INTRODUCTION
The theory of singular integral equations has been developed significant importance during the last years, it is arise in many problems of mathematical physics, such as the theory of elasticity, hydrodynamics, biological problems, population genetics and others. Also, nonlinear singular integral equations with Hilbert and Cauchy kernel and its related RimannHilbert problems have been studied by Pogorzelski (1966), Guseinov and Mukhtarov (1980), Wolfersdorf (1985) and Wegert (1992).
Existence results and approximate solutions for certain classes of nonlinear singular integral equations are studied by Amer and Nagdy (1999 and 2000), Amer and Dardery (2005), Amer (1996, 2001 and 2005), Jinyuan (2000) and Junghanns and Weber (1993). The theory of approximation methods and its application for the solution of linear and nonlinear singular integral equations has been developed by Guseinov and Mukhtarov (1980), Kravchenko and Akilov (1982), Ladopoulos and Zisis (1996) and Mikhlin and Prossdorf (1980).
It is well known that the nonlinear singular integral equations are the muchcomplicated
forms of the nonlinear integral equations. The mechanical quadrature method
is one of the basic tools to investigate the approximate solutions of many classes
of nonlinear and linear equations involving integral operator. In this research
we applied the mechanical quadrature method to a certain class of Nonlinear
Singular Integral Equation (NSIE) with Hilbert kernel in generalized Hölder
spaces. The method has been applied to a nonlinear and a linear Singular Integral
Equation (SIE) with known exact solution and the error has been calculated.
The obtained results of the mechanical quadrature method of SIE are compared
with the obtained results of the Toeplitz matrix method and the product Nystrom
method that have been applied (Abdou et al., 2002), to obtain the approximate
solution of the same problem.
FORMULATION OF THE PROBLEM
This study is devoted to investigate the approximate solution of the following nonlinear singular integral equation:
where:
in generalized Hölder spaces H_{φ,m} and is
a numerical parameter, under the following assumptions:
Assumption 1: Suppose that the function g(t, τ, u, (τ)) is defined on the domain
that has partial derivatives up to (m1)  order and satisfy the following HölderLipschitz condition for arbitrary
where, φ, φ* are nondecreasing functions belong to the class Φ, i + j + k = β and β = 0, 1, 2, ..., m1 and (η)β is a constant depends on β.
Assumption 2: Suppose that the function F (t, u(t), v(t)) is defined on the domain:
that has partial derivatives up to (m1) order and satisfy the following condition for arbitrary
for p + q + r = v, v = 0, 1, 2, ..., m1, where, φ_{1} εΦ and ξ (v) is a constant depends on v. Equation 1 with Cauchy kernel has been studied by the collocation method (Amer, 1966), the special cases of Eq. 1 have been found (Ladopoulos and Zisis, 1996; Saleh and Amer, 1987).
Some basic definitions and auxiliary results: In this section we introduce some definitions and results which will be used in the sequel.
Definition 1: (Guseinov and Mukhatarov, 1980; Mikhlin and Prossdorf, 1986).
• 
We denote by Φ to the class of all continuous almost
increasing functions φ defined on (0, π] such that 
• 
We denote by Φ^{m} to the class of all functions
φ ε Φ such that
implies ,
where, c(m) is a constant depends on m. 
• 
We denote by c_{2π} to the space of 2πperiodic
continuous functions with the norm

