Solving a multi-order fractional differential equation using homotopy analysis method

H. Jafari; S. Das; H. Tajadodi

doi:10.1016/j.jksus.2010.06.023

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ORIGINAL ARTICLE

23 (

); 151-155

doi:

10.1016/j.jksus.2010.06.023

Solving a multi-order fractional differential equation using homotopy analysis method

H. Jafari^a,⁎, S. Das^b, H. Tajadodi^a

a Department of Mathematics, University of Mazandaran, Babolsar, Iran

b Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi 221 005, India

⁎Corresponding author. Tel.: +98 9112113567. jafari@umz.ac.ir (H. Jafari), jafari_h@math.com (H. Jafari),

Received: 2010-6-7, Accepted: 2010-6-28, Published: 2010-04-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Available online 1 July 2010

Peer-review under responsibility of King Saud University.

Abstract

In this paper we have used the homotopy analysis method (HAM) to obtain solution of multi-order fractional differential equation. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented.

Keywords

Fractional differential equation

Homotopy analysis method

Caputo fractional derivative

Riemann-Liouville fractional derivative

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1 Introduction

Fractional differential equations have been found to be effective to describe some physical phenomena such as damping laws, rheology, diffusion processes, and so on. Several methods have been used to solve Fractional differential equations, such as Laplace transform method (Podlubny, 1999), Fourier transform method (Kemple and Beyer, 1997), Adomians decomposition method (ADM) (Daftardar-Gejji and Jafari, 2005; Daftardar-Gejji and Jafari, 2007; Jafari and Daftardar-Gejji, 2006 ), Homotopy analysis method (Liao, 2003; Momani and Odibat, 2008) and so on. For nonlinear FDE, however, one mainly resorts to numerical methods (Diethelm, 1997; Diethelm and Ford, 2002; Diethelm and Ford, 2004; Edwards et al., 2002 ). These numerical methods involve discretization of the variables, which gives rise to rounding off errors. Another drawback of numerical methods stems from the requirement of large computer memory.

In this paper, the homotopy analysis method (Liao, 1992) is applied to solve the multi-order fractional differential equation studied by Diethelm and Ford (2004): $D_{*}^{α} y (t) = f (t, y (t), D_{*}^{β_{1}} y (t), \dots, D_{*}^{β_{n}} y (t)), y^{(k)} (0) = c_{k}, k = 0, \dots, m,$ where $m < α ⩽ m + 1, 0 < β_{1} < β_{2} < \dots < β_{n} < α$ and $D_{*}^{α}$ denotes Caputo fractional derivative of order $α$ .

Liao (1992) employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely homotopy analysis method (HAM), Liao (1992, 2004, 2003). This method (HAM) (Liao, 2003) provides an effective procedure for explicit and numerical solutions of a wide and general class of differential systems representing real physical problems. Based on homotopy of topology, the validity of the HAM is independent of whether or not there exist small parameters in the considered equation. Therefore, the HAM can overcome the foregoing restrictions and limitations of perturbation techniques so that it provides us with a possibility to analyze strongly nonlinear problems. Jafari and seifi have been solved diffusion-wave equation and partial differential equations and system of nonlinear fractional partial differential equations using homotopy analysis method (Jafari and Seifi, 2009; Jafari and Seifi, 2009). This method has been successfully applied to solve many types of nonlinear problems (Hayat et al., 2004; Momani and Odibat, 2008). These authors have discussed the analytical questions of existence and uniqueness of solutions and investigated how the solutions depend on the given initial data. Further they have presented an algorithm to convert the multi-order FDE into a system of FDE under some conditions and have developed numerical method to solve the system of FDE. In this paper we present an algorithm to convert the multi-order FDE into a system of FDE, without putting any of the restrictions. Thus our algorithm is valid in the most general case and yields fewer number of equations in a system compared to those in Diethelm–Ford algorithm. Consequently the solutions of the system of FDE have been obtained by employing the HAM approach. The paper has been organized as follows. Section 2 describes how to convert a multi-order FDE. In Section 3, HAM is developed to solve the system of FDE. Section 4 presents some illustrative examples. Discussion and conclusions are summarised in the Section 5.

2 Preliminaries

We enlist below some definitions (Luchko and Gorenflo, 1999; Podlubny, 1999) and basic results.

