- Research Article
- Open Access
- Published:

# On Efficient Method for System of Fractional Differential Equations

*Advances in Difference Equations*
**volume 2011**, Article number: 303472 (2011)

## Abstract

The present study introduces a new version of homotopy perturbation method for the solution of system of fractional-order differential equations. In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the nonlinear problem. The systems include fractional-order stiff system, the fractional-order Genesio system, and the fractional-order matrix Riccati-type differential equation. The new approximate analytical procedure depends only on two components. Comparing the methodology with some known techniques shows that the present method is relatively easy, less computational, and highly accurate.

## 1. Introduction

Fractional differential equations have received considerable interest in recent years and have been extensively investigated and applied for many real problems which are modeled in different areas. One possible explanation of such unpopularity could be that there are multiple nonequivalent definitions of fractional derivatives [1]. Another difficulty is that fractional derivatives have no evident geometrical interpretation because of their nonlocal character. However, during the last 12 years fractional calculus starts to attract much more attention of scientists. It was found that various, especially interdisciplinary, applications [2–6] can be elegantly modeled with the help of the fractional derivatives.

The homotopy perturbation method is a powerful devise for solving nonlinear problems. This method was introduced by He [7–9] in the year 1998. In this method, the solution is considered as the summation of an infinite series that converges rapidly. This technique is used for solving nonlinear chemical engineering equations [10], time-fractional Swift-Hohenberg (S-H) equation [11], viscous fluid flow equation [12], Fourth-Order Integro-Differential equations [13], nonlinear dispersive equations [14], Long Porous Slider equation [15], and Navier-Stokes equations [16]. It can be said that He's homotopy perturbation method is a universal one, which is able to solve various kinds of nonlinear equations. The new homotopy perturbation method (NHPM) was applied to linear and nonlinear ODEs [17].

In this paper, we construct the solution of system of fractional-order differential equations by extending the idea of [17, 18]. This method leads to computable and efficient solutions to linear and nonlinear operator equations. The corresponding solutions of the integer-order equations are found to follow as special cases of those of fractional-order equations.

We consider the system of fractional-order equations of the form

## 2. Basic Definitions

We give some basic definitions, notations, and properties of the fractional calculus theory used in this work.

Definition 2.1.

The Riemann-Liouville fractional integral operator of order on the usual Lebesgue space is given by

It has the following properties:

(i) exists for any ,

(ii),

(iii),

(iv),

(v),

where , and .

Definition 2.2.

The Caputo definition of fractal derivative operator is given by

where . It has the following two basic properties for and :

## 3. Analysis of New Homotopy Perturbation Method

Let us consider the system of nonlinear differential equations

where are the operators, are known functions and are sought functions. Assume that operators can be written as

where are the linear operators and are the nonlinear operators. Hence, (3.1) can be rewritten as follows:

We define the operators as

where is an embedding or homotopy parameter, and are the initial approximation of solution of the problem in (3.3) can be written as

Clearly, the operator equations and are equivalent to the equations and , respectively. Thus, a monotonous change of parameter from zero to one corresponds to a continuous change of the trivial problem to the original problem. Operator is called a homotopy map. Next, we assume that the solution of equation can be written as a power series in embedding parameter , as follows:

Now, let us write (3.5) in the following form:

By applying the inverse operator, to both sides of (3.7), we have

Suppose that the initial approximation of (3.3) has the form

where , are unknown coefficients and , are specific functions on the problem. By substituting (3.6) and (3.9) into (3.8), we get

Equating the coefficients of like powers of , we get the following set of equations:

Now, we solve these equations in such a way that . Therefore, the approximate solution may be obtained as

## 4. Applications

Application 1

Consider the following linear fractional-order 2-by-2 stiff system:

with the initial conditions

where and are constants. To obtain the solution of (4.1) by NHPM, we construct the following homotopy:

Applying the inverse operator, of both sides of the above equation, we obtain

The solution of (4.1) to has the following form:

Substituting (4.5) in (4.4) and equating the coefficients of like powers of , we get the following set of equations:

Assuming , , , , and and solving the above equation for and lead to the result

Vanishing and lets the coefficients by taking the following values:

Therefore, we obtain the solutions of (4.1) as

Our aim is to study the mathematical behavior of the solution and for different values of . This goal can be achieved by forming Pade' approximants, which have the advantage of manipulating the polynomial approximation into a rational function to gain more information about and . It is well known that Pade' approximants will converge on the entire real axis, if and are free of singularities on the real axis. It is of interest to note that Pade' approximants give results with no greater error bounds than approximation by polynomials. To consider the behavior of solution for different values of , we will take advantage of the explicit formula (4.9) available for and consider the following two special cases.

