A new analytical method for solving systems of linear integro-differential equations

Hossein Aminikhah

doi:10.1016/j.jksus.2010.07.016

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ORIGINAL ARTICLE

23 (

); 349-353

doi:

10.1016/j.jksus.2010.07.016

A new analytical method for solving systems of linear integro-differential equations

Hossein Aminikhah^*

Department of Mathematics, School of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht, Iran

*Tel./fax: +98 131 3233509 hossein.aminikhah@gmail.com (Hossein Aminikhah)

Received: 2010-6-24, Accepted: 2010-7-10, Published: 2010-10-01

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

Available online 15 July 2010

Peer-review under responsibility of King Saud University.

Abstract

In this paper, we introduce a new modification of homotopy perturbation method (NHPM) to obtain exact solutions of systems of linear integro-differential equations. Theoretical considerations are discussed. Some examples are presented to illustrate the efficiency and simplicity of the method.

Keywords

New homotopy perturbation method

Systems of integro-differential equations

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1 Introduction

Integro-differential equation has attracted much attention and solving this equation has been one of the interesting tasks for mathematicians. These equations have been found to describe various kind of phenomena such as wind ripple in the desert, nono-hydrodynamics, dropwise consideration and glass-forming process (Bo et al., 2007; Sun et al., 2007; Wang et al., 2007; Xu et al., 2007 ).

The homotopy perturbation method is a powerful device for solving functional equations. The method has been used by many authors to handle a wide variety of scientific and engineering applications to solve various functional equations. In this method the solution is considered as the summation of an infinite series which converges rapidly to the accurate solutions. Considerable research works have been conducted recently in applying this method to a class of linear and nonlinear equations. This method was further developed and improved by He and applied to nonlinear oscillators with discontinuities (He, 1999), nonlinear wave equations (He, 2000), boundary value problems (He, 2004), limit cycle and bifurcation of nonlinear problems (He, 2003), and many other subjects (He, 2004, 2005, 2006, 2005 ). It can be said that He's homotopy perturbation method is a universal one, and is able to solve various kinds of nonlinear functional equations. For examples it was applied to nonlinear Schrödinger equations Biazar and Ghazvini (2007), to nonlinear equations arising in heat transfer (Ganji, 2006), to the quadratic Ricatti differential equation (Abbasbandy, 2006), and to other equations (Odibat and Momani, 2008; Siddiqui et al., 2008; Ganji and Sadighi, 2007; Golbabai and Javidi, 2007; Golbabai and Keramati, 2008; Shakeri and Dehghan, 2008; Beléndez et al., 2008 ). Biazar and Ghazvini (2009) and Biazar and Ghazvini (2008) employed He's homotopy perturbation method to compute an approximation to the solution of system of Volterra integral equations and nonlinear Fredholm integral equation of second kind. In Mohyud-Din et al. (2010), Raftari and Yildirim (2010), Yildirim et al. (2010) and Yildirim and Gülkanat (2010), some recent non-perturbative methods have been studied to solve various nonlinear problems.

In this article a new homotopy perturbation method is introduced to obtain exact solutions of systems of integro-differential equations. To demonstrate this method, some examples are given.

2 New homotopy perturbation method (NHPM) for systems of integro-differential equations

A system of integro-differential can be considered in general as follows:

(1)

\begin{matrix} \frac{dx (t)}{dt} = F (t, x (t)) + \int_{t_{0}}^{t} K (s, t, x (s)) ds, t_{0} ⩾ 0, \\ x (t_{0}) = x_{0}, \end{matrix}

where

\begin{matrix} x (t) = (x_{1} (t), x_{2} (t), \dots x_{n} (t))^{T}, \\ K (s, t, x (t)) = (k_{1} (s, t, x (s)), k_{2} (s, t, x (s)), \dots, k_{n} (s, t, x (s)))^{T} . \end{matrix}

If $K (s, t, x (t))$ and $F (t, x (t))$ be linear, the system (1) can be represented as the following simple form:

(2)

\frac{{dx}_{i}}{dt} = f_{i} (t) + \sum_{j = 1}^{n} (w_{i, j} (t) x_{j} (t) + \int_{t_{0}}^{t} k_{i, j} (s, t) x_{j} (s) ds), x_{i} (t_{0}) = α_{i}, i = 1, 2, \dots, n .

