- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

International Journal of Differential Equations

VolumeÂ 2012Â (2012), Article IDÂ 346089, 17 pages

http://dx.doi.org/10.1155/2012/346089

## On the Solutions Fractional Riccati Differential Equation with Modified Riemann-Liouville Derivative

Department of Mathematics Engineering, GĂĽmĂĽĹźhane University, 29100 GĂĽmĂĽĹźhane, Turkey

Received 10 December 2011; Accepted 6 March 2012

Academic Editor: EbrahimÂ Momoniat

Copyright Â© 2012 Mehmet Merdan. 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.

#### Abstract

Fractional variational iteration method (FVIM) is performed to give an approximate analytical solution of nonlinear fractional Riccati differential equation. Fractional derivatives are described in the Riemann-Liouville derivative. A new application of fractional variational iteration method (FVIM) was extended to derive analytical solutions in the form of a series for these equations. The behavior of the solutions and the effects of different values of fractional order are indicated graphically. The results obtained by the FVIM reveal that the method is very reliable, convenient, and effective method for nonlinear differential equations with modified Riemann-Liouville derivative

#### 1. Introduction

In recent years, fractional calculus used in many areas such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics, engineering, acoustics, viscoelasticity, material science and signal processing can be successfully modelled by linear or nonlinear fractional order differential equations [1â€“10]. As it is well known, Riccati differential equations concerned with applications in pattern formation in dynamic games, linear systems with Markovian jumps, river flows, econometric models, stochastic control, theory, diffusion problems, and invariant embedding [11â€“17]. Many studies have been conducted on solutions of the Riccati differential equations. Some of them, the approximate solution of ordinary Riccati differential equation obtained from homotopy perturbation method (HPM) [18â€“20], homotopy analysis method (HAM) [21], and variational iteration method proposed by He [22]. The Heâ€™s homotopy perturbation method proposed by He [23â€“25] the variational iteration method [26] and Adomian decomposition method (ADM) [27] to solve quadratic Riccati differential equation of fractional order.

The variational iteration method (VIM), which proposed by He [28, 29], was successfully applied to autonomous ordinary and partial differential equations and other fields. He [30] was the first to apply the variational iteration method to fractional differential equations. In recent years, a new modified Riemann-Liouville left derivative is suggested by Jumarie [31â€“35]. Recently, the fractional Riccati differential equation is solved with help of new homotopy perturbation method (HPM) [23].

In this paper, we extend the application of the VIM in order to derive analytical approximate solutions to nonlinear fractional Riccati differential equation: subject to the initial conditions where is fractional derivative order, is an integer, , , and are known real functions, and is a constant.

The goal of this paper is to extend the application of the variational iteration method to solve fractional nonlinear Riccati differential equations with modified Riemann-Liouville derivative.

The paper is organized as follows: In Section 2, we give definitions related to the fractional calculus theory briefly. In Section 3, we define the solution procedure of the fractional variational iteration method to show inefficiency of this method, we present the application of the FVIM for the fractional nonlinear Riccati differential equations with modified Riemann-Liouville derivative and numerical results in Section 4. The conclusions are then given in the final Section 5.

#### 2. Basic Definitions

Here, some basic definitions and properties of the fractional calculus theory which can be found in [31â€“35].

*Definition 2.1. *Assume denote a continuous (but not necessarily differentiable) function, and let the partition in the interval . Jumarieâ€™s derivative is defined through the fractional difference [34]:
where . Fractional derivative is defined as the following limit form [1, 7]:
This definition is close to the standard definition of derivatives (calculus for beginners), and as a direct result, the th derivative of a constant, , is zero.

*Definition 2.2. *The left-sided Riemann-Liouville fractional integral operator of order , of a function is defined as
The properties of the operator can be found in [1, 7, 36].

*Definition 2.3. *The modified Riemann-Liouville derivative [33, 34] is defined as
where .

In addition, we want to give as in the following some properties of the fractional modified Riemann-Liouville derivative.

