INTRODUCTION
In recent years, interest has substantially increased in the solution
of singularly perturbed Volterra integral equations. In this study we
continue this trend and consider a new analytical technique, the homotopy
perturbation method (HPM) (He, 1999, 2000), for solving singularly perturbed
Volterra integral equations (SVIEs) of the form:
Here, ε > 0 is a small parameter that gives rise to singularly
perturbed nature of the problem (Alnasr, 2000, 1997; Lange and Smith,
1988; Angell and Olmstead, 1987). The kernel K and the data function g(x)
are given smooth functions. Under appropriate condition on g and K, for
every ε > 0, Eq. 1 has a unique continuous
solution on [0, X] (Alnasr, 1997; Brunner and Van Der Houwen, 1986) and
references cited therein.
The singularly perturbed nature of (1) arises when the properties of
the solution with ε > 0 are incompatible with those when ε
= 0. For ε > 0, (1) is an integral equation of the second kind
which typically is well posed whenever K is sufficiently well behaved.
When ε = 0, (1) reduced to an integral equation of the first kind
whose solution may well be incompatible with the case for ε >
0. The interest here is in those problems which do imply such an incompatibility
in the behavior of y near x = 0. This suggests the existence of boundary
layer near the origin where the solution undergoes a rapid transition
(Alnasr, 1997; Lange and Smith, 1988; Brunner and Van Der Houwen, 1986).
Lange and Smith (1988) and Angell and Olmstead (1987) developed a formal
methodology to obtain asymptotic solution for Eq. 1.
Alnasr (1997, 2000) applied a multistep method to solve the singular
perturbation problem in Volterra integral equation. The author studied
the stability of the multistep method for the following SVIEs:
or
of which the basic test equation (μ = 0) is special case. Here ξ
and η are positive real constants.
Hes homotopy perturbation technique (He, 1999, 2000) has recently been
used to solve singular boundary and initial value problems (AlKhaled,
2007; Ramos, 2006). It has been claimed that this techniques is valid
for nonlinear problems regardless of the presence or absence of a small
parameter in the problem. In particular, Hes homotopy perturbation method
(He, 1999, 2000) for singular linear boundaryvalue problems (AlKhaled,
2007) has been implemented by first reducing a secondorder ordinary differential
equation to a Volterra integral equation, introducing a homotopy parameter,
expanding the solution as a series of this parameter and then setting
this parameter to unity. In this present study we employ Hes homotopy
method to obtain explicit and numerical solutions of Eq.
1.
The homotopy perturbation method is a new approach which searches for
an analytical approximate solution of linear and nonlinear problems. The
homotopy perturbation method has been applied to Volterras integrodifferential
equation (ElShahed, 2005), to nonlinear oscillators (He, 2004a), bifurcation
of nonlinear problems (He, 2005a), bifurcation of delaydifferential equations
(He, 2005b), nonlinear wave equations (He, 2005c) and boundary value problems
(He, 2006a) and to other fields (He, 2003, 2004b, 2006b; Abbasbandy, 2006;
Siddiqui et al., 2006). Recently, the application of the method
has been extended to ordinary and partial differential equations of fractional
order (Odibat and Momani, 2008, 2007; Momani and Odibat, 2007a, b; Abdulaziz
et al., 2007).
HOMOTOPY PERTURBATION METHOD
The homotopy perturbation method (HPM) was first proposed by Chinese
mathematician J.H. (He, 1999, 2000). The essential idea of this method
is to introduce a homotopy parameter, say p, which takes the values from
0 to 1. When p = 0, the system of equations usually reduces to a sufficiently
simplified form, which normally admits a rather simple solution. As p
gradually increases to 1, the system goes through a sequence of deformation,
the solution of each of which is close to that at the previous stage of
deformation. Eventually at p = 1, the system takes the original form of
the equation and the final stage of `deformation` gives the desired solution.
For convenience of the reader, we will present a review of the HPM (He,
1999, 2000), then we apply the method to solve the nonlinear problem (1).
To achieve our goal, we consider the nonlinear integral equation:
with boundary conditions
where, L is a linear operator, while N is nonlinear operator, B is a
boundary operator, Γ is the boundary of the domain Ω and f(r)
is a known analytic function.
or
where, r ∈0 Ω and p ∈0[0,1] is an impeding parameter,
u_{0} is an initial approximation of Eq. 4
which satisfies the boundary conditions. Obviously, from Eq.
6 and 7, we have
The changing process of p from zero to unity is just that of v(r, p)
from u_{0}(r) to u(r). In topology, this called deformation, L(v)L(u_{0})
and L(v)+N(v)f(r), are called homotopic. The basic assumption is that
the solution of Eq. 6 and 7 can be
expressed as a power series in p:
The approximate solution of Eq. 4, therefore, can
be readily obtained:
The convergence of the series (11) has been proved in He (1999, 2000).
In view of the homotopy perturbation method, we can construct the following
homotopy for Eq. 1:
where, p ∈0[0,1]. Substituting y = y_{0} + py_{1}
+ p^{2}y_{2} +..., into (12) and equating coefficients
of like power of yields the following equations:
Finally, the approximate solution for Eq. 1 is given
by
APPLICATIONS AND NUMERICAL RESULTS
Example 1: Consider the following linear problem
which has the exact solution
Table 1: 
Numerical results compared to the exact solution for
Example 1 