For solving system (2), by new homotopy perturbation method, we construct the following homotopy

(3)

(1 - p) (\frac{{dX}_{i}}{dt} - x_{i, 0}) + p (\frac{{dX}_{i}}{dt} - f_{i} (t) - \sum_{j = 1}^{n} (w_{i, j} (t) x_{j} (t) + \int_{t_{0}}^{t} k_{i, j} (s, t) x_{j} (s) ds)) = 0,

or equivalently,

(4)

\frac{{dX}_{i}}{dt} = x_{i, 0} - p (x_{i, 0} - f_{i} (t) - \sum_{j = 1}^{n} (w_{i, j} (t) x_{j} (t) + \int_{t_{0}}^{t} k_{i, j} (s, t) x_{j} (s) ds)) .

Applying the inverse operator,

L^{- 1} = \int_{t_{0}}^{t} (\cdot) dt

to both sides of Eq. (4), we obtain

(5)

X_{i} (t) = α_{i} + \int_{t_{0}}^{t} x_{i, 0} (s) ds - p (\int_{t_{0}}^{t} x_{i, 0} (s) ds - \int_{t_{0}}^{t} f_{i} (s) ds - \sum_{j = 1}^{n} (\int_{t_{0}}^{t} w_{i, j} (s) x_{j} (s) ds + \int_{t_{0}}^{t} \int_{t_{0}}^{τ} k_{i, j} (s, τ) x_{j} (s) dsd τ)) .

Suppose the solutions of system (5) have the following form:

(6)

X_{i} (t) = X_{i, 0} (t) + {pX}_{i, 1} (t) + p^{2} X_{i, 2} (t) + \dots, i = 1, 2, \dots, n .

where

X_{i, j} (t), i = 1, 2, \dots, n

and

j = 0, 1, 2, \dots

are functions which should be determined.

Now suppose that the initial approximations to the solutions $X_{i, 0} (t)$ or $x_{i, 0} (t)$ have the form

(7)

X_{i, 0} (t) = x_{i, 0} (t) = \sum_{j = 0}^{\infty} α_{i, j} P_{j} (t), i = 1, 2, \dots, n,

where

α_{i, j}

are unknown coefficients and

P_{0} (x), P_{1} (x), P_{2} (x), \dots

are specific functions.

Substituting (6) into (5) and equating the coefficients of p with the same power leads to

(8)

\begin{matrix} p^{0} : X_{i, 0} (t) = α_{i} + \sum_{j = 0}^{\infty} α_{i, j} \int_{t_{0}}^{t} P_{j} (s) ds, \\ p^{1} : X_{i, 1} (t) = - \sum_{j = 0}^{\infty} α_{i, j} \int_{t_{0}}^{t} P_{j} (s) ds + \int_{t_{0}}^{t} f_{i} (s) ds \\ + \sum_{j = 1}^{n} (\int_{t_{0}}^{t} w_{i, j} (s) X_{j, 0} (s) ds \\ + \int_{t_{0}}^{t} \int_{t_{0}}^{τ} k_{i, j} (s, τ) X_{j, 0} (s) dsd τ), \\ p^{2} : X_{i, 2} (t) = \sum_{j = 1}^{n} (\int_{t_{0}}^{t} w_{i, j} (s) X_{j, 1} (s) ds \\ + \int_{t_{0}}^{t} \int_{t_{0}}^{τ} k_{i, j} (s, τ) X_{j, 1} (s) dsd τ), \\ ⋮ \\ p^{j} : X_{i, j - 1} (t) = \sum_{j = 1}^{n} (\int_{t_{0}}^{t} w_{i, j} (s) X_{j, j - 1} (s) ds \\ + \int_{t_{0}}^{t} \int_{t_{0}}^{τ} k_{i, j} (s, τ) X_{j, j - 1} (s) dsd τ), \\ ⋮ \end{matrix}

Now if these equations be solved in a way that

X_{i, 1} (t) = 0

, then Eq. (8) result in

X_{i, 2} (t) = X_{i, 2} (t) = \dots = 0

, therefore the exact solution can be obtained by using

(9)

x_{i} (t) = X_{i, 0} (t) = α_{i} + \sum_{j = 0}^{\infty} α_{i, j} \int_{t_{0}}^{t} P_{j} (s) ds .

