Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method

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Abstract

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical superdiffusive problems in fluid flow, finance and other areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by two-step Adomian Decomposition Method (TSADM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form and then their solutions have been represented graphically. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present technique. The solutions obtained by the standard decomposition method have been numerically evaluated and presented in the form of tables and then compared with those obtained by TSADM. The present TSADM performs extremely well in terms of efficiency and simplicity.

Introduction

Fractional diffusion equations are used to model problems in Physics [1], [2], [3], Finance [4], [5], [6], [7], and Hydrology [8], [9], [10], [11], [12]. Fractional space derivatives may be used to formulate anomalous dispersion models, where a particle plume spreads at a rate that is different than the classical Brownian motion model. When a fractional derivative of order 1 < α < 2 replaces the second derivative in a diffusion or dispersion model, it leads to a superdiffusive flow model. Nowadays, fractional diffusion equation plays important roles in modeling anolmalous diffusion and subdiffusion systems, description of fractional random walk, unification of diffusion and wave propagation phenomenon, see, e.g., the reviews in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], and references therein.

Consider a one-dimensional fractional diffusion equation considered in [17]u(x,t)t=d(x)αu(x,t)xα+q(x,t)on a finite domain xL < x < xR with 1 < α  2. We assume that the diffusion coefficient (or diffusivity) d(x) > 0. We also assume an initial condition u(x, t = 0) = s(x) for xL < x < xR and Dirichlet boundary conditions of the form u(xL, t) = 0 and u(xR, t) = bR(t). Eq. (1.1) uses a Riemann fractional derivative of order α.

Consider a two-dimensional fractional diffusion equation considered in [18]u(x,y,t)t=d(x,y)αu(x,y,t)xα+e(x,y)βu(x,y,t)yβ+q(x,y,t)on a finite rectangular domain xL < x < xH and yL < y < yH, with fractional orders 1 < α  2 and 1 < β  2, where the diffusion coefficients d(x, y) > 0 and e(x, y) > 0. The ‘forcing’ function q(x, y, t) can be used to represent sources and sinks. We will assume that this fractional diffusion equation has a unique and sufficiently smooth solution under the following initial and boundary conditions. Assume the initial condition u(x, y, t = 0) = f(x, y) for xL < x < xH, yL < y < yH, and Dirichlet boundary condition u(x, y, t) = B(x, y, t) on the boundary (perimeter) of the rectangular region xL  x  xH, yL  y  yH, with the additional restriction that B(xL, y, t) = B(x, yL, t) = 0. In physical applications, this means that the left/lower boundary is set far away enough from an evolving plume that no significant concentrations reach that boundary. The classical dispersion equation in two dimensions is given by α = β = 2. The values of 1 < α < 2, or 1 < β < 2 model a superdiffusive process in that coordinate. Eq. (1.2) also uses Riemann fractional derivatives of order α and β.

In this paper, we use the Adomian Decomposition Method (ADM) [19], [20] to obtain the solutions of the fractional diffusion equations (1.1), (1.2). Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the ADM [19], [20], [21], [22], [23], [24], [25], [26]. A reliable modification of ADM has been done by Wazwaz [27]. The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems [19], [20], [21], [22], [23], [24], [25]. Recently, the implementations of ADM for the solutions of generalized regularized long-wave (RLW) and Korteweg-de Vries (KdV) equations have been well established by the notable researchers [26], [27], [28], [29], [30], [31]. This method efficiently works for initial-value or boundary-value problems and for linear or nonlinear, ordinary or partial differential equations and even for stochastic systems. Moreover, we have the advantage of a single global method for solving ordinary or partial differential equations as well as many types of other equations. Recently, the solution of fractional differential equation has been obtained through ADM by the researchers [32], [33], [34], [35], [36], [37], [38], [39]. The application of ADM for the solution of nonlinear fractional differential equations has also been established by Shawagfeh, Saha Ray and Bera [35], [38].

Section snippets

Mathematical definition

The mathematical definition of fractional calculus has been the subject of several different approaches [40], [41]. The most frequently encountered definition of an integral of fractional order is the Riemann–Liouville integral, in which the fractional order integral is defined asDt-qf(t)=d-qf(t)dt-q=1Γ(q)0tf(x)dx(t-x)1-qwhile the definition of fractional order derivative isDtqf(t)=dndtnd-(n-q)f(t)dt-(n-q)=1Γ(n-q)dndtn0tf(x)dx(t-x)1-n+qwhere q(q > 0 and q  R) is the order of the operation and n

The fractional diffusion equation model and its solution

We adopt Adomian Decomposition Method for solving Eq. (1.1). In the light of this method we assume thatu=n=0unto be the solution of Eq. (1.1).

Now, Eq. (1.1) can be rewritten asLtu(x,t)=d(x)Dxαu(x,t)+q(x,t)where Ltt which is an easily invertible linear operator, Dxα() is the Riemann–Liouville derivative of order α.

Therefore, by Adomian Decomposition Method, we can writeu(x,t)=u(x,0)+Lt-1d(x)Dxαn=0un+Lt-1(q(x,t))Each term of series (3.1) is given by the standard Adomian Decomposition

Implementation of two-step Adomian Decomposition Method

Example 1

Let us consider a one-dimensional fractional diffusion equation for Eq. (1.1), as taken in [17]

u(x,t)t=d(x)1.8u(x,t)x1.8+q(x,t)on a finite domain 0 < x < 1, with the diffusion coefficientd(x)=Γ(2.2)x2.8/6=0.183634x2.8,the source/sink functionq(x,t)=-(1+x)e-tx3,the initial conditionu(x,0)=x3,for0<x<1and the boundary conditions u(0, t) = 0, u(1, t) = et, for t > 0.

Now, Eq. (4.1) can be rewritten in operator form asLtu(x,t)=d(x)Dx1.8u(x,t)+q(x,t)where Ltt symbolizes the easily invertible linear

Numerical results and discussion

From Table 1, Table 2, Table 3, we see that Standard Adomian Decomposition Method solution converges very slowly to the exact solution. On the other hand, TSADM requires only two iterations to achieve the exact solution. Therefore, TSADM is more effective and promising compared to standard Adomian Decomposition Method.

From Table 4 we see that for small values of x and y the absolute error for TSADM Solution and Standard Adomian Decomposition Method Solution ϕ3 is very small. But as the values

Conclusion

This paper presents an analytical scheme to obtain the solutions of the one-dimensional and two-dimensional fractional diffusion equations. Two typical examples have been discussed as illustrations. In our previous papers [36], [37], [38], [39] we have already established as well as successfully exhibited the applicability of Adomian Decomposition Method to obtain the solutions of different types of fractional differential equations. In this work, we demonstrate that TSADM is also well suited

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