Research Article | Open Access

Birol İbiş, Mustafa Bayram, "Approximate Solution of Time-Fractional Advection-Dispersion Equation via Fractional Variational Iteration Method", *The Scientific World Journal*, vol. 2014, Article ID 769713, 5 pages, 2014. https://doi.org/10.1155/2014/769713

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

**Academic Editor:**A. M. A. El-Sayed

#### Abstract

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.

#### 1. Introduction

Many problems in mechanical engineering, physics, biology, chemistry, control theory, fluid mechanics, signal processing, viscoelasticity, electromagnetism, electrochemistry, thermal engineering, and many other physical processes are modeled by fractional differential equations (FDEs) or fractional partial differential equations (FPDEs) [1–9]. The solution of FDEs has been recently studied. In most cases, FDEs do not have analytical solutions, so these equations have been solved by using various analytical and numerical methods. The time-fractional advection-dispersion equations, which are a special type of FPDEs, have been applied to many problems [10–18]. There may be several methods for solving FADEs such as variable transformation [19], finite element method [20], Adomian decomposition method (ADM) [21], implicit and explicit difference method [22], homotopy analysis method (HAM) [23], optimal homotopy asymptotic method [24], homotopy perturbation method (HPM) [25], Green function [26], and least-squares spectral method [27].

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.

#### 2. Preliminaries

Some necessary definitions, lemmas, and properties of the fractional calculus are reviewed in this section [28, 29].

*Definition 1. *The Riemann-Liouville fractional integral of order is defined as
where , , .

*Definition 2. *The modified Riemann-Liouville fractional derivative of order is defined as
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 ):
where is the derivative of order of .

When substituting and in (10), we get the fractional McLaurin series:

Lemma 7. *The solution of continuous function in (8) is
**
For example, when (12) is applied for function , one gets
*

#### 3. 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: 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 and are considered as restricted variations; that is, . Making the above functional stationary, with the property from (4) and (6), must satisfy Therefore, is determined as Substituting (19) into the functional (16) gives the iteration formulation as We start by selecting an appropriate initial function ; the consecutive approximations of can be easily achieved. Generally, the initial values are chosen as zeroth approximation . Consequently, the solution of (1) is obtained by .

#### 4. 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.

*Example 8. *Firstly, the following time FADE subject to the initial condition is considered [24]:
The corresponding iterative formula (20) for (21) can be derived as
Starting with , by the iterative formula (22), we derive the following results:
Consequently, the approximate solution is obtained as follows:
where is the Mittag-Leffler function.

*Example 9. *Now, the following time FADE subject to the initial condition is considered [21]:
The corresponding iterative formula (20) for (25) can be derived as
Starting with , by formula (26), we derive the following results:
which is the same solution as that given in [21] using the ADM.

*Example 10. *Finally, the following nonhomogeneous time FADE with variable coefficients with the initial condition is considered:
Using (20), the iteration formula for (28) is given by
Starting with , and using (29), we have
Consequently, the exact solution is obtained as follows:

