Direct Solution of nth-Order IVPs by Homotopy Analysis Method

Abstract

Direct solution of a class of nth-order initial value problems (IVPs) is considered based on the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge-Kutta method (RK78).

Full text

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Hindawi Publishing Corporation
Differential Equations and Nonlinear Mechanics
Volume 2009, Article ID 842094, 15 pages
doi:10.1155/2009/842094
Research Article
Direct Solution of nth-Order IVPs by
Homotopy Analysis Method
A. Sami Bataineh,1M. S. M. Noorani,2and I. Hashim2
1Department of Mathematics, Irbid National University, Irbid 2600, Jordan
2Centre for Modelling and Data Analysis, School of Mathematical Sciences,
Universiti Kebangsaan Malaysia (National University of Malaysia), 43600 Bangi Selangor, Malaysia
Correspondence should be addressed to I. Hashim, ishak h@ukm.my
Received 3 February 2009; Accepted 4 June 2009
Recommended by Tasawar Hayat
Direct solution of a class of nth-order initial value problems ?IVPs? is considered based on the
homotopy analysis method ?HAM?. The HAM solutions contain an auxiliary parameter which
provides a convenient way of controlling the convergence region of the series solutions. The HAM
gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-
order Runge-Kutta method ?RK78?.
Copyright q 2009 A. Sami Bataineh et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Higher-order initial value problems ?IVPs? arise in mathematical models for problems in
physics and engineering. Generally, second- and higher-order IVPs are more difficult to solve
than first-order IVPs. It is possible to integrate a special nth-order IVP by reducing it to a
first-order system and applying one of the established methods available for such system.
However, it seems more natural to provide direct numerical methods for solving the nth-
order IVPs.
It is the purpose of the present paper to present an alternative approach for the direct
solution of nth-order IVPs based on the homotopy analysis method ?HAM?. The analytic
homotopy analysis method ?HAM?, initially proposed by Professor Liao in his Ph.D. thesis
?1?, is a powerful method for solving both linear and nonlinear problems. ?The interested
reader can refer to the much-cited book ?2? for a systematic and clear exposition on this
method.? In recent years, this method has been successfully employed to solve many types
of nonlinear problems in science and engineering ?3–17?. All of these successful applications
verified the validity, effectiveness and flexibility of the HAM. More recently, Bataineh et al.
?18–25? employed the standard HAM to solve some problems in engineering sciences. HAM
yields a very rapid convergence of the solution series and in most cases, usually only a few

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2Differential Equations and Nonlinear Mechanics
iterations leading to very accurate solutions. Thus Liao’s HAM is a universal one which can
solve various kinds of nonlinear equations. Bataineh et al. ?18? first presented a modified
HAM called ?MHAM? to solve systems of second-order BVPs. Another new approach in
HAM was presented by Yabushita et al. ?26? who applied HAM not only to the governing
differential equations, but also to algebraic equation. We call this new variant of HAM as
NHAM.
In this work, we consider a class of nth-order IVPs of the form
x?n??t? ? f
?
t,x?t?,x??t?,...,x?n−1??t?
?
,
?1.1?
subject to the initial conditions
x?t0? ? x0,x??t0? ? x?
0,...,x?n−1??t0? ? x?n−1?
0
,
?1.2?
where f represents a continuous, real linear/nonlinear function, and x0, x?
are prescribed. Some of the more recent direct ?purely? numerical methods for solving
second-order IVPs were developed by Cash ?27?, Ramos and Vigo-Aguiar ?28, 29?. Recently
Yahaya et al. ?30? applied the seminumeric multistage modified Adomian decomposition
method to solve the nth-order IVPs ?1.1?-?1.2?. Very recently, Chowdhury and Hashim ?31?
demonstrated the applicability of the analytic homotopy-perturbation method for solving
nth-order IVPs.
The aim of this paper is to apply HAM and NHAM for the first time to obtain
approximate solutions of nth-order IVPs directly. We demonstrate the accuracy of the HAM
and NHAM through some test examples. Numerical comparison will be made against the
seven- and eight-order Runge-Kutta method ?RK78?.
0, ..., x?n−1?
0
2. Basic Ideas of HAM
To describe the basic ideas of the HAM, we consider the following differential equation:
N ?x?t?? ? 0,
?2.1?
where N is a nonlinear operator, t denotes the independent variable, x?t? is an unknown
function. By means of generalizing the traditional homotopy method, Liao ?2? constructs the
so-called zero-order deformation equation
?1 − q?L?φ?t;q?− x0?t??? q??N?φ?t;q???,
?2.2?
whereq ∈ ?0,1?isanembeddingparameter,?isanonzeroauxiliaryfunction,Lisanauxiliary
linear operator, x0?t? is an initial guess of x?t? and φ?t;q? is an unknown function. It is
important to note that one has great freedom to choose auxiliary objects such as ? and L
in HAM. Obviously, when q ? 0 and q ? 1, both
φ?t;0? ? x0?t?,φ?t;1? ? x?t?,
?2.3?