Fractional Leibniz product law:

Fractional Leibniz Formulation:

Fractional the integration of part:

*Definition 2.4. *Fractional derivative of compounded functions [33, 34] is defined as

*Definition 2.5. *The integral with respect to [33, 34] is defined as the solution of the fractional differential equation:

Lemma 2.6. *Let denote a continuous function [33, 34] then the solution of (2.5) is defined as
**
For example, in (2.10) one obtains
*

*Definition 2.7. *Assume that the continuous function has a fractional derivative of order , for any positive integer and any , ; then the following equality holds, which is
On making the substitution and , we obtain the fractional Mc-Laurin series:

#### 3. Fractional Variational Iteration Method

To describe the solution procedure of the fractional variational iteration method [31â€“35], we consider the following fractional Riccati differential equation: According to the VIM, we can build a correct functional for (3.1) as follows: Using (2.3), we obtain a new correction functional: It is obvious that the sequential approximations , can be established by determining , a general Lagrangeâ€™s multiplier, which can be identified optimally with the variational theory. The function is a restricted variation which means . Therefore, we first designate the Lagrange multiplier that will be identified optimally via integration by parts. The successive approximations of the solution will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function . The initial values are usually used for choosing the zeroth approximation . With determined, then several approximations , follow immediately [37]. Consequently, the exact solution may be procured by using

#### 4. Applications

In this section, we present the solution of two examples of the Riccati differential equations as the applicability of FVIM.

*Example 4.1. *Let usconsider the fractional Riccati differential equation, we get
with initial conditions:
Construction the following functional:
we have
Similarly, we can get the coefficients of to zero:
The generalized Lagrange multiplier can be identified by the above equations:
substituting (4.6) into (4.3) produces the iteration formulation as follows:
Taking the initial value , we can derive
Then, the approximate solutions in a series form are
As is
The exact solution of (4.1) is , when .

Figure 1 indicates the solution obtained using FVIM versus the exact solution when . Figure 2 is plotted for approximate solution of time-fractional Riccati differential equation for , 0.8, 0.9, and 1. Equation (4.1) is solved by using the homotopy perturbation method (HPM) [24]. FVIM solutions indicate that the present algorithm performs extreme efficiency, simplicity, and reliability. The results obtained from FVIM are fully compatible with those of the HPM.

Table 1 shows the approximate solutions for (4.1) obtained for different values of using the variational iteration method and HPM [24]. From the numerical results in Table 1, it is clear that the approximate solutions are in high agreement with the exact solutions, when , and the solution continuously depends on the time-fractional derivative. Example 4.1 has been solved using HAM [21], ADM [27], VIM [26], and HPM [23â€“25].

*Example 4.2. *Let usconsider the fractional Riccati differential equation, we get
with initial conditions
Construction the following functional:
we have
Similarly, we can get the coefficients of to zero:
The generalized Lagrange multiplier can be identified by the above equations:
substituting (4.16) into (4.13) produces the iteration formulation as follows:
Taking the initial value , we can derive
Then, the approximate solutions in a series form are
As is
The exact solution of (4.11) is , when .

Figure 3 is plotted for approximate solution of time-fractional Riccati differential equation found in Example 4.2. In Figure 4, we have shown the graphic of approximate solution of (4.11) for . Figures 2 and 4 show that a decrease in the fractional order causes an increase in the function.

Table 2 indicates the approximate solutions for (4.11) obtained for different values of using the variational iteration method and HPM [24]. From the numerical results in Table 2, it is clear that the approximate solutions are in high agreement with the exact solutions, when , and the solution continuously depends on the time-fractional derivative.

*Example 4.3. *Let usconsider the fractional Riccati differential equation, we get
with initial conditions:
Construction the following functional:
we have
Similarly, we can get the coefficients of to zero:
The generalized Lagrange multiplier can be identified by the above equations:
substituting (4.26) into (4.23) produces the iteration formulation as follows:
Taking the initial value , we can derive
Then, the approximate solutions in a series form are
As is
The exact solution of (4.21) is
where is the Bessel function of first kind, when .

Figure 5 is plotted for approximate solution of time-fractional Riccati differential equation found in Example 4.3. In Figure 6, we have shown the graphic of approximate solution of (4.21) for . Figures 2, 4, and 6 show that a decrease in the fractional order causes an increase in the function.

Table 3 indicates the approximate solutions for (4.21) obtained for different values of using the HPM [23]. From the numerical results in Table 3, it is clear that the approximate solutions are in substantial agreement with the exact solutions, when , and the solution continuously depends on the time-fractional derivative.