• 
The generalized Hölder space H_{φ,m}
is defined as

where, is
the modulus of continuity of order m of the function u and
where, is
a constant depends on m.
• 
For u ε H_{φ,m} we define 
and
as a subspace of H_{φ,m}
Definition 2: (Mosaev and Salaev, 1980; Saleh and Amer, 1987).
•  Let the generalized Hölder space ,be
the space of 2Ndimensional vectors z = (z_{0}, z_{1}, ...,
z_{2N1}) with the norm: 
where:
and
are the modulus of continuity of order m of the vector z with respect to the two sets X = {0, 2,..., 2N2}, Y = {1, 3, ..., 2N1} and p ε X,
For we
define:
as a subspace of
• 
We denote the norm in the space by 
Theorem 1 (Mosaev and Salaev, 1980; Saleh, 1984): Let φ ε HΦ^{m}, then the operator
Transforms H_{φ,m}(M) into
, where:
where, e_{1}(m), e_{2}(m) and are
constants depend on m.
Lemma 1 (Saleh and Amer, 1987): Let the condition (3) is satisfied and
then .
Lemma 2 (Saleh and Amer, 1990): Let the condition (4) is satisfied and
The approximate solution in the space H_{φ,m} Theorem 2: Let the function F(t, u(t), v(t)) satisfy the condition (4) and the function g(t, τ, u(τ)) satisfy the condition (3), then for λ<λ_{0}, (λ_{0} sufficiently small), the Eq. 1 has a unique solution in H_{φ,m}(M). The solution is uniformly convergent and can be obtained by the method of successive approximations.
Proof: Let u, v ε H_{φ,m}(M). Then by Lemmas 1, 2 and Theorem 1, the operator
transforms the space H_{φ,m}(M) into the space H_{φ,m} (λR). Therefore if λR≤M, the operator P transforms H_{φ,m}(M) into itself. Using M. Riesz’s Theorem (Kravchenko and Akilov, 1982; Saleh and Amer, 1987).
where:
and from the conditions (3), (4), we obtain
Choosing
and
then the operator P is a contraction mapping. From the completeness of H_{φ,m}(M) in L_{p}, p>1, the Eq. 1 has a unique solution in the subspace H_{φ,m}(M) and this solution can be found by the method of successive approximations.
The approximate solution in the space H^{(N)}_{φ, m}:
By the mechanical quadrature formula (Saleh, 1984), the integral
takes the following form:
where:
the formula (9) at node points t_{i} takes the form:
where:
Applying the quadrature formula (10) to the Eq. 1 at the node points, we obtain:
where, R_{N}(g, t_{j}) is the remainder term, .
If we put u(t_{j}) = z_{j} and the R_{N}(g, t_{j})
is negligible, we obtain the following system of nonlinear algebraic equations:
Lemma 3 (Guseinov and Mukhatarov, 1980; Amer, 2001): If the function g (t, τ, u) and its derivative g_{t} (t, τ, u) satisfy the condition (3), then the function:
satisfies the following condition
where:
Theorem 4.1: Let the function F(t, τ, u) satisfy the condition
(4) and the function g(t, τ, u) satisfy the condition (3), then the system
(11) has a unique solution in the space for
arbitrary N≥3 and this solution can be found by the method of successive
approximations.
Proof: From definition 2, we have
where:
Putting
since the space of
vectors of bounded norms is a closed subspace of and
the function g(t, τ, u) satisfies the condition of Lemmas 1, 2 and Theorem
2 hence .
Taking
where:
let
where:
and
(Saleh and Amer, 1987)
where, θ(m) is a constant depends on m. Thus we have:
Now, let
Hence,
since
where, q(p) is a constant depends on p.
Hence from condition (3), Lemma 3 and from (Saleh and Amer, 1987) we obtain:
substituting from inequality (15) into (13), we get:
From boundedness of the operator and
by using the principle of contraction mapping at
the system (11) has a unique solution in for
arbitrary N≥3, hence the theorem is proved.
The rate of convergence of the approximate solution: From inequality
(17), the Eq. 1 has a unique solution and
the system (11) has a unique solution
The relation
at t = t_{j} is called the approximate solution of the Eq. 1, . The norm of the difference of the vectors z* and u* where,
u* = (u (t_{0}), u(t_{1}), ... , u(t_{2N1})) in can
be determined as follows:
Applying the quadrature formula (10) to Eq. 1 at node points t_{j}, we obtain
putting in
(16) and using the inequality (17), we get:
consequently, we have
To evaluate ,
we state the following two lemmas:
Lemma 4 (Saleh, 1984): Let .
Then for arbitrary natural number , we get
Lemma 5 (Saleh, 1984):
Applying formula (9) on the Eq. 1 and from Eq. 18, we obtain:
Using Lemma 5, we have:
from Lemma 4 and inequality (20), we get:
since
therefore,
taking h = N^{α}, p¯1 < α < 1, then
consequently, from (21), (22) and (23), we obtain
Hence,
Now we present the following examples.
Example 1: Consider the integral equation:
where:
It is easy to check that u* (t) = t is the exact solution of Eq. 24 at λ = 1. Applying the quadrature formula (10) to Eq. 24 at node points, we obtain the following system of nonlinear algebraic equations:
where:
Table 1 displays the exact solution, the approximate solution
and error between them for the Eq. 24 by using the mechanical
quadrature method with N = 20, λ = 1 and at initial values
.
Now, we apply the mechanical quadrature method to a class of LSIE.
Example 2: Consider the integral equation:
under the condition
where:
It is clear that u*(t) = sin t is the exact solution of Eq. 25 at λ = 1. Applying the quadrature formula (10) to Eq. 25 under the condition (26) at the node points t_{j}, we obtain the following system of linear algebraic equations:
Table 1: 
The results for the Eq. 24 by using the
mechanical quadrature method 

u* = The exact solution, z* = The approximate solution, E
= The error 
where:
The condition u(±π) = 0 reduces the node points t_{j} to 2N1 points.
Table 2 displays the exact solution, the approximate solution and the error between them for the Eq. 25 under the condition (26) by using the mechanical quadrature method with N = 20, λ = 1.
CONCLUSIONS
• 
Table 1 and 2 display
that the mechanical quadrature method gives accurate results with respect
to NSIE and LSIE, these results are very acceptable compared to the exact
solution. 
Table 2: 
The results of Eq. 25 by using the mechanical
quadrature method 

u* = The exact solution, z* = The approximate solution, E
= The error 
Table 3: 
The results of Eq. 25 by using the Toeplitz
matrix method and the product Nystrom method 

u = The exact solution, = The approximate solution using product
Nystrom method and R^{(N)} = The error, = The approximate solution
using Toeplitz matrix method and R^{(T)} = The error 
• 
The Toeplitz matrix method and the product Nystrom method
have been applied to the same Eq. 25 under the condition
(26) by Abdou et al. (2002), Table 3 displays the
values of exact solution u(t) = sin t, approximate solution and
the error R^{(T)} at the interior points by using the Toeplitz matrix
method with N = 20, λ = 1. Also it shows the approximate solution and
the error R^{(T)} at the interior points by using the product Nystrom
method with N = 20, λ = 1. 
• 
Its found that, the obtained results of the mechanical quadrature
method are better than the obtained results of the Toeplitz matrix method
and the product Nystrom method that have been applied in Abdou et al.
(2002) to obtain the approximate solution of the same problem (25) under
the same condition (26). 