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Differential Equations and Nonlinear Mechanics3
hold. Thus as q increases from 0 to 1, the solution φ?t;q? varies from the initial guess x0?t? to
the solution x?t?. Expanding φ?t;q? in Taylor series with respect to q, one has
φ?t;q?? x0?t? ?
?∞
?
m?1
xm?t?qm,
?2.4?
where
xm?
1
m!
∂mφ?t;q?
∂qm
????
q?0
.
?2.5?
If the auxiliary linear operator, the initial guess, the auxiliary parameter ?, and the auxiliary
function are so properly chosen, then the series ?2.4? converges at q ? 1 and one has
φ?t;1? ? x0?t? ?
?∞
?
m?1
xm?t?,
?2.6?
which must be one of the solutions of the original nonlinear equation, as proved by Liao ?2?.
If ? ? −1, ?2.2? becomes
?1 − q?L?φ?t;q?− x0?t??? q?N?φ?t;q???? 0,
which is used mostly in the HPM ?32?.
According to ?2.5?, the governing equations can be deduced from the zero-order
deformation equations ?2.2?. We define the vectors
?2.7?
− →xi? {x0?t?,x1?t?,...,xi?t?}.
?2.8?
Differentiating ?2.2? m times with respect to the embedding parameter q and then setting
q ? 0 and finally dividing them by m!, we have the so-called mth-order deformation equation
L?xm?t? − χmxm−1?t??? ?Rm
?− →xm−1
?,
?2.9?
where
Rm
?− →xm−1
??
1
?m − 1?!
⎧
⎩
∂m−1?N?φ?t;q???
∂qm−1
?????q?0
,
χm?
⎨
0,m ≤ 1,
m > 1.
1,
?2.10?
It should be emphasized that xm?t? ?m ≥ 1? are governed by the linear equation ?2.9? with the
linear boundary conditions that come from the original problem, which can be easily solved
by symbolic computation softwares such as Maple and Mathematica.

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4Differential Equations and Nonlinear Mechanics
A new approach in the HAM was proposed by Yabushita et al. ?26?. We will call this
method NHAM. Yabushita et al. ?26? considered the following projectile problem:
du
dt? f ?t?u ? 0,
dv
dt? f ?t?v ? 1 ? 0,
?2.11?
where
f ?t? ?
?
u2? v2.
?2.12?
The standard HAM applied to this problem yields a divergent solution on some part of the
solutiondomain.InNHAM,thezeroth-orderdeformationequationswereconstructedfornot
only ?2.11?, but also for ?2.12?. This slight modification in the NHAM gives a more accurate
solution.
3. Numerical Experiments
To illustrate the effectiveness of the HAM we will consider four examples of nth-order IVPs
?1.1?-?1.2?.
3.1. Example 1
We first consider the nonlinear second-order IVP
x??? x?2? 0,
?3.1?
subject to the initial conditions
x?0? ? 1,x??0? ? 2.
?3.2?
The exact solution is
x?t? ? 1 ? ln?1 ? 2t?.
?3.3?
To solve ?3.1?-?3.2? by means of HAM, we choose the initial approximation
x0?t? ? 1 ? 2t
?3.4?
and the linear operator
L?Φ?t;q???∂2φ?t;q?
∂t2
,
?3.5?

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Differential Equations and Nonlinear Mechanics5
with the property
L?c1? tc2? ? 0,
?3.6?
where ci?i ? 1,2? are constants of integration. Furthermore, ?3.1? suggests that we define the
nonlinear operator as
N?φ?x;q???∂2φ?t;q?
∂t2
?∂φ2?t;q?
∂t
.
?3.7?
Using the above definition, we construct the zeroth-order deformation equation as in ?2.2? and
the mth-order deformation equation for m ≥ 1 is as in ?2.9? with the initial conditions
xm?0? ? 0,x?
m?0? ? 0,
?3.8?
where
Rm
?− →xm−1
?? x??
m−1?t? ?
m−1
?
j?0
x?
j?t?x?
m−1−j?t?,
?3.9?
now the solution of the mth-order deformation for m ≥ 1 is
xm?t? ? χmxm−1?t? ? ?L−1Rm
?− →xm−1
?.
?3.10?
We now successively obtain
x1?t? ? 2?t2,
x2?t? ? 2?t2? 2?2t2?8
3?2t3,
x3?t? ? 2?t2? 4?2t2? 2?2t2?16
3?2t3?16
3?3t3? 4?3t4,
...
?3.11?
Then the series solution expression can be written in the form
x?t? ? x0?t? ? x1?t? ? x2?t? ? ··· ,
?3.12?
and so forth. Hence, the series solution when ? ? −1 is
x?t? ? 1 ? 2t − 2t2?8
3t3− 4t4?32
5t5− ··· ,
?3.13?
which converges to the closed-form solution ?3.3?.