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^{2}

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Edited by: Dumitru Baleanu, University of Craiova, Romania

Reviewed by: Francisco Gomez, Centro Nacional de Investigación y Desarrollo Tecnológico, Mexico; Kolade Matthew Owolabi, Federal University of Technology, Nigeria

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The objective of this article is to present the computable 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.

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 [

In early 90s, Watugala [

for all real

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 [

For a function

and for the function ^{*}(η,

For more details of Fourier transform, see [Debnath and Bhatta [

Mittag-Leffler function of two parameters is studied by Wiman [

Mittag-Leffler function of three parameter introduced by Prabhakar [

Riemann-Liouville fractional integral (right-sided) of order α is defined in [

The right sided Riemann-Liouville fractional derivative of order α defined as

here [

Caputo [

The Sumudu transform of (10) is given in [

where ū(

Hilfer [

For ν = 0, equation (12) reduces into (9) and for ν = 1, equation (12) reduces into (10).

The Sumudu transform of (12) is given in [

Where the initial value term

A generalization of Hilfer derivate is given in [

Let μ ∈ (0, 1), ν ∈ [0, 1], and let

where γ, ω ∈

For details of this derivative, refer to the work in [

Brockmann and Sokolov [

where the operators are defined by

and

The Fourier transform of

Inverse Sumudu transform of the following function is directly applicable in this sequel:

In the complex plane C, for any

Here we will find, the solution of the generalized space-time Advection-Dispersion equation (18) under the conditions given in (19) and (20). Our main findings in the form of the following Theorem 3.1 and Corollary 3.2.

where λ ∈ (0, 2] ^{+}, μ ∈ (0, 1), ν ∈ [0, 1],

with initial condition,

and boundary condition

where

^{*}(

where

Solve equation (23), by using conditions (19)-(20), we get

On taking inverse Sumudu transform of equation (24), and after little simplification, apply result (17), it gives

Taking inverse Fourier transform of (25), get our required result (21).

This completes the proof of the theorem 3.1.

On taking

where λ ∈ (0, 2], ^{+}, μ ∈ (0, 1), ν ∈ [0, 1],

with initial condition

and boundary condition

where ^{−27}^{−21}

with initial condition

and boundary condition

where ^{2}^{−1}] and ν′ is the Darcy velocity [^{−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

with the initial condition

Here δ(

The solution of (34) can be obtained by same technique as we applied in proof of Theorem 3.1

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:

where (0 < λ ≤ 2), ^{+}, 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:

where (0 < λ ≤ 2), ^{+}, 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

In this paper, we have presented a solution of generalized space-time 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.

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

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors are grateful to referees for their suggestions and useful comments on this paper.

^{a}(x), (German)