It is worthwhile to note that if $f_{i} (t)$ and $x_{i, 0} (t)$ are analytic at $t = t_{0}$ , then their Taylor series

(10)

x_{i, 0} (t) = \sum_{n = 0}^{\infty} a_{n} (t - t_{0})^{n}, f_{i} (t) = \sum_{n = 0}^{\infty} a_{n}^{*} (t - t_{0})^{n},

can be used in Eq. (8), where

a_{0}^{*}, a_{1}^{*}, a_{2}^{*}, \dots

are known coefficients and

a_{0}^{,} a_{1}^{,} a_{2}^{,} \dots

are unknown ones, which must be computed.

We would explain this method by considering several examples.

3 Examples

In this section we present two examples. These examples are considered to illustrate the NHPM for systems of integro-differential equations.

Example 1

Consider the following system of integro-differential equations with the exact solutions $x_{1} (t) = e^{t}$ and $x_{2} (t) = e^{- t}$ ,

(11)

\begin{matrix} \frac{{dx}_{1} (t)}{dt} = t^{4} - t^{3} - 2 t^{2} - 6 + (3 t^{2} - 6 t + 7) x_{1} (t) \\ + 2 t^{2} (t + 1) x_{2} (t) \\ + \int_{0}^{t} ((s^{3} - t^{3}) x_{1} (s) + t^{2} (s^{2} - t^{2}) x_{2} (s)) ds, x_{1} (0) = 1, \\ \frac{{dx}_{2} (t)}{dt} = - t^{4} - 3 t^{2} + 2 + 2 (t - 1) x_{1} (t) + (2 t^{4} \\ + 2 t^{3} + 2 t^{2} - 1) x_{2} (t) \\ + \int_{0}^{t} ((s^{2} - t^{2}) x_{1} (s) + t^{2} (s^{2} + t^{2}) x_{2} (s)) ds, x_{2} (0) = 1 . \end{matrix}

For solving system (11), by NHPM, we construct the following homotopy:

(12)

\begin{matrix} \frac{{dX}_{1} (t)}{dt} = x_{1, 0} (t) - p (x_{1, 0} (t) - t^{4} + t^{3} + 2 t^{2} + 6 \\ - (3 t^{2} - 6 t + 7) X_{1} (t) - 2 t^{2} (t + 1) X_{2} (t) \\ - \int_{0}^{t} ((s^{3} - t^{3}) X_{1} (s) + t^{2} (s^{2} - t^{2}) X_{2} (s)) ds), \\ \frac{{dX}_{2} (t)}{dt} = x_{2, 0} (t) - p (x_{2, 0} (t) + t^{4} + 3 t^{2} - 2 - 2 (t - 1) X_{1} (t) \\ - (2 t^{4} + 2 t^{3} + 2 t^{2} - 1) X_{2} (t) \\ - \int_{0}^{t} ((s^{2} - t^{2}) X_{1} (s) + t^{2} (s^{2} + t^{2}) X_{2} (s)) ds . \end{matrix}

Assuming that, $x_{1, 0} (t) = \sum_{n = 0}^{\infty} α_{n} P_{n} (t), x_{2, 0} (t) = \sum_{n = 0}^{\infty} β_{n} P_{n} (t), P_{i} (t) = t^{i}, X_{1} (0) = X_{2} (0) = 1$ .

