An analytic algorithm for the space-time fractional reaction-diffusion equation

In this paper, we solve the space-time fractional reaction-diffusion equation by the fractional homotopy analysis method. Solutions of different examples of the reaction term will be computed and investigated. The approximation solutions of the studied models will be put in the form of convergent series to be easily computed and simulated. Comparison with the approximation solution of the classical case of the studied modeled with their approximation errors will also be studied.


Introduction
Nonlinear phenomena occur in a wide range of apparently different contexts in nature, for instance biological, economical, chemical and physical systems.Some of them can be described by nonlinear partial differential equations.A particular class of partial differential equations (PDE) is those modeling nonlinear reaction and diffusion phenomena [1,2].Nonlinear diffusion equations are important class of parabolic equations appearing in many physical problems like phase transition in mechanical, electrical and electronic engineering, biochemistry and dynamics of biological sciences and in many methods for image processing and computer vision [3,4,5].Reaction diffusion equations are applied to describe numerous problems such as the CO oxidation on Pt(110) [6], the study of Ca 2+ waves on Xenopus oocytes [7], the study of re-entry in heart tissues [8,9].In [10] the linear Convection Diffusion (CD) equation was solved with the homotopy analysis method (HAM).Transports of molecular oxygen from the blood plasma to the living tissue of the skeletal muscle or brain among capillary walls are very important topics in the medical field.The study of chemical reaction taking place the environment conditions or the living organisms are popular chemical problems.The spreading of an advantageous gene and the spatial temporal evolution of a population of individuals wich diffuses with a constant of diffusion or even according to a function and then grow according to a specific rule, are very important rules in biology [11,12].Similar types of propagation phenomena are widely occur in nature, especially in biology, chemistry and physics, see [13,14,15].The nonlinear reaction diffusion equation of the characteristic Cauchy type has successfully manged to model such problems, namely, where u (x,t) is the concentration, F (u) = r (x,t) u (x,t) is the reaction parameter and D > 0 is the diffusion coefficient.This equation has originally introduced by Fisher [12], and A. N. Kolmogrov, I. Petrovskii and N. Piskunov [11], to describe the evolution of the diffusion and the grow of population in a certain colony taking into consideration the conditions of the environments.Equation (1.1) has successfully modeled the reaction diffusion processes taking place in moving media such as fluids, atmosphere, chemistry, and ecology.Even the passive scalars substances which are simply transported by the flow can be modeled by equation (1.1).
In this paper, we consider a generalization of equation (1.1), we replace the space and time of equation (1.1) by fractional derivatives operators in the Caputo fractional derivative, namely subject to the initial conditions: where u (x,t) is the concentration, F (u) = r (x,t) u (x,t) is the reaction parameter and D > 0 is the diffusion coefficient, α and β are a parameter describing the order of the fractional derivative.Four problems will be solved with different r (x,t).For solving equation (1.2) we use the fractional homotopy analysis method (FHAM), to study the effect of the fractional orders α and β on the analytical and approximation solutions of equation (1.2).The homotopy analysis method is used in [18] to solve equation (1.2) for different r (x,t) but for the classical case as α = 1, β = 2.Our main aim is to find the approximation solutions of equation (1.2) for different r (x,t) and compare the simulated solutions by the approximation solutions of the classical case.The approximation errors will also simulated for different r (x,t), α and β .The organization of this paper is as follows.Section 2, is devoted to present some necessary definition and preliminary.The idea of the fractional homotopy analysis method (FHAM) is given in Section 3. In Section 4, applications of the fractional homotopy analysis method (FHAM) for the space-time fractional reaction-diffusion equation and numerical results are given.Finally, we give a summary to explain our results.

