Analytical Solution of Generalized Space-Time Fractional Advection-Dispersion Equation via Coupling of Sumudu and Fourier Transforms

The objective of this article is to present the computational solution of space-time advection-dispersion equation of fractional order associated with Hilfer-Prabhakar fractional derivative operator as well as fractional Laplace operator. The method followed in deriving the solution is that of joint Sumudu and Fourier transforms. The solution is derived in compact and graceful forms in terms of the generalized Mittag-Leffler function, which is suitable for numerical computation. Some illustration and special cases of main theorem are also discussed.


INTRODUCTION
In the last decade, considerable interest in fractional differential equations has been stimulated due to their numerous applications in the areas of physics, biology, engineering, and other areas. Several numerical and analytical methods have been developed to study the solutions of nonlinear fractional partial differential equations, for details, refer to the work in [1][2][3][4][5][6]. Fractional equations have enabled the investigation of the nonlocal response of multiple phenomena such as diffusion processes, electrodynamics, fluid flow, elasticity, and many more. Nowadays, fractional derivatives have gained a significant development to model some real life phenomena in the form of partial differential equations or the ordinary equations. Several researchers have performed the numerical simulation for fractional problems and revealed their applications in different directions include [7][8][9][10][11][12] and references therein. The exchange of heat, mass and momentum are considered to be the fundamental transfer phenomena in the universe. The mathematical framework for heat and mass transfer are of same kind, basically encompass by advection-dispersion equation. In recent work many authors have demonstrated the depth of mathematics and related physical issues of advection-dispersion equations. Schumer et al. [13] gave physical interpretation of space-time fractional advection-dispersion equation. Space-time fractional advection-dispersion equations are generalizations of classical advection-dispersion equations. The use of Hilfer-Prabhakar fractional derivative operator is gaining importance in physics because of their specific properties. The objective of this paper is to derive the solution of Cauchy type generalized fractional advection dispersion equation (18), associated with the Hilfer-Prabhakar fractional derivative. This paper provides an elegant extension of results, given earlier by Haung and Liu [14], Haubold et al. [15], Saxena et al. [16], and Agarwal et al. [17].

RESULTS REQUIRED IN THE SEQUEL
In early 90s, Watugala [18] introduced Sumudu transform, which is defined as, for all real t ≥ 0 the Sumudu transform of function f (t) ∈ A is defined as, inversion formula of (2), is given by where γ being a fixed real number. Among others, the Sumudu transform was shown to have units preserving properties, and hence may be used to solve problems without resorting to the frequency domain. Further details and properties about this transform can be found in Belgacem [19], Belgacem et al. [20], and Katatbeh and Belgacem [21].
For a function u (x, t) , the Fourier transform of with respect to x is defined by and for the function u * (η, t), inverse Fourier transform with respect to η is given by the formula For more details of Fourier transform, see [Debnath and Bhatta [22]]. Mittag-Leffler function of two parameters is studied by Wiman [23] as , α, β ∈ C, R (α) > 0. (6) Mittag-Leffler function of three parameter introduced by Prabhakar [24] as , α, β, γ ∈ C, R (α) > 0. (7) Riemann-Liouville fractional integral (right-sided) of order α is defined in [25] The right sided Riemann-Liouville fractional derivative of order α defined as here [x] is the integral part of x.
Brockmann and Sokolov [32], defined a fractional Laplace operator as: where the operators are defined by The Fourier transform of λ 2 is given in [32], as Inverse Sumudu transform of the following function is directly applicable in this sequel: In the complex plane C, for any R (α) > 0, R (β) > 0, and ω ∈ C

Example 4.1.
To describe solute transport in aquifers, consider the following generalized fractional advection dispersion equation with initial condition (31) and boundary condition where µ ′ = d ν ′ L and we consider a dimensionless parameter, called Peclet number, Pe = 1 µ ′ where L is the packing length. The Peclet number determines the nature of the problem, that is, the Peclet number is low for dispersion-dominated problems and is large for advective dominated problems, d is the dispersion coefficient L 2 T −1 and ν ′ is the Darcy velocity LT −1 .
Our interest is in the solution of (30), for this we follow same procedure, as we applied in the proof of Theorem 3.1, and after little simplification, finally we obtain Here u (x, t) represent the analytical expression of solute concentration and g k = 1 √ 2π e −(1+ik) −1 1+ik .

Example 4.2.
Consider the generalized fractional order spacetime advection-dispersion equation with the initial condition Here δ(x) is Dirac-delta function and boundary condition The solution of (34) can be obtained by same technique as we applied in proof of Theorem 3.1

SPECIAL CASES
Some interesting special cases of Theorem 3.1 are enumerated below: If we set γ = 0, in (14), then Hilfer-Prabhakar derivative reduces to Hilfer derivative (12), and the Theorem 3.1 reduces to: (I). Consider the generalized fractional order space-time advection-dispersion equation of Cauchy type where (0 < λ ≤ 2) , x ∈ R, t ∈ R + , µ ∈ (0, 1) , ν ∈ [0, 1] , with initial condition and boundary condition For obtaining the solution of (38), follow same procedure as we used in the proof of theorem 3.1, and use (13), after little simplification, obtain the following Again, use convolution theorem of the Fourier transform to (41), then we get solution of (38), in term of Green's function as Here Green's function is given as If we set ν = 1 in (12), then Hilfer fractional derivative reduces to Caputo fractional derivative operator (10) and the equation (38), yields the following: (II). Consider the generalized fractional order space-time advection-dispersion equation of Cauchy type where (0 < λ ≤ 2) , x ∈ R, t ∈ R + , µ ∈ (0, 1) , with initial condition and boundary condition For obtaining the solution of (42), follow same procedure as we used in the proof of theorem 3.1, and use (11), after little simplification, obtain the following Again, use convolution theorem of the Fourier transform to (45) then we get solution of (42), in term of Green's function as Here Green's function is given as (III). On giving suitable value to the parameters involved in Theorem 3.1, we can obtained same results, earlier given by Haung and Liu [14], Haubold et al. [15], Saxena et al. [16], and Agarwal et al. [17].

CONCLUSION
In this paper, we have presented a solution of generalized spacetime fractional advection-dispersion equation. The solution has been developed in terms of Mittag-Leffler function with the help of Sumudu transform and Fourier transform. We can develop the efficient numerical techniques to find solution of various fractional partial differential equations arising in various fields by considering these analytic solutions as base. For future research, the methodology presented in this paper can serve as a good working template to solve any fractional advection-dispersion equations in higher dimensions.

AUTHOR CONTRIBUTIONS
VG, JS, and YS designed the study, developed the methodology, collected the data, performed the analysis, and wrote the manuscript.