Numerical Treatment of Nonlinear Volterra-Fredholm Integral Equation with a Generalized Singular Kernel ()
Received 11 July 2016; accepted 11 September 2016; published 14 September 2016
1. Introduction
The V-FIE arises from parabolic boundary value problems. In practical applications one frequently encounters the V-FIE with singular kernel of the form
(1)
The functions
, 
and 
are given and called the kernel of Fredholm integral term, Volterra integral term and the free term respectively and 
is a real parameter (may be complex and has physical meaning). Also, Ω is the domain of integration with respect to position, and the time t,
. While 
is the unknown function to be determined in the space
. In [1] Abdou et al. studied the existence and uniqueness of solution of V-FIE.
There are several techniques that have been utilized to handle the integral Equation (1), in [2] - [5] a few techniques, for example, the projection method, time collocation method, the trapezoidal Nystrom method, and furthermore analytical or numerical techniques were utilized to treated this equation, but this techniques experienced troubles as far as computational work utilized. In [6] treated Maleknejad and Hadizadeh Equation (1) by using the ADM presented in [7] - [9] .
Many authors have studied solutions of two-dimensional linear and nonlinear integral equations by utilizing different techniques, such as Abdou et al. in [10] discussed the solution of linear and nonlinear Hammerstien integral equations with continuous kernel and used two different methods (Adomian decomposition method and homotopy analysis method). Abdou et al. in [11] - [13] considered the integral equation with singular kernel and used Toeplitz matrix and product Nystrom methods to obtain the solution. In [14] El-Kalla and Al-Bugami used ADM and degenerate kernel method for solving nonlinear V-FIE with continuous kernel.
In this paper, we will discuss the combined (LADM) to approximate solutions with high degree of accuracy for V-FIE with a generalized singular kernel.
2. The Adomian Decomposition Method for Solving V-FIE
Consider the integral equation
(2)
The (ADM) introduces the following expression
(3)
for the solution 
of Equation (2), where the components 
will be determined recurrently. Moreover, the method defines the nonlinear function 
by an infinite series of polynomials
(4)
where 
are the so-called Adomian polynomials that represent the nonlinear term ![]()
and can be calculated for various classes of nonlinear operators according to specific algorithms set by Adomian [8] [9] . A new algorithm for calculating these polynomials was established by Wazwaz [15] [16] .
Substituting Equation (3) and Equation (4) into Equation (2) yields
![]()
(5)
The components ![]()
are computed using the following recursive relations
![]()
(6)
![]()
(7)
Relations (6,7) will enable us to determine the components ![]()
recurrently, and as a result, the series solution of ![]()
is readily obtained.
3. Laplace Adomian Decomposition Method Applied to V-FlE with Singular Kernel
3.1. Carleman Kernel
We assume that the kernel ![]()
of Equation (1) takes the form![]()
, [17] then integral Equation (1) can be expressed as:
![]()
(8)
Applying the Laplace transform to both sides of Equation (8) gives:
![]()
(9)
The ADM can be used to handle Equation (9). We represent the linear term ![]()
from Equation (3) and the nonlinear term ![]()
will be represented by the Adomian polynomials from Equation (4).
Substituting Equation (3) and Equation (4) into Equation (9) leads to
![]()
(10)
The ADM introduces the recursive relation
![]()
![]()
(11)
Applying the inverse Laplace transform to the first part of Equation (11) gives![]()
. Utilizing ![]()
will empower us to evaluate![]()
, and so on. This will prompt the complete determination of the components of ![]()
upon utilizing the second part of Equation (11). The series solution follows promptly after utilizing Equation (3). The obtained series solution may converge to an exact solution if such a solution exists.
3.2. Logarithmic Kernel
We assume that the kernel ![]()
of Equation (1) takes the form![]()
, [17] then integral Equation (1) can be expressed as:
![]()
(12)
Applying the Laplace transform to both sides of Equation (12) gives:
![]()
(13)
Using the same method we shall find at the end the required solution by the inverse of Laplace transform.
4. Numerical Examples
4.1. Application for Carleman Kernel and Logarithmic Kernel
We consider two examples for the integral equation
![]()
(14)
![]()
, the exact solution ![]()
We consider the linear and nonlinear cases:![]()
respectively, for the Carleman kernel
![]()
and the computing results are obtained when ![]()
where ![]()
is called Poisson’s coefficient, ![]()
, while the kernel in the second example takes the logarithmic kernel ![]()
and the results are computing, using Maple 17 at ![]()
and![]()
.
Example 1 [(see 1)]: Consider the V-FIE with Carleman kernel
![]()
(15)
![]()
, the exact solution ![]()
Using Maple 17, we obtain Table 1 and Table 2
Example 2 [(see 1)]: Consider the V-FIE with logarithmic kernel
![]()
(16)
![]()
, the exact solution ![]()
Using Maple 17, we obtain Table 3 and Table 4
![]()
Table 1. Results obtained for example 1 and error (Linear case,![]()
).
![]()
Table 2. Results obtained for example 1 and error (Nonlinear case,![]()
).
![]()
Table 3. Results obtained for example 2 and error (Linear case,![]()
).
4.2. Application for a Generalized Carleman Kernel and Logarithmic Kernel
Example 3 [(see 11, 13)]: Consider the V-FIE with generalized Carleman kernel
![]()
(17)
![]()
, the exact solution ![]()
Using Maple 17, we obtain Table 5.
Example 4 [(see 11,13)]: Consider the V-FIE with generalized logarithmic kernel
![]()
(18)
![]()
, the exact solution![]()
,
Using Maple 17, we obtain Table 6.
![]()
Table 4. Results obtained for example 2 and error (Nonlinear case,![]()
).
![]()
Table 5. Results obtained for example 3 and error.
![]()
Table 6. Results obtained for example 4 and error.
5. Conclusion
In this study, we considered linear and nonlinear integral equations of type Volterra-Fredholm with singular kernel. We have proven that the (LADM) is effective and useful technique for solving these kinds of integral equations with singular kernel and many nonlinear problems, efficiency and accuracy of the introduced method are illustrated by four numerical examples which showed simplicity of this method.
Acknowledgements
The authors would like to thank the King Abdulaziz city for science and technology.