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This article is part of the series Progress in Functional Differential and Difference Equations.

Open Access Research

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

Najeeb Alam Khan1*, Asmat Ara2 and Nadeem Alam Khan1

Author Affiliations

1 Department of Mathematical Sciences, University of Karachi, Karachi, 75270, Pakistan

2 Department of Mathematical Sciences, Federal Urdu University Arts, Science and Technology, Karachi, 75300, Pakistan

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Advances in Difference Equations 2013, 2013:185  doi:10.1186/1687-1847-2013-185


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/185


Received:8 January 2013
Accepted:29 May 2013
Published:26 June 2013

© 2013 Khan et al.; licensee Springer

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

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:
Adomian decomposition method (ADM); Mittag-Leffler function; Padé approximation; Riccati equation

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

Caputo’s fractional derivative of a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M1">View MathML</a> is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M2">View MathML</a>

(1)

The Laplace transform to Caputo’s fractional derivative gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M3">View MathML</a>

(2)

The Mittag-Leffler function and its generalized forms have played a special role in solving the fractional differential equations. The so-called Mittag-Leffler function with two parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M4">View MathML</a> was introduced by Agarwal [20]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M5">View MathML</a>

(3)

Its kth derivative is given by [20]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M6">View MathML</a>

(4)

We find it convenient to introduce the function

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M7">View MathML</a>

(5)

Its Laplace transform was evaluated by Podlubny [4]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M8">View MathML</a>

(6)

Hence

(7)

Another convenient property of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M10">View MathML</a>, which has been used in this paper, is its simple fractional differentiation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M11">View MathML</a>

(8)

3 Implementation of LAPM

Consider the fractional-order Riccati differential equation of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M12">View MathML</a>

(9)

subject to the initial condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M13">View MathML</a>

(10)

The nonlinear term in Eq. (9) is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M14">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M15">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M16">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M17">View MathML</a> are known functions. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18">View MathML</a>, the fractional-order Riccati equation converts into the classical Riccati differential equation. Applying the Laplace transform on both sides of Eq. (9),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M19">View MathML</a>

(11)

Using the property of the Laplace transform, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M20">View MathML</a>

(12)

Using the initial condition from Eq. (10), the outcome is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M21">View MathML</a>

(13)

Equation (13) can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M22">View MathML</a>

(14)

The method assumes the solution as an infinite series:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M23">View MathML</a>

(15)

The nonlinearity <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M14">View MathML</a> is decomposed as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M25">View MathML</a>

(16)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M26">View MathML</a> are the so-called Adomian polynomials given as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M27">View MathML</a>

(17)

Substituting Eqs. (15) and (17) into Eq. (14), the result is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M28">View MathML</a>

(18)

Matching both sides of Eq. (18) yields the following iterative algorithm:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M29">View MathML</a>

(19)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M30">View MathML</a>

(20)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M31">View MathML</a>

(21)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M32">View MathML</a>

(22)

The aim is to study the mathematical behavior of the solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33">View MathML</a> for different values of α. By applying the inverse Laplace transform to both sides of Eq. (19), the value of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34">View MathML</a> is obtained. Substituting these values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36">View MathML</a> into Eq. (20), the first component <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M37">View MathML</a> is obtained. The other terms <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M38">View MathML</a> . 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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M39">View MathML</a>. It is well known that Padé approximants will converge on the entire real axis, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33">View MathML</a> 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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M41">View MathML</a>.

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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M42">View MathML</a>

(23)

with the initial condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M43">View MathML</a>

(24)

The exact solution for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18">View MathML</a> was found to be

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M45">View MathML</a>

(25)

First, applying the Laplace transform on both sides of Eq. (23), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M46">View MathML</a>

(26)

Using the property of the Laplace transform, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M47">View MathML</a>

(27)

Using the initial condition from Eq. (24), it becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M48">View MathML</a>

(28)

Substituting Eqs. (15) and (16) into Eq. (28), the result is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M49">View MathML</a>

(29)

Matching both sides of Eq. (29) yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M50">View MathML</a>

(30)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M51">View MathML</a>

(31)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M52">View MathML</a>

(32)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M53">View MathML</a>

(33)

Applying the inverse fractional Laplace transform to Eq. (30), hence we can write it as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M54">View MathML</a>

(34)

By applying the inverse Laplace transform to Eq. (34), the value <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34">View MathML</a> is obtained as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M56">View MathML</a>

(35)

Now, considering the few terms of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M58">View MathML</a>