Definition 2.1

A real function $f (x), x > 0$ is said to be in the space $C_{α}, α \in R$ if there exists a real number $p (> α)$ such that $f (x) = x^{p} f_{1} (x)$ where $f_{1} (x) \in C [0, \infty)$ . Clearly $C_{α} \subset C_{β}$ if $β ⩽ α$ .

Definition 2.2

A function $f (x), x > 0$ is said to be in the space $C_{α}^{m}, m \in N ⋃ {0}$ if $f^{(m)} \in C_{α}$ .

Definition 2.3

The (left sided) Riemann-Liouville fractional integral of order $μ > 0$ of a function $f \in C_{α}, α ⩾ - 1$ is defined as:

(1)

\begin{matrix} I^{μ} f (t) = \frac{1}{Γ (μ)} \int_{0}^{t} \frac{f (τ)}{{(t - τ)}^{1 - μ}} d τ, μ > 0, t > 0, \\ I^{0} f (t) = f (t) . \end{matrix}

Definition 2.4

The (left sided) Riemann-Liouville fractional derivative of $f, f \in C_{- 1}^{m}, m \in IN ⋃ {0}$ of order $α > 0$ is defined as:

(2)

D^{μ} f (t) = \frac{d^{m}}{{dt}^{m}} I^{m - μ} f (t), m - 1 < μ ⩽ m, m \in IN .

Definition 2.5

The (left sided) Caputo fractional derivative of $f, f \in C_{- 1}^{m}, m \in IN ⋃ {0}$ is defined as:

(3)

D_{*}^{μ} f (t) = \{\begin{matrix} [I^{m - μ} f^{(m)} (t)] & m - 1 < μ < m, m \in IN, \\ \frac{d^{m}}{{dt}^{m}} f (t) & μ = m . \end{matrix}

Note that

\begin{matrix} (i) & I^{μ} t^{γ} = \frac{Γ (γ + 1)}{Γ (γ + μ + 1)} t^{γ + μ}, μ > 0, γ > - 1, t > 0 . \\ (ii) & I^{μ} D_{*}^{μ} f (t) = f (t) - \sum_{k = 0}^{m - 1} f^{(k)} (0 +) \frac{t^{k}}{k!}, m - 1 < μ ⩽ m, m \in IN . \\ (iii) & D_{*}^{μ} f (t) = D^{μ} (f (t) - \sum_{k = 0}^{m - 1} f^{(k)} (0 +) \frac{t^{k}}{k!}), m - 1 < μ ⩽ m, m \in IN . \\ (iv) & D^{β} I^{α} f (t) = \{\begin{matrix} I^{α - β} f (t), & if α > β, \\ f (t), & if α = β, \\ D^{β - α} f (t), & if α < β . \end{matrix} \\ (v) & D_{*}^{α} D^{m} f (t) = D_{*}^{α + m} f (t), m = 0, 1, 2, \dots, n - 1 < α < n . \end{matrix}

Definition 2.6

The Mittag-Leffler function $E_{α} (z)$ with $α > 0$ is defined by the following series representation valid in the whole complex plane (Mainardi, 1994):

(4)

E_{α} (z) = \sum_{n = 0}^{\infty} \frac{z^{n}}{Γ (α n + 1)}, α > 0, z \in C .

Lemma 2.7

Lemma 2.7 Diethelm and Ford, 2002

Let $y (t) \in C^{k} [0, T]$ for some $T > 0$ and $k \in IN$ and let $q \notin IN$ be such that $0 < q < k$ . Then $D_{*}^{q} y (0) = 0$ .

3 Multi-order FDE as a system of FDE

Daftardar-Gejji and Jafari have solved multi-order FDE by Adomian decomposition method (Daftardar-Gejji and Jafari, 2007):

(5)

D_{*}^{α} y (t) = f (t, y (t), D_{*}^{β_{1}} y (t), \dots, D_{*}^{β_{n}} y (t)), y^{(k)} (0) = c_{k}, k = 0, \dots, m,

where

m < α ⩽ m + 1, 0 < β_{1} < β_{2} < \dots < β_{n} < α

and

D_{*}^{α}

denotes Caputo fractional derivative of order

α

. It should be noted that f can be non linear in general.

(6)

α, β_{j} \in Q, α - β_{n} ⩽ 1, β_{j} - β_{j - 1} ⩽ 1, \forall j, and 0 ⩽ β ⩽ 1 .