Case 1.

Setting in (4.9), we obtain the approximate solution in a series form as

Case 2.

In this case, we will examine the linear fractional stiff equation (4.1). Setting in (4.9) gives

For simplicity, let , then

Calculating the [10/11] Pade' approximants and recalling that , we get

Application 2

Consider the following nonlinear fractional-order 2-by-2 stiff system:

with the initial conditions

To obtain the solution of (4.14) by NHPM, we construct the following homotopy:

Applying the inverse operator, of both sides of the above equation, we obtain

The solution of (4.14) to have the following form:

Substituting (4.18) in (4.17) and equating the coefficients of like powers of , we get the following set of equations:

Assuming , , , , and and solving the above equation for and lead to the result

Vanishing and lets the coefficients to take the following values:

Therefore, we obtain the solution of (4.14) as

The exact solution of (4.14) for is .

Application 3

Consider the following nonlinear Genesio system with fractional derivative:

with the initial conditions

where , , and are constants. To obtain the solution of (4.23) by NHPM, we construct the following homotopy:

Applying the inverse operator, of both sides of the above equation, we obtain

The solution of (4.23) to have the following form:

Substituting (4.27) in (4.26) and equating the coefficients of like powers of , we get the following set of equations:

Assuming , , , , , , , , and , and solving the above equation for and lead to the result

Vanishing , and lets the coefficients to take the following values:

Therefore, we obtain the solutions of (4.23) as

Application 4

Finally, we consider the following nonlinear matrix Riccati differential equation with fractional derivative:

where . To find the solution of this equation by NHPM, we will treat the matrix equation as a system of fractional-order differential equations

with the initial conditions

Therefore, we obtain the solution of (4.33) as

## 5. Concluding Remarks

The NHPM for solving system of fractional-order differential equations are based on two component procedure and polynomial initial condition. The NHPM applied on fractional-order Stiff equation, fractional Genesio equation, and the matrix Riccati-type differential equation. The Applications in problems 1–4 are plotted in Figures 1, 2, 3, and 4, which show the accuracy of NHPM. The computations associated with the applications discussed above, were performed by MATHEMATICA. The NHPM is very simple in application and is less computational more accurate in comparison with other mentioned methods. By using this method, the solution can be obtained in bigger interval. Unlike the ADM [19], the NHPM is free from the need to use Adomian polynomials. In this method, we do not need the Lagrange multiplier, correction functional, stationary conditions, and calculating integrals, which eliminate the complications that exist in the VIM [20]. In contrast to the HPM and HAM, in this method, it is not required to solve the functional equations in each iteration. The efficiency of HAM is very much depended on choosing auxiliary parameter . All the applications are taken from [20] with fractional derivatives.

## References

- [1]
Kilbas HM, Srivastava HM, Trujillo JJ:

*Theory and Applications of Fractional Differential Equations*. Elsevier, Amsterdam, The Netherlands; 2007. - [2]
Bagley RL, Calico RA:

**Fractional order state equations for the control of viscoelastically damped structures.***Journal of Guidance, Control, and Dynamics*1991,**14**(2):304–311. 10.2514/3.20641 - [3]
Mahmood A, Parveen S, Ara A, Khan NA:

**Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model.***Communications in Nonlinear Science and Numerical Simulation*2009,**14**(8):3309–3319. 10.1016/j.cnsns.2009.01.017 - [4]
Mahmood A, Fetecau C, Khan NA, Jamil M:

**Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders.***Acta Mechanica Sinica*2010,**26**(4):541–550. 10.1007/s10409-010-0353-4 - [5]
He JH:

**Nonlinear oscillation with fractional derivative and its applications.***Proceedings of the International Conference on Vibrating Engineering, 1998, Dalian, China*288–291. - [6]
Khan NA, Khan N-U, Ara A, Jamil M:

**Approximate analytical solutions of fractional reaction-diffusion equations.***Journal of King Saud University—Science*. In press - [7]
He J-H:

**Homotopy perturbation technique.***Computer Methods in Applied Mechanics and Engineering*1999,**178**(3–4):257–262. 10.1016/S0045-7825(99)00018-3 - [8]
He J-H:

**A coupling method of a homotopy technique and a perturbation technique for non-linear problems.***International Journal of Non-Linear Mechanics*2000,**35**(1):37–43. 10.1016/S0020-7462(98)00085-7 - [9]
He J-H:

**Homotopy perturbation method: a new nonlinear analytical technique.***Applied Mathematics and Computation*2003,**135**(1):73–79. 10.1016/S0096-3003(01)00312-5 - [10]
Khan NA, Ara A, Mahmood A:

**Approximate solution of time-fractional chemical engineering equations: a comparative study.***International Journal of Chemical Reactor Engineering*2010.,**8, article A19:** - [11]
Khan NA, Khan N-U, Ayaz M, Mahmood A:

**Analytical methods for solving the time-fractional Swift-Hohenberg (S-H) equation.***Computers and Mathematics with Applications*. In press - [12]
Khan NA, Ara A, Ali SA, Jamil M:

**Orthognal flow impinging on a wall with suction or blowing.***International Journal of Chemical Reactor Engineering*. In press - [13]
Yıldırım A:

**Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method.***Computers & Mathematics with Applications*2008,**56**(12):3175–3180. 10.1016/j.camwa.2008.07.020 - [14]
Koçak H, Öziş T, Yıldırım A:

**Homotopy perturbation method for the nonlinear dispersive K(m,n,1) equations with fractional time derivatives.***International Journal of Numerical Methods for Heat & Fluid Flow*2010,**20**(2):174–185. 10.1108/09615531011016948 - [15]
Khan Y, Faraz N, Yildirim A, Wu Q:

**A series solution of the long porous slider.***Tribology Transactions*2011,**54**(2):187–191. 10.1080/10402004.2010.533818 - [16]
Khan NA, Ara A, Ali SA, Mahmood A:

**Analytical study of Navier-Stokes equation with fractional orders using He's homotopy perturbation and variational iteration methods.***International Journal of Nonlinear Sciences and Numerical Simulation*2009,**10**(9):1127–1134. 10.1515/IJNSNS.2009.10.9.1127 - [17]
Aminikhah H, Biazar J:

**A new HPM for ordinary differential equations.***Numerical Methods for Partial Differential Equations*2010,**26**(2):480–489. - [18]
Aminikhah H, Hemmatnezhad M:

**An efficient method for quadratic Riccati differential equation.***Communications in Nonlinear Science and Numerical Simulation*2010,**15**(4):835–839. 10.1016/j.cnsns.2009.05.009 - [19]
Khan Y, Faraz N:

**Modified fractional decomposition method having integral w.r.t .***Journal of King Saud University—Science*. In press - [20]
Bataineh AS, Noorani MSM, Hashim I:

**Solving systems of ODEs by homotopy analysis method.***Communications in Nonlinear Science and Numerical Simulation*2008,**13**(10):2060–2070. 10.1016/j.cnsns.2007.05.026

## Acknowledgment

M. Jamil is highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan, the Department of Mathematics & Basic Sciences, NED University of Engineering & Technology, Karachi-75270, Pakistan, and also the Higher Education Commission of Pakistan for generously supporting and facilitating this research work.

## Author information

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Khan, N.A., Jamil, M., Ara, A. *et al.* On Efficient Method for System of Fractional Differential Equations.
*Adv Differ Equ* **2011, **303472 (2011). https://doi.org/10.1155/2011/303472

Received:

Accepted:

Published:

### Keywords

- Fractional Derivative
- Fractional Calculus
- Fractional Differential Equation
- Inverse Operator
- Homotopy Perturbation Method