- Research Article
- Open Access

# On Efficient Method for System of Fractional Differential Equations

- Najeeb Alam Khan
^{1}Email author, - Muhammad Jamil
^{2, 3}, - Asmat Ara
^{1}and - Nasir-Uddin Khan
^{1}

**2011**:303472

https://doi.org/10.1155/2011/303472

© Najeeb Alam Khan et al. 2011

**Received:**14 December 2010**Accepted:**5 February 2011**Published:**9 March 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.

## Keywords

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

## 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.

## 2. Basic Definitions

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

Definition 2.1.

It has the following properties:

(i) exists for any ,

(ii) ,

(iii) ,

(iv) ,

(v) ,

where , and .

Definition 2.2.

## 3. Analysis of New Homotopy Perturbation Method

## 4. Applications

Application 1

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.

Case 2.

Application 2

The exact solution of (4.14) for is .

Application 3

Application 4

## 5. Concluding Remarks

## Declarations

### 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.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.