Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations

The space-time fractional advection dispersion equations are linear partial pseudo-differential equations with spatial fractional derivatives in time and in space and are used to model transport at the earth surface. The time fractional order is denoted by   0,1   and   0,2   1 is devoted to the space fractional order. The time fractional advection dispersion equations describe particle motion with memory in time. Space-fractional advection dispersion equations arise when velocity variations are heavy-tailed and describe particle motion that accounts for variation in the flow field over entire system. In this paper, I focus on finding the precise explicit discrete approximate solutions to these models for some values of 0    with 0 1    , 1 2    while the Cauchy case as 1   and the classical case as 2   with 1   are studied separately. I compare the numerical results of these models for different values of  and  and for some other related changes. The approximate solutions of these models are also discussed as a random walk with or without a memory depending on the value of  . Then I prove that the discrete solution in the Fourierlaplace space of theses models converges in distribution to the Fourier-Laplace transform of the corresponding fractional differential equations for all the fractional values of  and  .


Introduction
The development which has happened on the last twentyfive years on the fractional calculus opened many new applications on many fields such as physics, hydrodynamics, chemistry, financial mathematics, and some other fields. Actually a growing number of articles and books which are interesting on this field and its applications have appeared in these last 25 years (see for example: [1][2][3][4][5] and see also my thesis [6]. Fractional in time means that the first-order time derivative is replaced by the Caputo derivative of order , see [4]. Fractional in space means replacing the second order space-derivative is replaced by the Feller operator [7] in the symmetric case with order    . The behaviour of particles in transport under the earth surface is an important problem. For examples, the transport of solute and contaminant particles in surface and subsurface water flows, the behaviour of soil particles and associated soil particles, and the transport of sediment particles and sediment-borne substances in turbulent flow. There are many other examples in this field. The classical advection dispersion equation, ade, has been used to formulate such problems. The generalized fractional advection-dispersion equation, fade, has recently gotten an increasing interest from many scientists because it has many applications specially on studying the transport of passive tracers carried by fluid flow in a porous medium, see Benson, Meerschaert et al. [8][9][10][11][12]. In their work they gave applications and experimental results for the space-fade.
There is no unique solution for the space-time fractional diffusion processes but there are some attempts using different forms of the hyper geometric functions, as for example: in [13] the authors attempted to find an analytical solution for other special form of the fractional, see also [14,15]. Therefore authors who study modelling of fractional processes use some developed methods to descritize the fractional operators. For examples, in [16], the authors used their own method of descretization to find the approximate solution of the space-fade and gave some numerical results. In [17], the authors studied the approximate solution of the space-fade, for 1 2    and 1   , only using the backward Grünwald-Letnikov Scheme. The backward Grünwald-Letnikov Scheme has been successfully adopting by Gorenflo, Mainrdi, and etal, see [4,18,19] for modelling space-fractional diffusion processes. Also has been used by Gorenflo and E. A. Abdel-Rehim, see [20][21][22][23][24] for modelling time-fractional Fokker-Planck equations and their convergence in the Fourier Laplace domain. I am interested in this paper to find the approximate solutions of the space-time fractional advection equation, space-time fade, by adopting the backward Grünwald-Letnikov Scheme joined with the common finite difference methods. The space-time fade is considered as a diffusion process under the action of a constant force in a fractal medium with a memory. I study and numerically investigate the effect of the time fractional on the path of the particle motion as well as the effect of the spacefractional order for the three cases as: 0 < α < 1, 1 < α < 2, and 1   . I compare between all these cases numerically. My numerical results are consistent with the results of [17] for the studied case 1 2    and 1   .
The approximate solutions according to the values 1 < α < 2, and 1   joined with 0 1    are firstly studied on this paper. The proof of the convergence in distribution for each case is also considered. Therefore this paper is organized as follows: Section 1 is denoted to the introduction; Section 2 is devoted to the definitions of the used fractional operators and their Laplace-Fourier transformations; and Section 3 introduces the classical case 2 is studied at Section 5.2. Section 6 is devoted to the caseα = 1, 0 1    . Finally, the numerical results will be displayed and explained in Section 7 and one compares these results with the results of the given references.

Important Definitions and the Outline of the Proof of Convergence in Distribution
The generalized fade reads Here a, and b are positive constants representing the dispersion coefficient, and the average fluid velocity and it acts as the drift term to the right respectively. My aim is to give the approximation solutions of the space-time fad equations for all values of  and  . I study also the convergence of the approximation solutions to the solutions of the corresponding analytical solutions of the space-time fad equations in the Fourier-Laplace domain. The used time-fractional derivative operator is called Caputo fractional operator, see [4] to know the relation between Caputo fractional derivative and the famous Riemann-Liouville fractional derivative operators. Caputo fractional derivative in the Laplace domain reads This equation is important for solving the fractional differential equations because it show the dependence on the initial conditions. Here called the Riesz space-fractional differentiation operator. I adopt here the notation introduced by [25]. It is formally a power of the positive definitive operator x   and must not be confused with a power of the first order differential operator (see [4] for a 1 0 x D detailed theory of this operator and related operators). I need to adopt the Fourier transform of a (generalized) For the proof of convergence in distribution I need to use the Fourier transform of This means, in the Zaslavski' s notations, From (2.2) -(2.4), one easily sees that in the case 1 which proves that the Riesz derivative is a symmetric fractional generalization of the second derivative. For more information about the Fourier transform and the pseudo-differential operators as semi groups of linear operators, see e.g. [26,27]. In my paper, I discuss the approximate solution of the Equation (2.1) for all values of  and  , to do so, I descretize x and t by the Here , and 0 h  0   are the steps in space and in time, respectively, and is the number of steps at the x direction. Treating as a density of an extensive quantity (like mass, charge, solute concentration, or probability), the approximation of where . To proceed on the proof of convergence in distribution, one needs to use the method of generating functions, see [22] for more information about the procedures used to prove the convergence. Therefore, for Now, introduce the bivariate (two-fold) generating function 0 . (2.9) Introduce the function and apply Our aim now is to prove that is related asymptotically to the Fourier-Laplace transform of , u x t which represents the analytical solution of Equation (2.1) for any values of  and  , and for a fixed    and , as . So far, I will prove 0 for each case.