By integration of Eq. (12) we have

(13)

X_{1} (t) = 1 + \sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1} - p (\sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1} - \frac{t^{5}}{5} + \frac{t^{4}}{4} + \frac{2 t^{3}}{3} + 6 t - \int_{0}^{t} ((3 s^{2} - 6 s + 7) X_{1} (s) + 2 s^{2} (s + 1) X_{2} (s)) ds - \int_{0}^{t} \int_{0}^{τ} ((s^{3} - τ^{3}) X_{1} (τ) + τ^{2} (s^{2} - τ^{2}) X_{2} (τ)) dsd τ),

(14)

X_{2} (t) = 1 + \sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1} - p (\sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1} + \frac{t^{5}}{5} + t^{3} - 2 t - \int_{0}^{t} (2 (s - 1) X_{1} (s) + (2 s^{4} + 2 s^{3} + 2 s^{2} - 1) X_{2} (s)) ds - \int_{0}^{t} \int_{0}^{τ} ((s^{2} - τ^{2}) X_{1} (τ) + τ^{2} (s^{2} + τ^{2}) X_{2} (τ)) dsd τ) .

Suppose the solutions of system (13) have the following form:

(15)

X_{i} (t) = X_{i, 0} (t) + {pX}_{i, 1} (t) + p^{2} X_{i, 2} (t) + \dots, i = 1, 2,

where

X_{i, j} (t), i = 1, 2

and

j = 0, 1, 2, \dots

are functions which should be determined.

Substituting (14) into (13) and equating the coefficients of p with the same powers leads to $\begin{matrix} p^{0} : \{\begin{matrix} X_{1, 0} (t) = 1 + \sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1}, \\ X_{2, 0} (t) = 1 + \sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1}, \end{matrix} \\ p^{1} : \{\begin{matrix} X_{1, 1} (t) = - \sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1} + \frac{t^{5}}{5} - \frac{t^{4}}{4} - \frac{2 t^{3}}{3} - 6 t \\ + \int_{0}^{t} ((3 s^{2} - 6 s + 7) X_{1, 0} (s) \\ + 2 s^{2} (s + 1) X_{2, 0} (s)) ds \\ + \int_{0}^{t} \int_{0}^{τ} ((s^{3} - τ^{3}) X_{1, 0} (τ) \\ + τ^{2} (s^{2} - τ^{2}) X_{2, 0} (τ)) dsd τ), \\ X_{2, 1} (t) = - \sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1} - \frac{t^{5}}{5} - t^{3} \\ + 2 t + \int_{0}^{t} (2 (s - 1) X_{1, 0} (s) \\ + (2 s^{4} + 2 s^{3} + 2 s^{2} - 1) X_{2, 0} (s)) ds \\ + \int_{0}^{t} \int_{0}^{τ} ((s^{2} - τ^{2}) X_{1, 0} (τ) \\ + τ^{2} (s^{2} + τ^{2}) X_{2, 0} (τ)) dsd τ), \end{matrix} \\ p^{m} : \{\begin{matrix} X_{1, m} (t) = \int_{0}^{t} ((3 s^{2} - 6 s + 7) X_{1, m - 1} (s) \\ + 2 s^{2} (s + 1) X_{2, m - 1} (s)) ds \\ + \int_{0}^{t} \int_{0}^{τ} ((s^{3} - τ^{3}) X_{1, m - 1} (τ) \\ + τ^{2} (s^{2} - τ^{2}) X_{2, m - 1} (τ)) dsd τ), \\ X_{2, m} (t) = \int_{0}^{t} (2 (s - 1) X_{1, m - 1} (s) + (2 s^{4} \\ + 2 s^{3} + 2 s^{2} - 1) X_{2, m - 1} (s)) ds \\ + \int_{0}^{t} \int_{0}^{τ} ((s^{2} - τ^{2}) X_{1, m - 1} (τ) \\ + τ^{2} (s^{2} + τ^{2}) X_{2, m - 1} (τ)) dsd τ), \\ m = 2, 3, \dots . \end{matrix} \end{matrix}$