(36)

The first Adomian polynomial <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36">View MathML</a> is obtained from Eq. (17), then we substitute <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36">View MathML</a> in Eq. (31). Evaluating the Laplace transform of the quantities on the right-hand side of Eq. (31) and then applying the inverse Laplace transform, the value of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M37">View MathML</a> can be obtained. The other terms <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M63">View MathML</a> can be computed recursively in a similar calculation. By using LAPM, a power series solution is essentially a truncated series solution. The LAPM solution coincides with the Maclaurin series of the exact solution in the initial value case and diverges rapidly as the applicable domain increases. The next two components of the solution are

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M64">View MathML</a>

(37)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M65">View MathML</a>

(38)

Therefore the truncated series solution obtained from LAPM is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M66">View MathML</a>

(39)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M67">View MathML</a>

(40)

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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M39">View MathML</a>. To consider the behavior of a solution of different values of α, we will take advantage of Eq. (40) available for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M69">View MathML</a> and consider the following three special cases.

Case I: Setting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18">View MathML</a> in Eq. (40), we reproduce the approximate solution obtained in Eq. (40), given by the Taylor expansion of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33">View MathML</a> at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M72">View MathML</a> of the LAPM solution, as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M73">View MathML</a>

(41)

The Taylor expansion of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33">View MathML</a> at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M72">View MathML</a> of the exact solution (25) is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M76">View MathML</a>

(42)

It indicates that both the Taylor expansions at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M77">View MathML</a> of the LAPM solution and the exact solution coincide very well. In order to improve the LAPM solution, the Padé approximant is introduced. It is known that there exists the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M78">View MathML</a> Padé approximant which satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M79">View MathML</a>

(43)

By using Mathematica, the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M80">View MathML</a> Padé approximant gives that the rational approximation obtained from the solution in Eq. (42) is determined to be

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M81">View MathML</a>

(44)

Figures 1-2 represent the comparisons between the exact solution, the LAM and the LAPM solutions in problem 1. They show that the LAM solutions diverge rapidly after <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M82">View MathML</a>. However, they represent that the LAPM solution demonstrates a good convergence through the applicable domain. Table 1 shows the absolute errors of the LAPM solution in comparison with the exact and GEM solutions in problem 1.

thumbnailFigure 1. The approximate solutions solved by different methods in problem 1 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M83">View MathML</a>.

thumbnailFigure 2. The approximate solutions solved by different methods in problem 1 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M83">View MathML</a>.

Table 1. Numerical results of the Riccati equation in problem 1

Case II: Let us examine the case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85">View MathML</a>, the approximate solution obtained in Eq. (40) given by the Taylor expansion of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33">View MathML</a> at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M72">View MathML</a> has reproduced as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M88">View MathML</a>

(45)

For simplicity, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M89">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M90">View MathML</a>

(46)

Calculating the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M91">View MathML</a> Padé approximation and recalling that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M92">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M93">View MathML</a>

(47)

Figure 3 represents the LAPM solution in problem 1 for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85">View MathML</a>.

thumbnailFigure 3. The approximate curve of LAPM in problem 1 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M95">View MathML</a>.

Case III: Here, taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M96">View MathML</a> in Eq. (40), the approximate solution has been replicated by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M97">View MathML</a>

(48)

For simplicity, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M98">View MathML</a>; then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M99">View MathML</a>

(49)

Calculating the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M100">View MathML</a> Padé approximants and recalling that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M101">View MathML</a>, we achieve

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M102">View MathML</a>

(50)

Figure 4 shows the LAPM solution in problem 1 for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M96">View MathML</a>.

thumbnailFigure 4. The approximate curve of LAPM in problem 1 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M104">View MathML</a>.

Table 2 shows the results of the fractional Riccati equation in test problem 1 of the LAPM approximant solution in comparison with the different values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105">View MathML</a>. The technique described above was translated into a Mathematica program and run on a Pentium-4 PC to investigate the effects of various values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105">View MathML</a> on the fractional Riccati differential equation. The graphical results are in good agreement with the results of the exact solution.