#### 5. Conclusions

In this paper, variational iteration method having integral w.r.t. has been successfully implemented to finding approximate analytical solution of fractional Riccati differential equations. Variational iteration method known as very powerful and an effective method for solving fractional Riccati differential equation. It is also a promising method to solve other nonlinear equations. In this paper, we have discussed modified variational iteration method having integral w.r.t. used for the first time by Jumarie. The obtained results indicate that this method is powerful and meaningful for solving the nonlinear fractional differential equations. Three examples indicate that the results of variational iteration method having integral w.r.t. are in excellent agreement with those obtained by HPM, ADM, and HAM, which is available in the literature.

#### References

- K. B. Oldham and J. Spanier,
*The Fractional Calculus*, Academic Press, New York, NY, USA, 1974. - I. Podlubny,
*Fractional Differential Equations*, Academic Press, New York, NY, USA, 1999. - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999. - M. Caputo, â€śLinear models of dissipation whose Q is almost frequency independent, part II,â€ť
*Geophysical Journal International*, vol. 13, no. 5, pp. 529â€“539, 1967. View at Publisher Â· View at Google Scholar - A. A. Kilbas, H. H. Srivastava, and J. J. Trujillo,
*Theoryand Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. - 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. - G. M. Zaslavsky,
*Hamiltonian Chaosand Fractional Dynamics*, Oxford University Press, 2005. - M. Merdan, A. Yıldırım, and A. Gökdoğan, â€śNumerical solution of time-fraction Modified Equal Width Wave Equation,â€ť
*Engineering Computations*. In press. - G. A. Einicke, L. B. White, and R. R. Bitmead, â€śThe use of fake algebraic Riccati equations for co-channel demodulation,â€ť
*IEEE Transactions on Signal Processing*, vol. 51, no. 9, pp. 2288â€“2293, 2003. View at Publisher Â· View at Google Scholar Â· View at Scopus - M. Gerber, B. Hasselblatt, and D. Keesing, â€śThe riccati equation: pinching of forcing and solutions,â€ť
*Experimental Mathematics*, vol. 12, no. 2, pp. 129â€“134, 2003. View at Google Scholar Â· View at Scopus - R. E. Kalman, Y. C. Ho, and K. S. Narendra, â€śControllability of linear dynamical systems,â€ť
*Contributions to Differential Equations*, vol. 1, pp. 189â€“213, 1963. View at Google Scholar - S. Bittanti, P. Colaneri, and G. De Nicolao, â€śThe periodic Riccati equation,â€ť in
*The Riccati Equation*, Communications and Control Engineering, pp. 127â€“162, Springer, Berlin, Germany, 1991. View at Google Scholar - S. Bittanti, P. Colaneri, and G. O. Guardabassi, â€śPeriodic solutions of periodic Riccati equations,â€ť
*IEEE Transactions on Automatic Control*, vol. 29, no. 7, pp. 665â€“667, 1984. View at Google Scholar Â· View at Scopus - B. D. O. Anderson and J. B. Moore,
*Optimal Filtering*, Prentice-Hall, Englewood Cliffs, NJ, USA, 1979. - W. T. Reid,
*Riccati Differential Equations: Mathematics in Science and Engineering*, vol. 86, Academic Press, New York, NY, USA, 1972. - H. Aminikhah and M. Hemmatnezhad, â€śAn efficient method for quadratic Riccati differential equation,â€ť
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 4, pp. 835â€“839, 2010. View at Publisher Â· View at Google Scholar Â· View at Scopus - S. Abbasbandy, â€śHomotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method,â€ť
*Applied Mathematics and Computation*, vol. 172, no. 1, pp. 485â€“490, 2006. View at Publisher Â· View at Google Scholar Â· View at Scopus - S. Abbasbandy, â€śIterated He's homotopy perturbation method for quadratic Riccati differential equation,â€ť
*Applied Mathematics and Computation*, vol. 175, no. 1, pp. 581â€“589, 2006. View at Publisher Â· View at Google Scholar Â· View at Scopus - Y. Tan and S. Abbasbandy, â€śHomotopy analysis method for quadratic Riccati differential equation,â€ť
