) and (18) and using (15), we have: ;

Results and discussion
In this section, the numerical solution of examples obtained by HPSTM and HPTM through a graphical representation are studied. The surface graphs of Keller-Segel equation for respective cases (I & II) at di erent values of β are represented in Figures 1-4. For graphical representation of solution we take m = . , n = . , a = . , b = , c = , d = . Figure 1 represents solution v (x, t) at β = . , β = . , β = . , β = , respectively, whereas Figure 2 indicates ρ (x, t) corresponding to di erent values of β for Case-I. Figures 3 and 4 show surface graphs of solution v (x, t) and ρ (x, t) for Case-II at di erent values of β. Figure 5 represents solution v (x, t) and ρ (x, t) obtained from HPTM for both cases. It is clear from the graphs that results of HPSTM and HPTM are in good harmony with each other for β = .

Conclusion
In this work, homotopy perturbation transform method (HPTM) combined with sumudu transform has been successfully applied to approximate solution for a system of nonlinear partial di erential equations derived from an attractor for a one-dimensional Keller-Segel dynamics system. On comparing the results of this method with HPTM, it is observed HPSTM is extremely simple, straightforward and easy to handle the nonlinear terms. Maple 13 package is used to calculate series obtained from iteration. Further, the method needs much less computational work which shows fast convergent for solving nonlinear system of partial di erential equations.