and the asymptotic solution
In the view of the homotopy perturbation method; the homotopy for Eq.
14 can be constructed as
Substituting (10) into (17) and equating the terms with identical powers
of p, we have
and so on, in this manner the rest of components of the homotopy perturbation
solution can be obtained. The twentiethterm approximate solution for
Eq. 14 is given by
As the number of terms involved increase, one can observe that the series
solution obtained using the homotopy perturbation method converges to
the series expansion of the exact solution. Comparison of numerical results
with the exact solution (15) for ε = 1.0, 0.75, 0.5 and 0.25 are
shown in Table 1.
Example 2: We second consider the following linear problem
The exact solution for this problem is
where, the parameters γ_{1} and γ_{2} are defined
as:
and the asymptotic solution is given by
In view of homotopy technique, we can construct the following homotopy
Substituting (10) into (21) and equating the terms with identical powers
of p, we have
Table 2: 
Numerical results compared to the exact solution for
Example 2 

So, the approximate solution for Eq. 19 is given by
Table 2 shows the approximate solutions for Eq.
19 obtained for different values of using the homotopy perturbation method.
From the numerical results in Table 2, it is clear that the
approximate solutions are in high agreement with the exact solutions and the
solutions continuously depend on the parameter ε. It is to be noted that
only the twentiethorder term of the homotopy perturbation solution were used
in evaluating the approximate solutions for Table 2. It is
evident that the efficiency of this approach can be dramatically enhanced by
computing further terms or further components of y(x).
Example 3: Consider the following nonlinear integral equation:
which has the exact solution
and the asymptotic solution is given by
In view of homotopy technique, we can construct the following homotopy
for Eq. 23

Fig. 1: 
Plots of Eq. 23 when ε = 1. Exact
solution (—); HPM solution (− − −) 
Substituting (10) into (25) and equating the terms with identical powers
of p, we have
and so on, in this manner the rest of components of the homotopy perturbation
solution can be obtained. The tenthterm approximate solution for Eq.
23 is given by:
The evolution results for the exact solution (24) and the approximate
solution obtained using the homotopy perturbation method, for different
values ε, are shown in Fig. 13. It can be seen
from Fig. 13 that the solution obtained by the present
method is nearly identical with the exact solution when ε = 1 and
the error increases as approaches zero. It is evident that the efficiency
of this approach can be dramatically enhanced by computing further terms
of y(x) when the homotopy perturbation method is used.

Fig. 2: 
Plots of Eq. 23 when ε = 0.75.
Exact solution (—); HPM solution (− − −) 

Fig. 3: 
Plots of Eq. 23 when ε = 0.5.
Exact solution (—); HPM solution (− − −) 
CONCLUSIONS
The homotopy perturbation method was employed successfully for solving
singularly perturbed Volterra integral equations. The work emphasized
our belief that the method is a reliable technique to handle linear and
nonlinear singularly perturbed Volterra integral equations. It provides
the solutions in terms of convergent series with easily computable components
in a direct way without using linearization, discretization or restrictive
assumptions