Analytical modeling for fractional multi-dimensional diffusion equations by using Laplace transform

In this paper, we propose a simple numerical algorithm for solving multi-dimensional diffusion equations of fractional order which describes density dynamics in a material undergoing diffusion by using homotopy analysis transform method. The fractional derivative is described in the Caputo sense. This homotopy analysis transform method is an innovative adjustment in Laplace transform method and makes the calculation much simpler. The technique is not limited to the small parameter, such as in the classical perturbation method. The scheme gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive.


Introduction
Fractional differential equations are a generalization of differential equation through the application of fractional calculus.Fractional differential equations have gained importance and popularity, mainly due to its demonstrated applications in science and engineering.For example, these equations are increasingly used to model problems in fluid flow, rheology, diffusion, relaxation, oscillation, anomalous diffusion, reaction-diffusion, turbulence, diffusive transport akin to diffusion, electric networks, polymer physics, chemical physics, electrochemistry of corrosion, relaxation processes in complex systems, propagation of seismic waves, dynamical processes in self-similar and porous structures and many other physical processes.The most important advantage of using fractional differential equations in these and other applications is their non-local property.It is well known that the integer order differential operator is a http://www.ispacs.com/journals/cna/2015/cna-00220/International Scientific Publications and Consulting Services local operator but the fractional order differential operator is non-local.This means that the next state of a system depends not only upon its current state but also upon all of its historical states.This is more realistic and it is one reason why fractional calculus has become more and more popular [1][2][3][4][5][6][7][8][9].The diffusion equation is a partial differential equation which describes density dynamics in a material undergoing diffusion.It is also used to describe processes exhibiting diffusive-like behaviour, for instance the 'diffusion' of alleles in a population in population genetics.The equation can be written as )), , ) ), , ( ( r r t u D denotes the collective diffusion coefficient for density u at location r and  represents the vector differential operator del.Previously such type of problems has been studied by many researchers by using numerical method [10], variational iteration method and homotopy perturbation method [11], Adomian's decomposition method [12] and references therein.In this paper we consider the fractional generalization of multi-dimensional diffusion equations.There exists a wide class of literature dealing with the problems of approximate solutions to linear and nonlinear fractional differential equations with various different methodologies, called perturbation methods.But, the perturbation methods have some limitations e.g., the approximate solution involves series of small parameters which poses difficulty since majority of nonlinear problems have no small parameters at all.Although appropriate choices of small parameters some time leads to ideal solution but in most of the cases unsuitable choices lead to serious effects in the solutions.Therefore, an analytical method is welcome which does not require a small parameter in the equation modeling the phenomenon.The homotopy analysis method (HAM) was first proposed and applied by Liao [13][14][15][16][17] based on homotopy, a fundamental concept in topology and differential geometry.The HAM has been successfully applied by many researchers for solving linear and non-linear partial differential equations [18][19][20][21][22][23][24].In recent years, many authors have paid attention to study the solutions of linear and nonlinear partial differential equations by using various methods combined with the Laplace transform.Among these are Laplace decomposition method (LDM) [25][26][27], homotopy perturbation transform method (HPTM) [28][29][30][31][32][33][34][35] and homotopy analysis transform method (HATM) [36][37][38][39].In this article, we implement the homotopy analysis transform method (HATM) for obtaining analytical and numerical solutions of the two-and three-dimensional fractional diffusion equations.The HATM is an elegant combination of the Laplace transform and HAM.The advantage of this technique is its capability of combining two powerful methods for obtaining exact and approximate analytical solutions for nonlinear equations.It is worth mentioning that the proposed method is capable of reducing the volume of the computational work as compared to the classical methods while still maintaining the high accuracy of the numerical result; the size reduction amounts to an improvement of the performance of the approach.Recently, Agarwal et al. [40][41][42][43][44] have obtained few results on fractional calculus.

Definition1.1. The Laplace transform of the function
 is defined by the following series representation, valid in the whole complex plane [45]:

Basic idea of homotopy analysis transform method (HATM)
To illustrate the basic idea of this method, we consider a general fractional nonlinear nonhomogeneous partial differential equation with the initial condition of the form: ), , ( is the linear differential operator, N represents the general nonlinear differential operator and ), , ( t x g is the source term. By applying the Laplace transform on both sides of Eq. (2.6), we get Using the differentiation property of the Laplace transform, we have We define the nonlinear operator  is a real function of x, t and q.We construct a homotopy as follows If the auxiliary linear operator, the initial guess, the auxiliary parameter ,  and the auxiliary function are properly chosen, the series (2.13) converges at q = 1, then we have , ) which must be one of the solutions of the original nonlinear equations.According to the definition (2.15), the governing equation can be deduced from the zero-order deformation (2.11).Define the vectors )}., ( ),..., , ( ), , ( { Differentiating the zeroth-order deformation Eq. (2.11) m-times with respect to q and then dividing them by m! and finally setting , 0  q we get the following mth-order deformation equation: Applying the inverse Laplace transform, we have

Numerical Examples
In this section, we discuss the implementation of our proposed method and investigate its accuracy by applying the HAM with coupling of the Laplace transform method.The simplicity and accuracy of the proposed algorithm is illustrated through the following numerical examples.Applying the Laplace transform on both sides of Eq. (3.21) subject to the initial condition, we have We define a nonlinear operator as The th m -order deformation equation is given by ).
Applying the inverse Laplace transform, we have Solving the Eq.(3.27)The numerical solutions for the exact solution (3.31) and the approximate solution (3.30)  Applying the Laplace transform on both sides of Eq. (3.32) subject to the initial condition, we have We define a nonlinear operator as The th m -order deformation equation is given by ). ( )] , ( ) , ( [ Applying the inverse Laplace transform, we have Solving the Eq.(3.38) This solution is equivalent to the exact solution in closed form . ) , , (  Applying the Laplace transform on both sides of Eq. (3.43) subject to the initial condition, we have The th m -order deformation equation is given by ).
Applying the inverse Laplace transform, we have This solution is equivalent to the exact solution in closed form . ) 1 ( ) , , , (

Concluding remarks
In this work, our main concern has been to study the two-and three-dimensional fractional diffusion equations.An approximation to the analytic solution for two-and three-dimensional fractional diffusion equations was obtained by applying the HATM and symbolic calculations.The technique provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, perturbation or restrictive assumptions.The method is reliable and easy to use.The technique is very powerful and efficient in finding analytical as well as numerical solutions for wide classes of fractional differential equations.In conclusion, the HATM may be considered as a nice refinement in existing numerical techniques and might find the wide applications in science and engineering.

Figure 4
Figure 4: Plots of ) , , ( t y x u vs. y at x z =1 and y=0.5 and for different values of α.