The singular integral equations are considered to be of more interest than the others and a close form of solution is generally not available. Therefore, great attention must be considered for the numerical solution of these equations. Abdou in , studied Fredholm-Volterra integral equation with singular kernel. Al-Bugami, in , studied some numerical methods for solving singular and nonsingular integral equations. Abdou, El-Sayed and Deebs, in , obtained a solution of nonlinear integral equation. Also in , Abdou and Hendi used numerical solution for solving Fredholm integral equation with Hilbert kernel. In , Al-Bugami used TMM and Volterra-Hammerstein integral equation with a generalized singular kernel. In , Abdou, Borai, and El-Kojok used TMM and nonlinear integral equation of Hammerstein type. Al-Bugami, in , studied the error analysis for numerical solution of HIE with a generalized singular kernel. A. Shahsavaran in , studied Lagrange functions method for solving nonlinear F-VIE. In , Darwish, studied the nonlinear Fredholm-Volterra integral equations with hysteresis. In , Mirzaee used numerical solution of nonlinear F-VIEs via Bell polynomials. In , Raad studied linear F-VIE with logarithmic kernel and solved the linear system of Fredholm integral equations numerical with logarithmic form.
2. Existence and Uniqueness of the Solution of H-VIE
This formula is measured in
, where the FI term is measured with respect to position. While the VI term is considered in time, and
is known function.
is the parameter, while
defines the kind of the integral Equation (1).
, and satisfies:
is a constant)
is continuous in
, satisfies for the constant
, the following conditions:
In other words, we prove that the solution exists using the successive approximation method, also called the Picard method, that we pick up any real continuous function
, we assume
, then construct a sequence
Using the properties of the norm, we obtain:
, we get
Using Cauchy Schwarz inequality and from conditions (i)-(iv-a) with
, we get
, and then we have:
In general, we get:
This bound makes the sequence
The result (4), leads us to say that the formula (2) has a convergent solution. So let
, we have:
The infinite series of (5) is convergent, and
represents the convergent solution of Equation (1). Also each of
is continuous, therefore
is also continuous.
To show that
is unique, we assume that
is also a continuous solution of (1) then, we write
which leads us to the following:
Using conditions (iv-b), then we have:
Finally, with the aid of conditions (i) and (ii):
is necessarily non-negative, and
It follows that if (1) has a solution it must be unique.
Equation (13) becomes:
The formula (7) represents HIE of the second kind at
. Divide the interval
, then using the quadrature formula, the Volterra integral term in (6) becomes:
Using (8) in (6), we have:
The formula (10) represents SHIEs of the second kind, and we have N unknown
4. Some Numerical Techniques for Solving SHIEs
4.1. The TMM
In this section, we present the TMM to obtain numerical solution for HIE of the second kind with singular kernel. Consider:
Write the integral term in the form:
Approximate the integral in the right hand side of Equation (12) by:
are two arbitrary functions. Putting
in Equation (13), where in this case we choose
. By solving the result, then we take:
The relation (12), becomes:
The IE (11) becomes:
, we have:
may be written as
Is a Toeplitz matrix of order
. The solution of the formula (20):
4.2. The PNM
where p and
are badly behaved and well-behaved functions of their arguments, respectively. Then, we get:
, N even and
are the weights. When
, we write:
Form relation (25) through (27) we find:
Then, we obtain:
We now introduce the change of variable
thus the system (30) becomes:
If we define:
, we have:
. If we assume
Hence, the system (29) becomes:
Therefore, the integral Equation (25) is reduced to SLAEs as in (26) or:
Which has the solution:
The PNM is said to be convergent of order r in
. If for N sufficiently large, there exists a constant
independent of N such that:
5. Numerical Applications
We using TMM and PNM at
. In Tables 1-4:
® Exact solution,
® appro. sol. of TMM,
® the absolute error of TMM,
® appro. sol. of PNM,
® the absolute error of PNM.
Table 1. The values of exact, approximate solutions, and errors by using TMM, PNM at N = 20.
Table 2. The values of exact, approximate solutions, and errors by using TMM, PNM at N = 40.
Table 3. The values of exact, approximate solutions, and errors by using TMM, PNM at N = 20.
Table 4. The values of exact, approximate solutions, and errors by using TMM, PNM at N = 40.
The goal of this work is to study the H-VIE with singular kernel of the second kind. TMM and PNM are successive to solve this equation numerically. As
is increasing, the errors are decreasing. As
is increasing, the errors are increasing.