# Fractional-order Riccati differential equation: Analytical approximation and numerical results

- Najeeb Alam Khan
^{1}Email author, - Asmat Ara
^{2}and - Nadeem Alam Khan
^{1}

**2013**:185

https://doi.org/10.1186/1687-1847-2013-185

© Khan et al.; licensee Springer 2013

**Received: **8 January 2013

**Accepted: **29 May 2013

**Published: **26 June 2013

## Abstract

The aim of this article is to introduce the Laplace-Adomian-Padé method (LAPM) to the Riccati differential equation of fractional order. This method presents accurate and reliable results and has a great perfection in the Adomian decomposition method (ADM) truncated series solution which diverges promptly as the applicable domain increases. The approximate solutions are obtained in a broad range of the problem domain and are compared with the generalized Euler method (GEM). The comparison shows a precise agreement between the results, the applicable one of which needs fewer computations.

## Keywords

## 1 Introduction

In recent years, it has turned out that many phenomena in biology, chemistry, acoustics, control theory, psychology and other areas of science can be fruitfully modeled by the use of fractional-order derivatives. That is because of the fact that a reasonable modeling of a physical phenomenon having dependence not only on the time instant but also on the prior time history can be successfully achieved by using fractional calculus [1]. Fractional differential equations (FDEs) have been used as a kind of model to describe several physical phenomena [2–6] such as damping laws, rheology, diffusion processes, and so on. Moreover, some researchers have shown the advantageous use of the fractional calculus in the modeling and control of many dynamical systems. Besides modeling, finding accurate and proficient methods for solving FDEs has been an active research undertaking. Exact solutions for the majority of FDEs cannot be found easily, thus analytical and numerical methods must be used. Some numerical methods for solving FDEs have been presented and they have their own advantages and limitations.

Many physical problems are governed by fractional differential equations (FDEs), and finding the solution of these equations have been the subject of many investigations in recent years. Recently, there have been a number of schemes devoted to the solution of fractional differential equations. These schemes can be broadly classified into two classes, numerical and analytical. The Adomian decomposition method [7], homotopy perturbation method [8–11], homotopy analysis method [12, 13], Taylor matrix method [14] and Haar wavelet method [15] have been used to solve the fractional-order Riccati differential equation. However, the convergence region of the corresponding results is rather small.

In this work, the nonlinear fractional-order Riccati differential equations will be approached analytically by combining the Laplace transform, the Adomian decomposition method (ADM), and the Padé approximation. The Laplace-Adomian-Padé approximation was proposed by Tsai and Chen [16] for solving Ricatti differential equations. The method was extended by Zeng *et al*. [17] to derive the analytical approximate solutions of fractional differential equations. Khan *et al*. [18] applied the Laplace transformation coupled with the decomposition method in fractional order seepage flow and telegraph equations. We applied the idea of refs. [16, 17] for solving a fractional-order Riccati differential equation. The Laplace-Adomian-Padé method (LAPM) is illustrated by applications, and the results obtained are compared with those of the exact and numerical solutions by the generalized Euler method. Odibat and Momani [19] derived the generalized Euler method that was developed for the numerical solution of initial value problems with Caputo derivatives.

## 2 Definitions and preliminaries

### Caputo’s fractional derivative

*k*th derivative is given by [20]

## 3 Implementation of LAPM

The aim is to study the mathematical behavior of the solution $y(t)$ for different values of *α*. By applying the inverse Laplace transform to both sides of Eq. (19), the value of ${y}_{0}$ is obtained. Substituting these values of ${y}_{0}$ and ${A}_{0}$ into Eq. (20), the first component ${y}_{1}$ is obtained. The other terms ${y}_{2},{y}_{3},{y}_{4},\dots $ . can be calculated recursively in a similar way by Eqs. (20)-(22). The LAPM solution coincides with the Taylor series solution in the initial value case and diverges rapidly as the applicable domain increases. This goal can be achieved by forming Padé approximants, which have the advantage of manipulating the polynomial approximation into a rational function to gain more information about $y(t)$. It is well known that Padé approximants will converge on the entire real axis, if $y(t)$ is free of singularities on the real axis. To consider the behaviors of a solution for different values of *α*, we will take advantage of Eq. (15) available for $0<\alpha \le 1$.

## 4 Test problems

In this section, we implement LAPM to the nonlinear fractional Riccati differential equations. Two examples of nonlinear fractional Riccati differential equations are solved with real coefficients.