Now if we set $X_{1, 1} (t) = 0$ , then $(1 - α_{0}) t + (\frac{7 α_{0}}{2} - 3 - \frac{α_{1}}{2}) t^{2} + (\frac{7 α_{1}}{6} - 2 α_{0} - \frac{α_{2}}{3} + 1) t^{3} + (\frac{7 α_{2}}{12} + \frac{3 α_{0}}{4} - \frac{3 α_{1}}{4} - \frac{α_{3}}{4} + \frac{1}{4} + \frac{β_{0}}{2}) t^{4} + (\frac{7 α_{3}}{20} - \frac{2 α_{2}}{5} + \frac{3 α_{1}}{10} - \frac{α_{4}}{5} + \frac{2 β_{0}}{5} + \frac{β_{1}}{5} + \frac{1}{20}) t^{5} + \dots = 0,$ and if we set $X_{2, 1} (t) = 0$ , then $- (1 + β_{0}) t + (1 - \frac{β_{1}}{2} - \frac{β_{0}}{2} - α_{0}) t^{2} + (\frac{2 α_{0}}{3} - \frac{α_{1}}{3} - \frac{β_{1}}{6} - \frac{β_{2}}{3} - \frac{1}{3}) t^{3} + (\frac{α_{1}}{4} - \frac{α_{2}}{6} + \frac{β_{0}}{2} - \frac{β_{3}}{4} - \frac{β_{2}}{12} + \frac{1}{3}) t^{4} + (\frac{β_{1}}{5} + \frac{2 β_{0}}{5} + \frac{2 α_{2}}{15} - \frac{α_{0}}{20} - \frac{β_{4}}{5} - \frac{α_{3}}{10} - \frac{β_{3}}{20} + \frac{1}{5}) t^{5} + \dots = 0 .$

It can be easily shown that $\begin{matrix} α_{0} = 1, α_{1} = 1, α_{1} = \frac{1}{2!}, α_{3} = \frac{1}{3!}, \\ α_{4} = \frac{1}{4!}, α_{5} = \frac{1}{5!}, \dots, \\ β_{0} = - 1, β_{1} = 1, β_{2} = - \frac{1}{2!}, β_{3} = \frac{1}{3!}, \\ β_{4} = - \frac{1}{4!}, β_{5} = \frac{1}{5!}, \dots . \end{matrix}$

Therefore, the exact solutions of the system of integral-differential equation (11) can be expressed as $\begin{matrix} x_{1} (t) = 1 + \sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1} = \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} = e^{t}, \\ x_{2} (t) = 1 + \sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1} = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{t^{n}}{n!} = e^{- t} . \end{matrix}$

Example 2

Consider the following system of integro-differential equations with the exact solutions $x_{1} (t) = \cosh t$ and $x_{2} (t) = \sinh t$ ,

(16)

\begin{matrix} \frac{{dx}_{1} (t)}{dt} = - t^{3} - 6 t - 1 + x_{1} (t) + (7 - 2 t) x_{2} (t) \\ + \int_{0}^{t} ((s + t) x_{1} (s) + (s - t)^{3} x_{2} (s)) ds, x_{1} (0) = 1, \\ \frac{{dx}_{2} (t)}{dt} = - 3 t^{2} + t - 6 + (7 - 2 t) x_{1} (t) + x_{2} (t) \\ + \int_{0}^{t} ((s - t)^{3} x_{1} (s) + (s + t) x_{2} (s)) ds, x_{2} (0) = 0 . \end{matrix}

For solving system (15) by NHPM, we construct the following homotopy:

(17)

\begin{matrix} \frac{{dX}_{1} (t)}{dt} = x_{1, 0} (t) - p (x_{1, 0} (t) + t^{3} + 6 t + 1 - X_{1} (t) - (7 - 2 t) X_{2} (t) \\ - \int_{0}^{t} ((s + t) X_{1} (s) + (s - t)^{3} X_{2} (s)) ds), \\ \frac{{dX}_{2} (t)}{dt} = x_{2, 0} (t) - p (x_{2, 0} (t) + 3 t^{2} - t + 6 - (7 - 2 t) X_{1} (t) - X_{2} (t) \\ - \int_{0}^{t} ((s - t)^{3} X_{1} (s) + (s + t) X_{2} (s)) ds) . \end{matrix}

Assuming that, $x_{1, 0} (t) = \sum_{n = 0}^{\infty} α_{n} P_{n} (t), x_{2, 0} (t) = \sum_{n = 0}^{\infty} β_{n} P_{n} (t), P_{i} (t) = t^{i}, X_{1} (0) = 1, X_{2} (0) = 0$ .