The Classical ade
I describe in this section the classical partial differential ade and its proof of convergence in distribution. It is well known that the classical ade is a partial differential equation describing the solute transport in aquifers and it reads


With the initial condition . The classical ade is also interpreted as a deterministic equation with the probability function which describes the particle spreading away from the plume center of mass. The stochastic process described by Equation (3.1) is a Brownian motion with a constant drift [11].
then one has the diffusion of a free particle, that is, a particle in which no forces other than those due to the molecules of the surrounding medium are acting, which reads and hence fourth in the Fourier-Laplace domain, see [28] Now descretizing (3.1) by the central symmetric difference in space and forward in time, one gets Introduce the scaling relation and for the positivity of all the coefficients of   n j y , one must put 0 12 The discrete solution at Equation (3.5) describes also a random walk with sojourn probability   n j y of a particle at the point j x at the instant n and it may jump either to the points [29]. Utilizing this concept, Equation (3.5) can be rewritten as The transition probabilities 1 jj , p  jj and p 1 jj p  in Equation (3.6) satisfy the essential condition Now one can use these transition probabilities to constitute a tridiagonal, P matrix, in which 0, 2 Introduce the row vector , defined as In order to find the explicit discrete solution of Equation (3.1), I have to take the transpose of each sides of the matrix Equation (3.7) and rewrite it as and for the numerical calculations, it is convenient to write the stochastic matrix in the form P  , here I is the unit matrix and H is a matrix whose rows are summed to zero. In Section 7, I give the evolution of for different values of . Now I am going to prove that the discrete solution at Equation (3.5) converges to the Fourier-Laplace transform of Equation (3.1). Rewrite Equation (3.5) as Then multiplying both sides by j z and summing over all , to get j Multiplying both sides by n  and summing over all The choice of the initial condition of the column vector satisfying that , guarantees that

The Time-Fractional ade
In this section, I replace the first-order time derivative in Equation (3.1) by the Caputo fractional derivative, For more information about the Caputo fractional derivative and its relations to the Riemann-Liouville, see [6] and the list of references therein. Now, Taking the Fourier-Laplace transform, see Section 2, one gets To descretize * t , I utilize the backward Grünwald-Letnikov scheme which has been successfully utilizing at [19][20][21][22][23][24] for modelling and simulating the timefractional diffusion processes and the time-fractional Fokker-Planck equations.
where for ease of writing, I use n and which has been originally introduced in [19] as where 0 1 c  , see [19]. For all the coefficients of   This equation can be interpreted as a random walk with a memory, see [30]. In this case, the particle is sitting at the position j x at the time instant n and can move to either Now proceed further, multiply each sides by m  and sum over all , to get m After using this rule, and put , then apply Taylor expansion, and take the limit as , one gets

The Space-Time-Fractional ade
In this section I consider the space-time fade, Equation (2.1). It is known that the space-fractional ade arises when velocity variations are heavy tailed and describe particle motion that accounts for variation in the flow field over entire system. The time fractional ade arises as a result of power law particle residence time distributions and describe particle motion with memory in time, see [12]. The used space-fractional operator 0 x D  , is the sym-metric Feller operator, see [7]. This operator represents the negative inverse of the Riesz Potential 0 I  whose symbol is where the symmetric Riesz Potential operator is defined as The  [1] and see also [6], in which one can find a long list of related references. The inverse of the Riemann-Liouville integrals can formally be obtained as the limit 0 lim , where h I    denotes the approximating Grünwald-Letnikov scheme which reads, see [4,18,19] a) 0 1 The shift in the index in Equation (5.5) is required to obtain a scheme with all coefficients are non-negative in the final formula for j  1 j n y t  which gives schemes for simulating particle paths which results after replacing the second order space-derivative in Equation (4.1) by the Feller operator [7]. One can adopt, for simplicity, the notation introduced by Zaslavski [ Now we adjoin the descretization of Adopting the scaling relation To ensure that the coefficients of all , it requires that Let us write Equation (5.11) in the form of a random walk, in which the walker is sitting at j x at and jumps to at  [32] for more information about the discrete random walk of space-fractional diffusion processes. Then by using this notations, Equation (5.11) can be written in the form of random walk as To ensure that all the coefficients of   n j y are positive, the scaling relation must satisfy To have all the coefficients of   . Following [33], it can be proved that the summations of the transition probabilities of Equation (6.4) are summed to one and one can easily write it in the form of a random walk as the previous cases. The last equation could also be written on the same matrix form (5.13). Where A is a fifth diagonal matrix whose diagonal elements are defined as           discussed by many authors. The pates as 1   1 are different than the paths corresponding to   . The pathes for 1   need a huge number of time steps to calculate them as I use the explicit difference methods.