*Communications in Nonlinear Science and Numerical Simulation*, vol. 13, no. 3, pp. 539â€“546, 2008. View at Publisher Â· View at Google Scholar Â· View at Scopus - S. Abbasbandy, â€śA new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials,â€ť
*Journal of Computational and Applied Mathematics*, vol. 207, no. 1, pp. 59â€“63, 2007. View at Publisher Â· View at Google Scholar Â· View at Scopus - N. A. Khan, A. Ara, and M. Jamil, â€śAn efficient approach for solving the Riccati equation with fractional orders,â€ť
*Computers and Mathematics with Applications*, vol. 61, no. 9, pp. 2683â€“2689, 2011. View at Publisher Â· View at Google Scholar Â· View at Scopus - Z. Odibat and S. Momani, â€śModified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,â€ť
*Chaos, Solitons and Fractals*, vol. 36, no. 1, pp. 167â€“174, 2008. View at Publisher Â· View at Google Scholar Â· View at Scopus - J. Cang, Y. Tan, H. Xu, and S. J. Liao, â€śSeries solutions of non-linear Riccati differential equations with fractional order,â€ť
*Chaos, Solitons and Fractals*, vol. 40, no. 1, pp. 1â€“9, 2009. View at Publisher Â· View at Google Scholar Â· View at Scopus - H. Jafari and H. Tajadodi, â€śHe’s variational iteration method for solving fractional Riccati differential equation,â€ť
*International Journal of Differential Equations*, vol. 2010, Article ID 764738, 8 pages, 2010. View at Publisher Â· View at Google Scholar - S. Momani and N. Shawagfeh, â€śDecomposition method for solving fractional Riccati differential equations,â€ť
*Applied Mathematics and Computation*, vol. 182, no. 2, pp. 1083â€“1092, 2006. View at Publisher Â· View at Google Scholar Â· View at Scopus - J. H. He, â€śVariational iteration method—a kind of non-linear analytical technique: some examples,â€ť
*International Journal of Non-Linear Mechanics*, vol. 34, no. 4, pp. 699â€“708, 1999. View at Google Scholar Â· View at Scopus - J. H. He and X. H. Wu, â€śVariational iteration method: new development and applications,â€ť
*Computers and Mathematics with Applications*, vol. 54, no. 7-8, pp. 881â€“894, 2007. View at Publisher Â· View at Google Scholar Â· View at Scopus - J. H. He, â€śSome applications of nonlinear fractional differential equations and their approximations,â€ť
*Bulletin of Science, Technology & Society*, vol. 15, no. 2, pp. 86â€“90, 1999. View at Google Scholar Â· View at Scopus - G. Jumarie, â€śStochastic differential equations with fractional Brownian motion input,â€ť
*International Journal of Systems Science*, vol. 24, no. 6, pp. 1113â€“1131, 1993. View at Publisher Â· View at Google Scholar - G. Jumarie, â€śNew stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations,â€ť
*Mathematical and Computer Modelling*, vol. 44, no. 3-4, pp. 231â€“254, 2006. View at Publisher Â· View at Google Scholar Â· View at Scopus - G. Jumarie, â€śLaplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative,â€ť
*Applied Mathematics Letters*, vol. 22, no. 11, pp. 1659â€“1664, 2009. View at Publisher Â· View at Google Scholar Â· View at Scopus - G. Jumarie, â€śTable of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,â€ť
*Applied Mathematics Letters*, vol. 22, no. 3, pp. 378â€“385, 2009. View at Publisher Â· View at Google Scholar Â· View at Scopus - G. Jumarie, â€śOn the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion,â€ť
*Applied Mathematics Letters*, vol. 18, no. 7, pp. 817â€“826, 2005. View at Publisher Â· View at Google Scholar Â· View at Scopus - M.-J. Jang, C.-L. Chen, and Y.-C. Liu, â€śTwo-dimensional differential transform for partial differential equations,â€ť
*Applied Mathematics and Computation*, vol. 121, no. 2-3, pp. 261â€“270, 2001. View at Publisher Â· View at Google Scholar Â· View at Scopus - N. Faraz, Y. Khan, H. Jafari, A. Yildirim, and M. Madani, â€śFractional variational iteration method via modified Riemann-Liouville derivative,â€ť
*Journal of King Saud University*, vol. 23, no. 4, pp. 413â€“417, 2010. View at Publisher Â· View at Google Scholar Â· View at Scopus