Applying the inverse operator, $L^{- 1} = \int_{t_{0}}^{t} (\cdot) dt$ to both sides of Eq. (17), we obtain

(18)

\begin{matrix} X_{1} (t) = 1 + \sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1} - p (\sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1} + \frac{t^{4}}{4} + 2 t^{3} + t \\ - \int_{0}^{t} (X_{1} (s) + (7 - 2 s) X_{2} (s)) ds \\ - \int_{0}^{t} \int_{0}^{τ} ((s + τ) X_{1} (s) + (s - τ)^{3} X_{2} (s)) dsd τ), \\ X_{2} (t) = \sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1} - p (\sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1} + t^{3} - \frac{t^{2}}{2} + 6 t \\ - \int_{0}^{t} ((7 - 2 s) X_{1} (s) + X_{2} (s)) ds \\ - \int_{0}^{t} \int_{0}^{τ} ((s - τ)^{3} X_{1} (s) + (s + τ) X_{2} (s)) dsd τ) . \end{matrix}

Suppose the solutions of system (17) have the form (14), substituting (14) into (17) and equating the coefficients of p with the same power leads to $\begin{matrix} p^{0} : \{\begin{matrix} X_{1, 0} (t) = 1 + \sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1}, \\ X_{2, 0} (t) = \sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1}, \end{matrix} \\ p^{1} : \{\begin{matrix} X_{1, 1} (t) = - \sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1} - \frac{t^{4}}{4} - 3 t^{2} - t \\ + \int_{0}^{t} (X_{1, 1} (s) + (7 - 2 s) X_{2, 1} (s)) ds \\ + \int_{0}^{t} \int_{0}^{τ} ((s + τ) X_{1, 1} (s) \\ + (s - τ)^{3} X_{2, 1} (s)) dsd τ), \\ X_{2, 1} (t) = - \sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1} - t^{3} + \frac{t^{2}}{2} - 6 t \\ + \int_{0}^{t} ((7 - 2 s) X_{1, 1} (s) + X_{2, 1} (s)) ds \\ + \int_{0}^{t} \int_{0}^{τ} ((s - τ)^{3} X_{1, 1} (s) \\ + (s + τ) X_{2, 1} (s)) dsd τ), \end{matrix} \\ p^{m} : \{\begin{matrix} X_{1, m} (t) = \int_{0}^{t} (X_{1, m - 1} (s) + (7 - 2 s) X_{2, m - 1} (s)) ds \\ + \int_{0}^{t} \int_{0}^{τ} ((s + τ) X_{1, m - 1} (s) \\ + (s - τ)^{3} X_{2, m - 1} (s)) dsd τ), \\ X_{2, m} (t) = \int_{0}^{t} ((7 - 2 s) X_{1, m - 1} (s) + X_{2, m - 1} (s)) ds \\ + \int_{0}^{t} \int_{0}^{τ} ((s - τ)^{3} X_{1, m - 1} (s) \\ + (s + τ) X_{2, m - 1} (s)) dsd τ), \\ m = 2, 3, \dots . \end{matrix} \end{matrix}$

