Approximate Solution of Time-Fractional Advection-Dispersion Equation via Fractional Variational Iteration Method

This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs.

In this study, the following time FADE with the initial condition is discussed: where ( , ), ( , ), and ]( , ) represent the solute concentration, the dispersion coefficient, and the average fluid velocity, respectively.

Preliminaries
Some necessary definitions, lemmas, and properties of the fractional calculus are reviewed in this section [28,29].
Definition 2. The modified Riemann-Liouville fractional derivative of order is defined as 2 The Scientific World Journal The properties of the modified Riemann-Liouville fractional derivative are (i) product rule for fractional derivatives: (ii) fractional Leibniz formula: (iii) integration by parts for fractional order: Definition 3. Limit form of the fractional derivative is defined as Definition 4. Fractional derivative is defined for compounded functions as follows: Definition 5. The integral with respect to ( ) is defined as Definition 6. The following equality is provided for the continuous function : → and → ( ) has a fractional derivative of order ( ∈ + and 0 < ≤ 1): where ( ) is the derivative of order of ( ). When substituting ℎ → and → 0 in (10), we get the fractional McLaurin series: For example, when (12) is applied for function ( ) = , one gets

Fractional Variation Iteration Method (FVIM)
Equation (1) with initial conditions is considered to describe the solution procedure of the FVIM. Using the VIM developed by He [30], a correction function for (1) can be set as follows: +1 ( , ) = ( , ) New correction functional is obtained as follows by combining (12) and (15): where is a Lagrange multiplier which can be determined optimally through the variational theory. Here with the property from (4) and (6), ( , ) must satisfy Therefore, ( , ) is determined as Substituting (19) into the functional (16) We start by selecting an appropriate initial function 0 ( , ); the consecutive approximations ( , ) of ( , ) can be easily achieved. Generally, the initial values are chosen as zeroth approximation 0 ( , ). Consequently, the solution ( , ) of (1) is obtained by ( , ) = lim → ∞ ( , ).

Approximate Solutions of Time FADEs
In this section, in order to show the applicability and efficiency of the FVIM for solving time FADEs, some illustrative examples are given.
Consequently, the approximate solution is obtained as follows: where ((1 + ) ) is the Mittag-Leffler function.

Conclusion
The fundamental aim of this study was to obtain an analytical approximate solution of time FADEs using the FVIM. The aforementioned implementation indicates that this method is powerful and efficient in solving the equation in an easier and a more accurate way. The method also provides an analytical approximation solution in a rapidly convergent series with easily calculable terms for various physical problems. Therefore, FVIM is a more effective, a more convenient, and a more accurate method than other methods mentioned in the introduction. The obtained results denote that this method can be considered as an alternative to the other methods in the literature in terms of the purpose of solving linear or nonlinear FDEs in general.