*Test problem 1.*Consider the nonlinear Riccati differential equation

The aim is to study the mathematical behavior of the result as the order of the fractional derivative changes. It was formally shown by Khan *et al*. [21] that this goal can be achieved by forming Padé approximants [22] which have the advantage of manipulating the polynomial approximation into a rational function to gain more information about $y(t)$. To consider the behavior of a solution of different values of *α*, we will take advantage of Eq. (40) available for $0<\alpha \le 1$ and consider the following three special cases.

*Case I:*Setting $\alpha =1$ in Eq. (40), we reproduce the approximate solution obtained in Eq. (40), given by the Taylor expansion of $y(t)$ at $t=0$ of the LAPM solution, as follows:

**Numerical results of the Riccati equation in problem 1**

t | GEM α = 1 | LAPM α = 1 | Exact solution | Absolute error |
---|---|---|---|---|

y(t) | y(t) | y(t) | ||

0.1 | 0.1000000000 | 0.1102952044 | 0.1102951969 | 7.5 × 10 |

0.2 | 0.2419000000 | 0.2419783394 | 0.2419767996 | 1.5 × 10 |

0.3 | 0.3580039000 | 0.3951442714 | 0.3951048487 | 0.00003942275 |

0.4 | 0.5167880007 | 0.5682377001 | 0.5678121663 | 0.00042553377 |

0.5 | 0.6934386 | 0.7588607194 | 0.7580143934 | 0.00084632599 |

*Case II:*Let us examine the case $\alpha =\frac{1}{2}$, the approximate solution obtained in Eq. (40) given by the Taylor expansion of $y(t)$ at $t=0$ has reproduced as

*Case III:*Here, taking $\alpha =\frac{3}{4}$ in Eq. (40), the approximate solution has been replicated by

**Numerical results of the Riccati equation in problem 1 for**
$\mathit{\alpha}\mathbf{=}\mathbf{1}\mathbf{,}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\frac{\mathbf{3}}{\mathbf{4}}$

t | $\mathit{\alpha}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$ GEM | $\mathit{\alpha}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$ LAPM | $\mathit{\alpha}\mathbf{=}\frac{\mathbf{3}}{\mathbf{4}}$ GEM | $\mathit{\alpha}\mathbf{=}\frac{\mathbf{3}}{\mathbf{4}}$ LAPM | α = 1 LAPM |
---|---|---|---|---|---|

0.1 | 0.3568251903 | 0.3568031433 | 0.1934884034 | 0.1934012434 | 0.1102952044 |

0.2 | 0.9228652311 | 0.9228654512 | 0.4546091238 | 0.4546025138 | 0.2419783394 |

0.3 | 1.6341391963 | 1.6341391234 | 0.7840321022 | 0.7840324522 | 0.3951442714 |

0.4 | 2.2044414876 | 2.2044414576 | 1.1619801122 | 1.1619856232 | 0.5682377001 |

0.5 | 2.4004512311 | 2.4004476111 | 1.5438814841 | 1.5438814521 | 0.7588607194 |

0.6 | 2.0414276521 | 2.0414345521 | 1.8736212813 | 1.873658343 | 0.8840411201 |

0.7 | 2.4142176521 | 2.414888821 | 2.1129313512 | 2.112943562 | 1.0827124311 |

0.8 | 2.4142456641 | 2.4142478941 | 2.2602500123 | 2.260134223 | 1.2820124311 |

0.9 | 2.4142456047 | 2.4142455667 | 2.339920199 | 2.339134229 | 1.4740612089 |

1 | 2.4142410607 | 2.4142312137 | 2.3795146712 | 2.37935612 | 1. 6515902374 |

*Test problem 2.*Consider the nonlinear Riccati differential equation

The plan is to study the mathematical performance of the solution of LAPM as the order of the fractional derivative changes. To consider the behavior of a solution of different values of *α*, we will take advantage of the explicit formula Eq. (69) available for $0<\alpha \le 1$ and consider the following three special cases.