In Daftardar-Gejji and Jafari (2007), it was proved that the Eq. (5) can be represented as a system of FDE, without any additional restrictions mentioned in Eq. (6). Here we present their approach. Set

y_{1} = y

and define:

(7)

D_{*}^{β_{1}} y_{1} = y_{2} .

Case (i) If

m - 1 ⩽ β_{1} < β_{2} ⩽ m

then define

(8)

D_{*}^{β_{2} - β_{1}} y_{2} = y_{3} .

Claim:

y_{3} = D_{*}^{β_{2}} y

. If

β_{1} = m - 1

, then

D_{*}^{β_{2} - β_{1}} y_{2} = D_{*}^{β_{2} - (m - 1)} y^{(m - 1)} = D_{*}^{β_{2}} y_{1}

Hence the claim. If $β_{1} \notin IN$ , then by Lemma 2.6, $D_{*}^{β_{1}} y_{1} (0) = 0$ and as $β_{2} - β_{1} < 1$ ,

(9)

D_{*}^{β_{2} - β_{1}} [D_{*}^{β_{1}} y_{1}] = D^{β_{2} - β_{1}} [D_{*}^{β_{1}} y_{1}] = {DI}^{1 + β_{1} - β_{2}} I^{m - β_{1}} y_{1}^{(m)} = {DI}^{1 + m - β_{2}} y_{1}^{(m)} = I^{m - β_{2}} y_{1}^{(m)} = D_{*}^{β_{2}} y_{1} = D_{*}^{β_{2}} y .

Therefore

y_{3} = D_{*}^{β_{2} - β_{1}} y_{2} = D_{*}^{β_{2}} y

Case (ii) Consider $m - 1 ⩽ β_{1} < m ⩽ β_{2}$ . If $β_{1} = m - 1$ , then define $D_{*}^{β_{2} - β_{1}} y_{2} = y_{3}$ . $D_{*}^{β_{2} - β_{1}} y_{2} = D_{*}^{β_{2} - m + 1} y_{1}^{(m - 1)} = D_{*}^{β_{2}} y_{1} .$ If $m - 1 < β_{1} < m ⩽ β_{2}$ , then define

(10)

D_{*}^{m - β_{1}} y_{2} = y_{3} .

Claim:

y_{3} = y^{(m)}

. As

β_{1} \notin IN, D^{β_{1}} y_{1} (0) = y_{2} (0) = 0

(in view of Lemma 2.6), and

0 < m - β_{1} < 1

(11)

D_{*}^{m - β_{1}} y_{2} = D^{m - β_{1}} y_{2} = {DI}^{1 + β_{1} - m} I^{m - β_{1}} y_{1}^{(m)} = {DIy}_{1}^{(m)} = y_{1}^{(m)} = y^{(m)} .

Hence

y_{3} = y^{(m)}

. Further define:

(12)

D_{*}^{β_{2} - m} y_{3} = y_{4} .

Claim

y_{4} = D_{*}^{β_{2}} y

. As

y_{4} = D_{*}^{β_{2} - m} y_{3} = D_{*}^{β_{2} - m} y^{(m)} = D_{*}^{β_{2}} y

. And continuing similarly we can convert the initial value problem (5) into a system of FDE. The following example will illlustrate the method. Consider

(13)

D_{*}^{3.6} y = f (x, y, D_{*}^{1.2} y, D_{*}^{1.7} y, D_{*}^{2.1} y, D_{*}^{3.5} y),

where

y (0) = c_{0}, y^{'} (0) = c_{1}, y^{″} (0) = c_{2} and y^{‴} (0) = c_{3}

. This initial value problem can be viewed as the following system of FDE.

(14)

\begin{matrix} D_{*}^{1.2} y_{1} (x) = y_{2} (x), y_{1} (0) = c_{0}, y_{1}^{'} (0) = c_{1}, \\ D_{*}^{0.5} y_{2} (x) = y_{3} (x) (= D_{*}^{1.7} y (x)), y_{2} (0) = 0, \\ D_{*}^{0.3} y_{3} (x) = y_{4} (x) (= y^{″} (x)), y_{3} (0) = 0, \\ D_{*}^{0.1} y_{4} (x) = y_{5} (x) (= D_{*}^{2.1} y (x)), y_{4} (0) = c_{2}, \\ D_{*}^{0.9} y_{5} (x) = y_{6} (x) (= y^{(3)} (x)), y_{5} (0) = 0, \\ D_{*}^{0.5} y_{6} (x) = y_{7} (x) (= D_{*}^{3.5} y (x)), y_{6} (0) = c_{3}, \\ D_{*}^{0.1} y_{7} (x) = f (x, y_{1}, y_{2}, y_{3}, y_{5}, y_{7}), y_{7} (0) = 0, \end{matrix}

where

y_{1} (x) = y (x) .