Table 2. Numerical results of the Riccati equation in problem 1 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105">View MathML</a>

Test problem 2. Consider the nonlinear Riccati differential equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M112">View MathML</a>

(51)

with the initial condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M113">View MathML</a>

(52)

The exact solution [9] was found to be

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M114">View MathML</a>

(53)

First, applying the Laplace transform to both sides of Eq. (51), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M115">View MathML</a>

(54)

Using the property of the Laplace transform yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M116">View MathML</a>

(55)

Utilizing the initial conditions from Eq. (52), it becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M117">View MathML</a>

(56)

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M118">View MathML</a>

(57)

Substituting Eqs. (15) and (16) into Eq. (57), the result is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M119">View MathML</a>

(58)

Matching both sides of Eq. (58) yields the following iterative algorithm:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M120">View MathML</a>

(59)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M121">View MathML</a>

(60)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M122">View MathML</a>

(61)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M123">View MathML</a>

(62)

Applying the inverse fractional Laplace transform to Eq. (59), hence the value <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34">View MathML</a> is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M125">View MathML</a>

(63)

Substituting the value of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34">View MathML</a> in Eq. (60), the first Adomian polynomial <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36">View MathML</a> is obtained, then substituting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M34">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M36">View MathML</a> in Eq. (60) and proceeding in a similar way, the other terms <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M38">View MathML</a> . can be computed recursively. The first twelve components of the solution are

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M131">View MathML</a>

(64)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M132">View MathML</a>

(65)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M133">View MathML</a>

(66)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M134">View MathML</a>

(67)

Therefore the truncated series solution is obtained as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M135">View MathML</a>

(68)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M136">View MathML</a>

(69)

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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M69">View MathML</a> and consider the following three special cases.

Case I: Setting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18">View MathML</a> in Eq. (69), we reproduce the approximate solution obtained in Eq. (69) given by the Taylor expansion of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33">View MathML</a> at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M72">View MathML</a> of the LAPM solution as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M141">View MathML</a>

(70)

It is known that there exists the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M78">View MathML</a> Padé approximant which satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M143">View MathML</a>

(71)

By using Mathematica, the Padé approximation gives that the truncated series obtained from the LAPM solution in Eq. (70) is determined to be

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M144">View MathML</a>

(72)

From Figure 5, the presented result is in a good agreement with the exact result for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18">View MathML</a>. Figure 5 represents the comparisons between the exact solution, the LAM, and the LAPM solutions for problem 2. It shows that the LAM solutions diverge rapidly after <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M146">View MathML</a>. However, it represents that the LAPM solution demonstrates a good convergence through the applicable domain. Table 3 shows the absolute errors of the LAPM solution in comparison with the exact solution.

thumbnailFigure 5. The approximate solutions solved by different methods in problem 2 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M83">View MathML</a>.

Table 3. Comparison results of the Riccati equation in problem 2 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18">View MathML</a>

Case II: Here we examine the case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85">View MathML</a> in Eq. (69), we replicate the approximate solution obtained in Eq. (69) given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M150">View MathML</a>

(73)

For simplicity, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M89">View MathML</a>; then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M152">View MathML</a>

(74)

Calculating the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M153">View MathML</a> Padé approximation and recalling that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M92">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M155">View MathML</a>

(75)

Figure 6 shows the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M153">View MathML</a> Pade approximants of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33">View MathML</a> in LAM and LAPM for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85">View MathML</a>. Figure 6 illustrates the comparisons between the LAM solution and the LAPM solution in problem 2 for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M85">View MathML</a>.

thumbnailFigure 6. The approximate solutions solved by different methods in problem 2 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M160">View MathML</a>.

Case III: In this case we examine the LAPM when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M96">View MathML</a> in Eq. (69)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M162">View MathML</a>

(76)

For simplicity, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M163">View MathML</a>; then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M164">View MathML</a>

(77)

Calculating the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M165">View MathML</a> Padé approximation and recalling that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M166">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M167">View MathML</a>

(78)

Figure 7 shows the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M165">View MathML</a> Padé approximants of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33">View MathML</a> in LAPM for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M96">View MathML</a>. Table 4 shows the comparison results of the fractional Riccati equation in test problem 2 of the LAPM solution in comparison with the different values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105">View MathML</a>. The procedure described above was translated into Mathematica program and run on a Pentium-4 PC to investigate the effects of special values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105">View MathML</a> for the fractional Riccati differential equation. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M33">View MathML</a> is evaluated up to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M174">View MathML</a> and plotted in Figure 8.

thumbnailFigure 7. The approximate curves of problem 2 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M104">View MathML</a>.

thumbnailFigure 8. The approximate curves by different methods in problem 2 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M176">View MathML</a>.

Table 4. Numerical results of the Riccati equation of problem 2 for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M105">View MathML</a>

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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/185/mathml/M18">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have equal contributions and they have approved the final version of the manuscript.

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.

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