*Case I:*Setting $\alpha =1$ in Eq. (69), we reproduce the approximate solution obtained in Eq. (69) given by the Taylor expansion of $y(t)$ at $t=0$ of the LAPM solution as follows:

**Comparison results of the Riccati equation in problem 2 for**
$\mathit{\alpha}\mathbf{=}\mathbf{1}$

t | LAPM α = 1 | Exact solution | Absolute error |
---|---|---|---|

y(t) | y(t) | ||

1.0 | 0.7615941560 | 0.7615941560 | 0.01235728510 × 10 |

2.0 | 0.9640275801 | 0.9640275801 | 0.00524212251 × 10 |

3.0 | 0.9950547537 | 0.9950547537 | 0.00816114713 × 10 |

4.0 | 0.9993292997 | 0.9993292997 | 1.15746178216 × 10 |

5.0 | 0.9999092043 | 0.9999092043 | 6.17826138530 × 10 |

6.0 | 0.9999877117 | 0.9999877117 | 4.55279023510 × 10 |

7.0 | 0.9999983377 | 0.9999983369 | 7.57246128510 × 10 |

8.0 | 0.9999997813 | 0.9999997749 | 6.33977368904 × 10 |

9.0 | 1.0000000063 | 0.9999999695 | 3.67962922016 × 10 |

10.0 | 1.0000001574 | 0.9999999958 | 1.61485463823 × 10 |

15.0 | 1.000000000 | 1.0000000000 | 0.00002076502676 |

20.0 | 1.0000000000 | 1.0000000000 | 0.00030117782151 |

25.0 | 1.0000000000 | 1.0000000000 | 0.00161702848408 |

30.0 | 1.0000000000 | 1.0000000000 | 0.00510751644687 |

*Case II:*Here we examine the case $\alpha =\frac{1}{2}$ in Eq. (69), we replicate the approximate solution obtained in Eq. (69) given by

*Case III:*In this case we examine the LAPM when $\alpha =\frac{3}{4}$ in Eq. (69)

**Numerical results of the Riccati equation of problem 2 for**
$\mathit{\alpha}\mathbf{=}\mathbf{1}\mathbf{,}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\frac{\mathbf{3}}{\mathbf{4}}$

t | $\mathit{\alpha}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$ GEM | $\mathit{\alpha}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$ LAPM | $\mathit{\alpha}\mathbf{=}\frac{\mathbf{3}}{\mathbf{4}}$ GEM | $\mathit{\alpha}\mathbf{=}\frac{\mathbf{3}}{\mathbf{4}}$ LAPM | α = 1 LAPM |
---|---|---|---|---|---|

1 | 1.1283791670 | 0.69873925716 | 1.08806525252 | 0.73683666979 | 0.7615941560 |

2 | 0.8200613571 | 0.78565566383 | 0.88785292084 | 0.87018299629 | 0.9640275801 |

3 | 1.1896048240 | 0.82585713776 | 1.11809163866 | 0.91495001137 | 0.9950547537 |

4 | 0.7211473382 | 0.85006608584 | 0.84593506046 | 0.93590393958 | 0.9993292997 |

5 | 1.2627089888 | 0.86673312218 | 1.15537415211 | 0.94734255879 | 0.9999092043 |

6 | 0.5919620577 | 0.87923035124 | 0.79099259853 | 0.95361548723 | 0.9999877117 |

7 | 1.3249356376 | 0.88919590450 | 1.19828883587 | 0.95638359853 | 0.9999983377 |

8 | 0.4724965813 | 0.89752861875 | 0.72400539873 | 0.95643868318 | 0.9999997813 |

9 | 1.3489616923 | 0.90476369883 | 1.24172445342 | 0.95425989627 | 1.0000000063 |

10 | 0.4240319436 | 0.91123881947 | 0.65212406998 | 0.95020154024 | 1.0000001574 |

## 5 Conclusions

Most of the real physical problems can be best modeled with fractional differential equations. Besides modeling, the solution techniques and their reliabilities are most important to catch critical points at which a sudden divergence or bifurcation starts. Therefore, high accuracy solutions are always needed. Here, we have implemented the Adomian decomposition method coupled with the Laplace transformation and the Padé approximation on the Ricatti differential equation with fractional order. From the test problems considered here, it can be easily seen that LAPM obtains results as accurate as possible. Thus, it can be concluded that the LAPM methodology is very dominant and efficient in finding approximate solutions, and comparison has been made with GEM. This paper can be used as a standard paradigm for other applications. The results of LAPM have been compared with exact solutions and ref. [16] for $\alpha =1$.

## Declarations

### Acknowledgements

The authors would like to express their sincere gratitude to the referees for their careful assessment and suggestions regarding the initial version of the manuscript. The author Najeeb Alam Khan is highly thankful and grateful to the Dean of Faculty of Sciences, University of Karachi, Karachi-75270, Pakistan for facilitating this research work.

## Authors’ Affiliations

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