The remark is in order

1. This algorithm is valid in the most general case, because we do not impose any of the restrictions on $α, β_{i}$ as mentioned in Eq. (6). We let $α, β_{i} \in R$ , whereas in Diethelm and Ford (2004) $α, β_{i} \in IQ$ .

4 Homotopy analysis method and a system of FDE

We can present the multi-order Eq. (5) as system of fractional differential equations:

(15)

\begin{matrix} D^{α_{i}} y_{i} (t) = y_{i + 1}, i = 1, 2, \dots, n - 1, \\ D^{α_{n}} y_{i} (t) = f (t, y_{1}, y_{2}, \dots, y_{n}), \\ y_{i}^{(k)} (0) = c_{k}^{i}, 0 ⩽ k ⩽ m_{i}, m_{i} < α_{i} ⩽ m_{i} + 1, 1 ⩽ i ⩽ n . \end{matrix}

According to the HAM, we construct the so called zero-order deformation equations:

(16)

\begin{matrix} (1 - q) D^{α_{i}} [φ_{i} (t; q) - y_{i 0} (t)] = {qh}_{i} H_{i} (t) [D^{α_{i}} φ_{i} (t; q) - φ_{i + 1} (t; q)], \\ i = 1, 2, \dots, n - 1, \\ (1 - q) D^{α_{n}} [φ_{n} (t; q) - y_{n 0} (t)] = {qh}_{i} H_{n} (t) [D^{α_{i}} φ_{n} (t; q) - \\ f (t, φ_{1}, φ_{2}, \dots, φ_{n})], \end{matrix}

where

q \in [0, 1]

is an embedding parameter,

h_{i} \neq 0

are non-zero auxiliary parameters for

H_{i} (t) \neq 0

i = 1, 2, \dots, n

are non-zero auxiliary functions,

y_{i 0} (t)

are initial guess of

y_{i} (t)

φ_{i} (t; q)

are unknown function, respectively. It is important that one has great freedom to choose auxiliary things in HAM. Obviously, when q = 0 and q = 1, it holds

(17)

φ_{i} (t; 0) = y_{i 0} (t), φ_{i} (t; 1) = y_{i} (t), i = 1, 2, \dots, n,

respectively. Thus as q increases from 0 to 1 the solution

φ_{i} (t; q)

varies from the initial guesses

y_{i 0} (t)

to the solution

y_{i} (t)

. Expanding

φ_{i} (t; q)

in Taylor series with respect to q, we have

(18)

φ_{i} (t; q) = y_{i 0} (t) + \sum_{m = 1}^{\infty} y_{im} (t) q^{m}, i = 1, 2, \dots, n,

where

(19)

y_{im} (t) = \frac{1}{m!} \frac{\partial^{m} φ_{i} (t; q)}{\partial q^{m}} |_{q = 0}, i = 1, 2, \dots, n,

If the auxiliary linear operator, the initial guess, the auxiliary parameter h and the auxiliary function are so properly chosen the series Eq. (18) converges at q = 1, then we have

(20)

y_{i} (t) = y_{i 0} (t) + \sum_{m = 1}^{\infty} y_{im} (t), i = 1, 2, \dots, n,

Define the vector

{\vec{y}}_{i} = {y_{i 0} (t), y_{i 1} (t), \dots, y_{in} (t)} .

Differentiating Eq. (16) m times with respect to the embedding parameter q and then setting

q = 0

and finally dividing them by m!, we obtain the mth-order deformation equation for

i = 1, 2, \dots, n,

(21)

\begin{matrix} D^{α_{i}} [y_{im} (t) - χ_{m} y_{im - 1} (t)] = h_{i} H_{i} (t) R_{im} ({\vec{y}}_{1 m - 1}, \dots, {\vec{y}}_{nm - 1}, t), \\ i = 1, 2, \dots, n - 1, \\ D^{α_{n}} [y_{nm} (t) - χ_{m} y_{nm - 1} (t)] = h_{i} H_{n} (t) R_{nm} ({\vec{y}}_{1 m - 1}, \dots, {\vec{y}}_{nm - 1}, t), \end{matrix}