If we set $X_{1, 1} (t) = 0$ , then $- α_{0} t + (\frac{α_{0}}{2} + \frac{7 β_{0}}{2} - 3 - α_{1}) t^{2} + (\frac{1}{2} + \frac{α_{1}}{6} + \frac{7 β_{1}}{6} - \frac{2 β_{0}}{3} - \frac{α_{2}}{3}) t^{3} + (\frac{α_{2}}{12} + \frac{5 α_{0}}{24} + \frac{7 β_{2}}{12} - \frac{β_{1}}{4} - \frac{1}{4} - \frac{α_{3}}{4}) t^{4} + (\frac{α_{3}}{20} - \frac{α_{4}}{5} + \frac{7 β_{3}}{20} - \frac{2 β_{2}}{15} + \frac{7 α_{1}}{120}) t^{5} + \dots = 0 .$

Further assume that $X_{2, 1} (t) = 0$ . Then we have $(1 - β_{0}) t + (\frac{7 α_{0}}{2} - \frac{1}{2} - \frac{β_{0}}{2} - \frac{β_{1}}{2}) t^{2} + (\frac{7 α_{1}}{6} - \frac{2 α_{0}}{3} + \frac{β_{1}}{6} - \frac{β_{2}}{3} - 1) t^{3} + (\frac{7 α_{2}}{12} + \frac{5 β_{0}}{24} - \frac{β_{3}}{4} - \frac{α_{1}}{4} + \frac{β_{2}}{12}) t^{4} + (\frac{7 α_{3}}{20} + \frac{7 β_{1}}{120} + \frac{β_{3}}{20} - \frac{2 α_{2}}{15} - \frac{β_{4}}{5} - \frac{1}{20}) t^{5} + \dots = 0 .$

It can be easily shown that $\begin{matrix} α_{0} = 0, α_{1} = 1, α_{2} = 0, α_{3} = \frac{1}{3!}, α_{4} = 0, α_{5} = \frac{1}{5!}, \dots, \\ β_{0} = 1, β_{1} = 0, β_{2} = \frac{1}{2!}, β_{3} = 0, β_{4} = \frac{1}{4!}, β_{5} = 0, \dots . \end{matrix}$ Thus $\begin{matrix} x_{1} (t) = 1 + \sum_{n = 0}^{\infty} \frac{α_{n}}{n + 1} t^{n + 1} = \sum_{n = 0}^{\infty} \frac{t^{2 n}}{(2 n)!} = \cosh t, \\ x_{2} (t) = \sum_{n = 0}^{\infty} \frac{β_{n}}{n + 1} t^{n + 1} = \sum_{n = 0}^{\infty} \frac{t^{2 n - 1}}{(2 n - 1)!} = \sinh t . \end{matrix}$ which are exact solutions.

4 Conclusion

In this work, we considered a new homotopy perturbation method for solving systems of linear integro-differential equations. New method is a powerful straightforward method. Using this method we obtained new efficient recurrent relations to solve these systems. The new homotopy perturbation method is apt to be utilized as an alternative approach to current techniques being employed to a wide variety of mathematical problems. The computations associated with the examples in this paper were performed using maple 10.

References

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Introduction

New homotopy perturbation method (NHPM) for systems of integro-differential equations

Examples

Conclusion

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[1] Abbasbandy S., . Numerical solutions of the integral equations: homotopy perturbation method and Adomian's decomposition method. Applied Mathematics and Computation. 2006;173:493-500.
[Google Scholar]

[2] Beléndez A., Beléndez T., Márquez A., Neipp C., . Application of He's homotopy perturbation method to conservative truly nonlinear oscillators. Chaos, Solitons and Fractals. 2008;37(3):770-780.
[Google Scholar]

[3] Biazar J., Ghazvini H., . Exact solutions for nonlinear Schrödinger equations by He's homotopy perturbation method. Physics Letters A. 2007;366:79-84.
[Google Scholar]

[4] Biazar J., Ghazvini H., . Numerical solution for special non-linear Fredholm integral equation by HPM. Applied Mathematics and Computation. 2008;195:681-687.
[Google Scholar]

[5] Biazar J., Ghazvini H., . He's homotopy perturbation method for solving system of Volterra integral equations of the second kind. Chaos, Solitons and Fractals. 2009;39:770-777.
[Google Scholar]

