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Abstract： In this work， the existence and uniqueness solution of the Hammerstein integral equation （HIE）， with a generalized singular kernel， is discussed and solved numerically using Toeplitz matrix method and Product Nystr？m method. Moreover， the error analysis for these methods is discussed. Finally， numerical results when the kernel takes a generalized logarithmic form， Carleman function and Cauchy kernel function are investigated. Also the error， in each case， is estimated.
Key words：： Hammerstein singular integral equation； Toeplitz matrix； Product Nystr？m method； Logarithmic form； Carleman function
Al-Bugami， A. M. （2013）. Error Analysis for Numerical Solutions of Hammerstein Integral Equation With a Generalized Singular Kernel. Progress in Applied Mathematics， 6（2）， 1-15. Available from http：//www.cscanada.net/index.php/ pam/ article/view/j.pam.1925252820130602.2578 DOI： 10.3968/j.pam.192525 2820130602. 2578
1. INTRODUCTION
Many methods are used to obtain the solution of the nonlinear integral equation. Abdou et al. [1] obtained numerically the solution of the singular Fredholm integral equation. Abdou et. al. [2] obtained a numerical solution for the nonlinear integral equation of Hammerstein type. Abdou and Hendi [3] solved the Fredholm integral equation with Hilbert kernel numerically. Abdou [4] obtained the solution of linear and nonlinear integral equation. Abdou and AL-Bugami [5] studied Fredholm integral equation with a generalized singular kernel in the linear case and solved this equation numerically. Emamzadeh and Kajani [6] solved nonlinear Fredholm integral equation by using quadrature Methods. Ahmad and Akbar [7] solved nonlinear Fredholm integral equation using Lagrange Functions.
In this paper， we consider the HIE with a generalized singular kernel of the form
2. EXISTENCE AND UNIQUENESS OF THE SOLUTION
In this section， we prove the existence of a unique solution of Eq. （1）， under the conditions （i-iii） by using successive approximation method：
Theorem 1：： The solution of the nonlinear two-dimensional VIEE （1） with continuous kernel is exist and a unique under the condition
Represents a matrix of order 2N+1 whose elements are zeros except the first and the last columns （rows）.
The solution of the system equation （20） can be obtained in the form
（b） The nonlinear algebraic system equation （24） has a unique solution， under the convergence condition： 5. NUMERICAL EXAMPLES
In this section， we apply the previous methods to obtain numerical results for equation （1）， with logarithmic kernel， Carleman function and Cauchy kernel. By using Maple 10 the approximate solution， and the absolute error in each case， is obtained and computed， respectively.
The Toeplitz matrix method and product Nystrom method are used to get the approximate solution for values of ？ = 1， ？ = 0.0001， 0.001， and N = 10， 20 units.
Here Toeplitz matrix method and product Nystrom method are used to get the approximate solution for values of ？ = 1， ？ = 0.001， ？ = 0.1， 0.2， and N = 10， 20 units.
Here， the Toeplitz matrix method and product Nystrom method are used to get the approximate solution for values of ？ = 1， ？ = 0.001， ？ = 0.1， 0.2 and N = 10， 20 units.
6. CONCLUSIONS
From the previous examples， we deduce the following：
a） When the values of N are increasing， the error is increasing.
b） The error increase when the values of is increasing.
c） In the Carleman function， the error increase when the values of N and are fixed and when values of increasing.
d） As t is increasing in interval [-1，1]， the errors due to Toeplitz matrix method and product Nystrom method are also increasing.
REFERENCES
[1] Abdou， M. A.， Mohamed， K. I.， & Ismail， A. S. （2002）. Toeplitz matrix and product Nystrom methods for solving the singular integral equation. Le Mathematical， 11（2）， 21-37.
[2] Abdou， M. A.， EL-Boraie， M. M.， El-Kojok， M. K. （2009）. Toeplitz matrix method for solving the nonlinear integral equation of Hammerstein type. J. Comp. Appl. Math.， 223， 765-776.
[3] Abdou， M. A.， & Hendi， F. A. The numerical solution of Fredholm integral equation with Hilbert kernel. JKSIAM， 9（1）， 111-123.
[4] Abdou， M. A. （2003）. On the solution of linear and nonlinear integral equation. Appl. Math. Comput， 146， 857-871.
[5] Abdou， M. A.， & Albugami， A. M. （2012）. Numerical Solution for Fredholm integral equation with a generalized singular kernel. International Journal of Computational and Applied Mathematics， 7（4）， 449-463.
[6] Jafari Emamzadeh， M.， & Tavassoli Kajani， M. （2010）. Nonlinear Fredholm integral equation of the second kind with quadrature methods. Journal of Mathematical Extension， 4（2）， 51-58.
[7] Shahsavaran， A.， & Shahsavaran， A. （2012）. Numerical approach to solve second kind nonlinear integral equations using lagrange functions. Applied Mathematical Sciences， 6（18）， 893-899.