where

(22)

\begin{matrix} R_{im} ({\vec{y}}_{1 m - 1}, \dots, {\vec{y}}_{nm - 1}, t) = \frac{1}{(m - 1)!} \frac{\partial^{m - 1} (D^{α_{i}} φ_{i} (t; q) - φ_{i + 1} (t; q))}{\partial q^{m - 1}} |_{q = 0}, \\ R_{nm} ({\vec{y}}_{1 m - 1}, \dots, {\vec{y}}_{nm - 1}, t) = \frac{1}{(m - 1)!} \frac{\partial^{m - 1} (D^{α_{i}} φ_{n} (t; q) - f (t, φ_{1}, φ_{2}, \dots, φ_{n}))}{\partial q^{m - 1}} |_{q = 0} . \end{matrix}

and

χ_{m} = \{\begin{matrix} 0, & m ⩽ 1, \\ 1, & m > 1 . \end{matrix}

Applying the Riemann-Liouville integral operator

I^{α i}

on both side of Eq. (21), we have

(23)

\begin{matrix} y_{im} (t) = χ_{m} y_{im - 1} (t) - χ_{m} \sum_{j = 0}^{n - 1} y_{im - 1}^{j} (0^{+}) \frac{t^{j}}{j!} \\ + I^{α i} h_{i} H_{i} (t) R_{im} ({\vec{y}}_{1 m - 1}, \dots, {\vec{y}}_{nm - 1}, t), \\ y_{nm} (t) = χ_{m} y_{nm - 1} (t) - χ_{m} \sum_{j = 0}^{n - 1} y_{nm - 1}^{j} (0^{+}) \frac{t^{j}}{j!} \\ + I^{α i} h_{i} H_{n} (t) R_{nm} ({\vec{y}}_{1 m - 1}, \dots, {\vec{y}}_{nm - 1}, t) . \end{matrix}

In this way, it is easily to obtain

y_{im}

for

m ⩾ 1

at Mth order, we have

(24)

y_{i} (t) = \sum_{m = 0}^{M} y_{im} (t), i = 1, 2, \dots, n .

When

M \to \infty

, we get an accurate approximation of the original Eq. (15).

5 Illustrative Examples

To demonstrate the effectiveness of the method we consider here some multi-order FDE. We transform multi-order FDE to a system of FDE and solve the system of FDE using HAM.