[6] Bo T.L., Xie L., Zheng X.J., . Numerical approach to wind ripple in desert. International Journal of Nonlinear Sciences and Numerical Simulation. 2007;8(2):223-228.
[Google Scholar]

[7] Ganji D.D., . The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer. Physics Letters A. 2006;355:337-341.
[Google Scholar]

[8] Ganji D.D., Sadighi A., . Application of homotopy perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. Journal of Computational and Applied Mathematics. 2007;207:24-34.
[Google Scholar]

[9] Golbabai A., Javidi M., . Application of homotopy perturbation method for solving eighth-order boundary value problems. Applied Mathematics and Computation. 2007;191(2):334-346.
[Google Scholar]

[10] Golbabai A., Keramati B., . Modified homotopy perturbation method for solving Fredholm integral equations. Chaos, Solitons and Fractals. 2008;37(5):1528-1537.
[Google Scholar]

[11] He J.H., . Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering. 1999;178:257-262.
[Google Scholar]

[12] He J.H., . A coupling method of homotopy technique and perturbation technique for nonlinear problems. International Journal of Non-Linear Mechanics. 2000;35(1):37-43.
[Google Scholar]

[13] He J.H., . Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation. 2003;135:73-79.
[Google Scholar]

[14] He J.H., . Comparison of homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation. 2004;156:527-539.
[Google Scholar]

[15] He J.H., . The homotopy perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics and Computation. 2004;151:287-292.
[Google Scholar]

[16] He J.H., . Limit cycle and bifurcation of nonlinear problems. Chaos, Solitons and Fractals. 2005;26(3):827-833.
[Google Scholar]

[17] He J.H., . Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons and Fractals. 2005;26:695-700.
[Google Scholar]

[18] He J.H., . Homotopy perturbation method for solving boundary value problems. Physics Letters A. 2006;350:87-88.
[Google Scholar]

[19] Mohyud-Din S.T., Yildirim A., Demirli G., . Traveling wave solutions of Whitham–Broer–Kaup equations by homotopy perturbation method. Journal of King Saud University-Science. 2010;22(3):173-176.
[Google Scholar]

[20] Odibat Z., Momani S., . Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order. Chaos, Solitons and Fractals. 2008;36(1):167-174.
[Google Scholar]

[21] Raftari B., Yildirim A., . The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets. Computers & Mathematics with Applications. 2010;59(10):3328-3337.
[Google Scholar]

[22] Shakeri Fatemeh, Dehghan Mehdi, . Solution of delay differential equations via a homotopy perturbation method. Mathematical and Computer Modelling. 2008;48:486-498.
[Google Scholar]

[23] Siddiqui A.M., Mahmood R., Ghori Q.K., . Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. Chaos, Solitons and Fractals. 2008;35:140-147.
[Google Scholar]

[24] Sun F.Z., Gao M., Lei S.H., . The fractal dimension of the fractal model of drop-wise condensation and its experimental study. International Journal of Nonlinear Sciences and Numerical Simulation. 2007;8(2):211-222.
[Google Scholar]

[25] Wang H., Fu H.M., Zhang H.F., . A practical thermodynamic method to calculate the best glass-forming composition for bulk metallic glasses. International Journal of Nonlinear Sciences and Numerical Simulation. 2007;8(2):171-178.
[Google Scholar]

[26] Xu L., He J.H., Liu Y., . Electrospun nano-porous spheres with Chinese drug. International Journal of Nonlinear Sciences and Numerical Simulation. 2007;8(2):199-202.
[Google Scholar]

[27] Yildirim A., Gülkanat Y., . Analytical approach to fractional Zakharov–Kuznetsov equations by He's homotopy perturbation method. Communications in Theoretical Physics. 2010;53(6):1005-1010.
[Google Scholar]

[28] Yildirim A., Mohyud-Din S.T., Zhang D.H., . Analytical solutions to the pulsed Klein–Gordon equation using Modified Variational Iteration Method (MVIM) and Boubaker Polynomials Expansion Scheme (BPES) Computers & Mathematics with Applications. 2010;59(8):2473-2477.
[Google Scholar]