[8] Kreysig， E. （1978）. Introductory functional analysis with applications. New York： John Wiley & Sons.
[9] Delves， L. M.， & Mohamed， J. L. （1985）. Computational methods for integral equations. New York： Cambridge University Press.
Key words：： Hammerstein singular integral equation； Toeplitz matrix； Product Nystr？m method； Logarithmic form； Carleman function
Al-Bugami， A. M. （2013）. Error Analysis for Numerical Solutions of Hammerstein Integral Equation With a Generalized Singular Kernel. Progress in Applied Mathematics， 6（2）， 1-15. Available from http：//www.cscanada.net/index.php/ pam/ article/view/j.pam.1925252820130602.2578 DOI： 10.3968/j.pam.192525 2820130602. 2578
1. INTRODUCTION
Many methods are used to obtain the solution of the nonlinear integral equation. Abdou et al. [1] obtained numerically the solution of the singular Fredholm integral equation. Abdou et. al. [2] obtained a numerical solution for the nonlinear integral equation of Hammerstein type. Abdou and Hendi [3] solved the Fredholm integral equation with Hilbert kernel numerically. Abdou [4] obtained the solution of linear and nonlinear integral equation. Abdou and AL-Bugami [5] studied Fredholm integral equation with a generalized singular kernel in the linear case and solved this equation numerically. Emamzadeh and Kajani [6] solved nonlinear Fredholm integral equation by using quadrature Methods. Ahmad and Akbar [7] solved nonlinear Fredholm integral equation using Lagrange Functions.
In this paper， we consider the HIE with a generalized singular kernel of the form
2. EXISTENCE AND UNIQUENESS OF THE SOLUTION
In this section， we prove the existence of a unique solution of Eq. （1）， under the conditions （i-iii） by using successive approximation method：
Theorem 1：： The solution of the nonlinear two-dimensional VIEE （1） with continuous kernel is exist and a unique under the condition
Represents a matrix of order 2N+1 whose elements are zeros except the first and the last columns （rows）.
The solution of the system equation （20） can be obtained in the form
（b） The nonlinear algebraic system equation （24） has a unique solution， under the convergence condition： 5. NUMERICAL EXAMPLES
In this section， we apply the previous methods to obtain numerical results for equation （1）， with logarithmic kernel， Carleman function and Cauchy kernel. By using Maple 10 the approximate solution， and the absolute error in each case， is obtained and computed， respectively.
The Toeplitz matrix method and product Nystrom method are used to get the approximate solution for values of ？ = 1， ？ = 0.0001， 0.001， and N = 10， 20 units.
Here Toeplitz matrix method and product Nystrom method are used to get the approximate solution for values of ？ = 1， ？ = 0.001， ？ = 0.1， 0.2， and N = 10， 20 units.
Here， the Toeplitz matrix method and product Nystrom method are used to get the approximate solution for values of ？ = 1， ？ = 0.001， ？ = 0.1， 0.2 and N = 10， 20 units.
6. CONCLUSIONS
From the previous examples， we deduce the following：
a） When the values of N are increasing， the error is increasing.
b） The error increase when the values of is increasing.
c） In the Carleman function， the error increase when the values of N and are fixed and when values of increasing.
d） As t is increasing in interval [-1，1]， the errors due to Toeplitz matrix method and product Nystrom method are also increasing.
REFERENCES
[1] Abdou， M. A.， Mohamed， K. I.， & Ismail， A. S. （2002）. Toeplitz matrix and product Nystrom methods for solving the singular integral equation. Le Mathematical， 11（2）， 21-37.
[2] Abdou， M. A.， EL-Boraie， M. M.， El-Kojok， M. K. （2009）. Toeplitz matrix method for solving the nonlinear integral equation of Hammerstein type. J. Comp. Appl. Math.， 223， 765-776.
[3] Abdou， M. A.， & Hendi， F. A. The numerical solution of Fredholm integral equation with Hilbert kernel. JKSIAM， 9（1）， 111-123.
[4] Abdou， M. A. （2003）. On the solution of linear and nonlinear integral equation. Appl. Math. Comput， 146， 857-871.
[5] Abdou， M. A.， & Albugami， A. M. （2012）. Numerical Solution for Fredholm integral equation with a generalized singular kernel. International Journal of Computational and Applied Mathematics， 7（4）， 449-463.
[6] Jafari Emamzadeh， M.， & Tavassoli Kajani， M. （2010）. Nonlinear Fredholm integral equation of the second kind with quadrature methods. Journal of Mathematical Extension， 4（2）， 51-58.
[7] Shahsavaran， A.， & Shahsavaran， A. （2012）. Numerical approach to solve second kind nonlinear integral equations using lagrange functions. Applied Mathematical Sciences， 6（18）， 893-899.
[8] Kreysig， E. （1978）. Introductory functional analysis with applications. New York： John Wiley & Sons.
[9] Delves， L. M.， & Mohamed， J. L. （1985）. Computational methods for integral equations. New York： Cambridge University Press.