Basic Definitions of Fractional Calculus
In this section, we give some basic definitions and properties of the fractional calculus theory which are used further in this paper [16].
Definition 2.2.The Riemann-Liouville fractional integral operator of order α ≥ 0, of a function f (x) ∈ C µ , and µ ≥ −1 is defined as Definition 2.3.For m to be the smallest integer that exceeds α, Caputo time fractional derivative operator of order α > 0 is defined as Definition 2.4.The fractional derivative of f (x) in the Caputo sense is defined as From Eq.(2.8), we get the following result 3 Basic idea of the homotopy analysis method (HAM) Here we give a brief description of the the HAM [17] to handle the general nonlinear problem, where NFD is a nonlinear fractional partial differential operator, x and t denote independent variables and u(x,t) is an unknown function.For simplicity, we ignore all boundary or initial conditions, which can be treated in the same way.
Based on the constructed zero-order deformation equation by Liao [17], we give the following zero-order deformation equation in the similar way where q ∈ [0, 1] is an embedding parameter, h is a nonzero auxiliary parameter, and v(x,t; q) is an unknown function.v(x,t; q)is related to u(x,t) as where u 0 (x,t) is an initial guess of u(x,t) By expanding v(x,t; q) in Taylor series with respect to q, one has where If the auxiliary linear non-integer order operator, the initial guess, and the auxiliary parameter h are so properly chosen, the series (3.13), converges at q = 1.Hence we have which must be one of the solution of the original nonlinear equation, as proved by [17].As h = −1, Eq.(3.11) becomes International Scientific Publications and Consulting Services which is used mostly in the homotopy perturbation method (HPM).By this choice the HPM is considered as a special case of the HAM.According to Eq.(3.13), the governing equation can be deduced from the zero-order deformation (3.11).To do so, define the vector Differentiating Eq.(3.11), m times with respect to the embedding parameter q and then setting q = 0 and finally dividing them by m!, we have the so-called mth-order deformation equation where and Finally, for the purpose of computation, we will approximate the HAM solution (3.15) by the following truncated series: The mth-order deformation (3.18), is linear and thus can be easily solved, especially by means of a symbolic computation software such as Mathematica, Maple, Matlab, and so on.
with the initial condition u (x, 0) = e −x + x (4.23) From Eqs. (3.17) and (3.18), we have the mth-order deformation equation as From Eqs. (4.23) and (4.24), we have International Scientific Publications and Consulting Services where [x] denotes the ceiling function and is equal to smallest integer greater than or equal to x.Similarly, we get the next iterations and truncating the series (3.15) by the aid of mathematica, the approximate solution of Eq. (4.22) for m = 6 is written as Note that the approximate solution ∼ u m (x,t) converges to the exact solution e −x + xe −t , as m → ∞ for h = −1, α = 1 and β = 2, since for these values of h, α and β , the various u m (x,t) are given by which is same as the results obtained from [18,19,20,21,22] for α = 1 and β = 2. Fig. 1 shows the approximate solution ∼ u 6 for Eq.(4.22) when h = −1, α = 1 and β = 2, Fig. 2 shows the absolute error E 6 (h), Figs. 3 and 4 show the approximate solution ∼ u 6 for different β and Figs. 5 and 6 show the approximate solution ∼ u 6 for different α.From Eqs. (3.17) and (3.18), we have the mth-order deformation equation as

International Scientific Publications and Consulting Services
From Eqs. (4.26) and (4.27), we have Similarly, we get next iterations and truncating the series (3.15) at m = 5, we obtain an approximate solution of Eq.( 4.25) as Note that the approximate solution ∼ u m (x,t) converges to the exact solution e x+t+t 2 , as m → ∞ for h = −1, α = 1 and β = 2, since for these values of h, α and β , the various u m (x,t) are given by u 0 (x,t) = e x u 1 (x,t) = e x ( t + t 2 ) ) . . .

Conclusion
The main interest of this paper is to use the FHAM to investigate the solution of the space-time fractional reactiondiffusion equation and to show its efficacy in contrast to the other reliable mathematical tools like the HPM and the VIM.Thus, it may be concluded that the HAM is a simple and a powerful analytical approach for handling fractional related PDEs.The results obtained from using the FHAM presented here agree well with the results obtained from [18] and [19].

1 .
In this example, the solution of equation (1.2) is given for r (x,t) =constant= −1.That means we solve

Figure 7 :
Figure 7: the approximate solution

Figure 8 :
Figure 8: the absolute error

Figure 9 :
Figure 9: the approximate solution for different β at x = 0.5

Figure 13 :
Figure 13: the approximate solution

Fig. 17 shows the approximate solution ∼ u 6
for Eq.(4.31) when h = −1 and α = 1, Fig.18shows the absolute error E 6 (h) and Figs.19 and 20show the approximate solution ∼ u 6 for different α.International Scientific Publications and Consulting Services

Figure 17 :
Figure 17: the approximate solution

Figure 18 :
Figure 18: the absolute error