#### 5. 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.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

- K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, Wiley, New York, NY, USA, 1993. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives: Theory and Applications*, Gordon and Breach,, Yverdon, Switzerland, 1993. - I. Podlubny,
*Fractional Differential Equations*, Academic Press, New York, NY, USA, 1999. - E. Çelik and M. Bayram, “The numerical solution of physical problems modeled as a systems of differential-algebraic equations (DAEs),”
*Journal of the Franklin Institute*, vol. 342, no. 1, pp. 1–6, 2005. View at: Publisher Site | Google Scholar - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. - N. Guzel and M. Bayram, “Numerical solution of differential-algebraic equations with index-2,”
*Applied Mathematics and Computation*, vol. 174, no. 2, pp. 1279–1289, 2006. View at: Publisher Site | Google Scholar - J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Eds.,
*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*, Springer, Dordrecht, The Netherlands, 2007. - M. Kurulay and M. Bayram, “Approximate analytical solution for the fractional modified KdV by differential transform method,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 7, pp. 1777–1782, 2010. View at: Publisher Site | Google Scholar - M. Kurulay, B. A. Ibrahimoǧlu, and M. Bayram, “Solving a system of nonlinear fractional partial differential equations using three dimensional differential transform method,”
*International Journal of Physical Sciences*, vol. 5, no. 6, pp. 906–912, 2010. View at: Google Scholar - G. H. Zheng and T. Wei, “Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation,”
*Journal of Computational and Applied Mathematics*, vol. 233, no. 10, pp. 2631–2640, 2010. View at: Publisher Site | Google Scholar - M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,”
*Journal of Computational and Applied Mathematics*, vol. 172, no. 1, pp. 65–77, 2004. View at: Publisher Site | Google Scholar - Z.-Q. Deng, V. P. Singh, and L. Bengtsson, “Numerical solution of fractional advection-dispersion equation,”
*Journal of Hydraulic Engineering*, vol. 130, no. 5, pp. 422–431, 2004. View at: Publisher Site | Google Scholar - R. Metzler and J. Klafter, “Accelerating Brownian motion: a fractional dynamics approach to fast diffusion,”
*Europhysics Letters*, vol. 51, no. 5, pp. 492–498, 2000. View at: Publisher Site | Google Scholar - M. E. Khalifa, “Some analytical solutions for the advection-dispersion equation,”
*Applied Mathematics and Computation*, vol. 139, no. 2-3, pp. 299–310, 2003. View at: Publisher Site | Google Scholar - N. Pochai, “A numerical computation of a non-dimensional form of stream water quality model with hydrodynamic advection-dispersion-reaction equations,”
*Nonlinear Analysis: Hybrid Systems*, vol. 3, no. 4, pp. 666–673, 2009. View at: Publisher Site | Google Scholar - A. R. Johnson, C. A. Hatfield, and B. T. Milne, “Simulated diffusion dynamics in river networks,”
*Ecological Modelling*, vol. 83, no. 3, pp. 311–325, 1995. View at: Publisher Site | Google Scholar - D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,”
*Water Resources Research*, vol. 36, no. 6, pp. 1403–1412, 2000. View at: Publisher Site | Google Scholar - G. J. Fix and J. P. Roof, “Least squares finite-element solution of a fractional order two-point boundary value problem,”
*Computers and Mathematics with Applications*, vol. 48, no. 7-8, pp. 1017–1033, 2004. View at: Publisher Site | Google Scholar - F. Liu, V. V. Anh, I. Turner, and P. Zhuang, “Time fractional advection-dispersion equation,”
*Journal of Applied Mathematics and Computing*, vol. 13, no. 1-2, pp. 233–245, 2003. View at: Google Scholar - Q. Huang, G. Huang, and H. Zhan, “A finite element solution for the fractional advection-dispersion equation,”
*Advances in Water Resources*, vol. 31, no. 12, pp. 1578–1589, 2008. View at: Publisher Site | Google Scholar - S. Momani and Z. Odibat, “Numerical solutions of the space-time fractional advection-dispersion equation,”
*Numerical Methods for Partial Differential Equations*, vol. 24, no. 6, pp. 1416–1429, 2008. View at: Publisher Site | Google Scholar - F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, “Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation,”
*Applied Mathematics and Computation*, vol. 191, no. 1, pp. 12–20, 2007. View at: Publisher Site | Google Scholar - R. K. Pandey, O. P. Singh, and V. K. Baranwal, “An analytic algorithm for the space-time fractional advection-dispersion equation,”
*Computer Physics Communications*, vol. 182, no. 5, pp. 1134–1144, 2011. View at: Publisher Site | Google Scholar - R. K. Pandey, O. P. Singh, V. K. Baranwal, and M. P. Tripathi, “An analytic solution for the space-time fractional advection-dispersion equation using the optimal homotopy asymptotic method,”
*Computer Physics Communications*, vol. 183, no. 10, pp. 2098–2106, 2012. View at: Google Scholar - A. Golbabai and K. Sayevand, “Analytical modelling of fractional advection-dispersion equation defined in a bounded space domain,”
*Mathematical and Computer Modelling*, vol. 53, no. 9-10, pp. 1708–1718, 2011. View at: Publisher Site | Google Scholar - F. Huang and F. Liu, “The fundamental solution of the space-time fractional advection-dispersion equation,”
*Journal of Applied Mathematics and Computing*, vol. 18, no. 1-2, pp. 339–350, 2005. View at: Google Scholar - A. R. Carella and C. A. Dorao, “Least-squares spectral method for the solution of a fractional advection-dispersion equation,”
*Journal of Computational Physics*, vol. 232, no. 1, pp. 33–45, 2013. View at: Google Scholar - G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,”
*Applied Mathematics Letters*, vol. 22, no. 3, pp. 378–385, 2009. View at: Publisher Site | Google Scholar - G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,”
*Computers and Mathematics with Applications*, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at: Publisher Site | Google Scholar - J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 167, no. 1-2, pp. 57–68, 1998. View at: Google Scholar

#### Copyright

Copyright © 2014 Birol İbiş and Mustafa Bayram. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.