Consider the following initial value problem in case of the inhomogeneous Bagley–Torvik equation (Diethelm and Ford, 2002):
(25)
$D_{*}^{2} y (t) + D_{*}^{1.5} y (t) + y (t) = 1 + t, y (0) = 1, y^{'} (0) = 1 .$ In view of the discussion in the Section 3, the Eq. (25) can be viewed as the following system of FDE: $\begin{matrix} D_{*}^{1.5} y_{1} = y_{2}, y_{1} (0) = y_{1}^{'} (0) = 1, \\ D_{*}^{0.5} y_{2} = - y_{2} - y_{1} + 1 + t, y_{2} (0) = 0 . \end{matrix}$ Using Eq. (23) we get the following scheme: $\begin{matrix} y_{10} = 1 + t, y_{20} = 0, \\ y_{1 m} = χ_{m} y_{1 m} + h_{1} I^{1.5} (D_{*}^{1.5} y_{1 m - 1} - y_{2 m - 1}), m ⩾ 1, \\ y_{2 m} = χ_{m} y_{2 m} + h_{2} I^{0.5} (D_{*}^{0.5} y_{2 m - 1} + y_{2 m - 1} + y_{1 m - 1} \\ - (1 - χ_{m}) (1 + t)), m ⩾ 1 . \end{matrix}$ Thus we get: $\begin{matrix} y_{11} = h_{1} I^{1.5} (D_{*}^{1.5} y_{10} - y_{20}) = 0, \\ y_{21} = h_{2} I^{0.5} (D_{*}^{0.5} y_{20} + y_{20} + y_{10} - (1 + t)) = 0, \end{matrix}$ and hence $y_{1 m} = 0, y_{2 m} = 0, m ⩾ 1 .$ In view of the above terms, we find $y_{1} (t) = 1 + t$ and $y_{2} (t) = 0$ . So $y (t) = 1 + t$ is the required solution of (25).
Consider the following initial value problem,
(26)
$D_{*}^{3} y (t) + D_{*}^{2.5} y (t) + y^{2} (t) = t^{4}, y (0) = y^{'} (0) = 0, y^{″} (0) = 2 .$ If we choose $y (t) = y_{1}$ and $D_{*}^{2.5} y (t) = y_{2}$ , then Eq. (26) can be reduced to the following system of nonlinear FDE: $\begin{matrix} D_{*}^{2.5} y_{1} = y_{2}, y_{1} (0) = y_{1}^{'} (0) = 0, y^{″} (0) = 2, \\ D_{*}^{0.5} y_{2} = t^{4} - y_{2} - y_{1}^{2}, y_{2} (0) = 0 . \end{matrix}$ Applying $I^{2.5}$ and $I^{0.5}$ to both sides of above system and using HAM Eq. (23) we get the following scheme:
(27)
$\begin{matrix} y_{10} = t^{2}, y_{20} = 0, \\ y_{1 m} = χ_{m} y_{1 m} + h_{1} I^{2.5} (D_{*}^{2.5} y_{1 m - 1} - y_{2 m - 1}), m ⩾ 1, \\ y_{2 m} = χ_{m} y_{2 m} + h_{2} I^{0.5} (D_{*}^{0.5} y_{2 m - 1} - t^{4} + y_{2 m - 1} + \sum_{i = 0}^{m - 1} y_{1 i} y_{1 m - 1 - i}), m ⩾ 1 . \end{matrix}$ Thus $\begin{matrix} y_{11} = h_{1} I^{2.5} (D_{*}^{2.5} y_{10}) = 0, \\ y_{21} = h_{2} I^{0.5} (D_{*}^{0.5} y_{20} - t^{4} + y_{20} + y_{10}^{2}) = 0 . \end{matrix}$ So $y_{1 m + 1} = y_{2 m + 1} = 0, for m ⩾ 1$ . Hence we find $y_{1} (t) = t^{2}$ and $y_{2} (t) = 0$ . Therefore $y (t) = t^{2}$ is the required solution.
Consider the following equation,
(28)
$D_{*}^{1.455} y (t) = - t^{0.1} \frac{E_{1.545} (- t)}{E_{1.445} (- t)} Exp (t) y (t) D_{*}^{0.555} y (t) + Exp (- 2 t) - {[D_{*}^{1} y (t)]}^{2} .$ Eq. (28) is equivalent to the following system of three FDE only.
(29)
$\begin{matrix} D_{*}^{0.555} y_{1} (x) = y_{2} (x), \\ D_{*}^{0.445} y_{2} (x) = y_{3} (x) (= y^{'} (x)), \\ D_{*}^{0.455} y_{3} (x) = - t^{0.1} \frac{E_{1.545} (- t)}{E_{1.445} (- t)} Exp (t) y_{1} (t) y_{2} (t) + Exp (- 2 t) - y_{3}^{2} (t), \end{matrix}$ where $y_{1} (t) = y (t)$ . In view of Eq. (29) and HAM we get $\begin{matrix} y_{10} = y_{1} (0), y_{20} = y_{2} (0), y_{30} = y_{3} (0), \\ y_{1 m} = χ_{m} y_{1 m} + h_{1} I^{0.555} (D_{*}^{0.555} y_{1 m - 1} - y_{2 m - 1}), \\ y_{2 m} = χ_{m} y_{2 m} + h_{2} I^{0.445} (D_{*}^{0.445} y_{2 m} - y_{3 m - 1}), \\ y_{3 m} = χ_{m} y_{3 m} + h_{3} I^{0.455} (D_{*}^{0.455} y_{3 m - 1} + t^{0.1} \frac{E_{1.545} (- t)}{E_{1.445} (- t)} Exp (t) y_{1 m - 1} y_{2 m - 1} (t) - Exp (- 2 t) + y_{3 m - 1}^{2}) . \end{matrix}$

In Fig. 1, we draw exact solution $(y = y_{1} = e^{- t})$ and solution obtained after 3 iterations $h_{1} = - 1.4$ .

Exact solution (solid line) and HAM solution obtained after 3 iterations ( y ≅ y 13 ) (dashed line).

The computations presented here have been carried out with the help of Mathematica 5.

6 Discussion and conclusions

Homotopy analysis method consequently has been utilized to solve the system of fractional differential equations generated by a multi-order fractional differential equation. Thus it has been demonestrated that HAM proves useful in solving linear as well as non linear multi-order fractional differential equations.

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