Runge-Kutta Method and Bolck by Block Method to Solve Nonlinear Fredholm-Volterra Integral Equation with Continuous Kernel ()
1. Introduction
Integral equations of various types and kinds play an important role in many branches of linear and nonlinear function analysis and their applications in the theory of elasticity, engineering, mathematical physical and contact mixed problems. Therefor, many different methods are used to obtain the solution of the Volterra integral equation. In [1] Linz, studied analytical and numerical methods of Volterra equation. In [2], Mirzaee and Rafei used the BBM for the numerical solution of the nonlinear two-dimensional Volterra integral equations. In the references [3] - [8] the authors considered many different methods to solve linear and nonlinear system of Volterra integral equations numerically with continuous and singular kernels. In [9], Al-waqdani studied linear F-VIE with continuous kernel and solved the linear SVIEs numerically with continuous kernel.
(1)
Equation (1) is called the NF-VIE in the space
. Here the Fredholm integral term is considered in position with a positive continuous kernel
for all
, while the Volterra integral term is considered in time with a positive continuous kernel
for all
. The free term
is known continuous function in the space
, while
is unknown function representing the solution of the nonlinear integral Equation (1). The numerical coefficient
is called the parameter of the integral equation, may be complex, and has physical meaning, while the constant parameter
defines the kind of the integral Equation (1).
2. Existence of Solution of NF-VIE
To prove the existence of a unique solution of Equation (1) using fixed point theorem.
We write it in the integral operator form:
(2)
where
(3)
(4)
Then, we assume the following conditions:
i) The kernel of Fredholm integral term satisfies:
(
is a constant).
ii) The kernel of Volterra integral term satisfies:
(
is a constant).
iii) The given function
with its partial derivatives is continuous in
where:
(
is a constant).
iv) The known continuous function
, for the constant
, the following conditions:
a)
b)
where
.
Theorem 1:
If the condition i)-iv) are verified, then Equation (1) has unique solution in the Banach space
.
The provement of this theorem depends on the following two lemmas:
Lemma 1:
Under the conditions i)-iv-a), the operator
defined by (2), maps the space
into itself.
Proof:
In view of Formula (2) and (3) we get:
Using the conditions (i)-(iii), then applying Cauchy-Schwarz inequality, we have:
In the light of the condition (iv-a), the above inequality take the form:
(5)
The lost inequality (5) shows that, the operator
maps the ball
into itself, where
(6)
Since
, therefore we have
. Moreover, the inequality (5) involves the boundedness of the operator W of Equation (2) where:
(7)
Also, the inequalities (5) and (7) define the boundedness of the operator
.
Lemma 2:
If the conditions (i),(ii) and (iv-b) are satisfied, then the operator
is contractive in space
.
Proof:
For two functions
and
in the space
Formula (2), (3) leads to:
Using the condition (iv-b), then apply Cauchy-Schwarz inequality we have:
Finally, with the aid of conditions (i), (ii), and (iv-b) we obtain:
(8)
In equality (8) shows that, the operator
is continuous in the space
, then
is a contraction operator under the condition
.
3. The SVIEs
Consider:
(9)
when
, Equation (9) becomes:
Then,
(10)
where
.
Formula (10) represents Volterra integral equation of the second kind at
. For representing (9) as a VIEs, we use the numerical method. Divide the interval
as
. Using the quadrature formula, Equation (9) becomes:
(11)
where
, and
.
Using (11) in (9), we have:
(12)
Then:
(13)
where
,
. Formula (13) represents a NSVIEs of the second kind, and we have N unknown functions
corresponding to time interval
.
4. Some Numerical Methods for Solving SVIEs
4.1. RKM
In this section, the RKM is used to solve NF-VIE of the second kind. By divide the interval
as
,
and using the quadrature formula, the integral Equation (1) represent a NSVIEs as:
where
.
To solve the NSVIEs:
(14)
where
and
Then, we get
(15)
Now, applying the RKM for solve (15):
Suppose that:
(16)
Substituting from (16) into (15),
Then, we have,
(17)
where
(18)
By derivative (18), we have,
Now, apply the RKM to this system of equations to give,
which lead to,
By using Equation (16) to give,
(19)
which is approximate solution for Equation (15).
Now, if
consider the Pouzet’s derivation, we define:
(20)
The function
is unknown function, such that
(21)
where
, where
.
when
, Equation (10) becomes:
such that
and
Since the function
is the approximate solution at
for Equation (1).
4.2. BBM
In this section, we use the BBM for solving the NF-VIE of the second kind. The interval
is divided into steps of width h,
and
. the approximate solution of
will be define at mesh-points
and denoted by
such as
is an approximation to
.
To solve the NSVIEs:
(22)
where
and
Then, we get
(23)
Rewrite Equation (23) as follows:
(24)
where
.
If the values
are known, then the first integral can be approximated by standard quadrature methods, and the second integral is obtain by a quadrature rule using values of the integrand at
.
Since the values of
at these points are unknown, we have a system of
nonlinear equations by applied the BBM:
(25)
For
,
, where
depend on the quadrature rule used.
Now, for the Modified method of two Blocks we take
, this integration over
can be accomplished by Simpson’s rule, and the integral over
by using a quadratic interpolation of the integrand at the point
, then Equation (23) becomes:
(26)
and
(27)
where
.
Therefore, by Equation (25) the approximate solution is computed by:
(28)
(29)
where
.
Thus, replace the second term in Equation (28) by using integration formulas, then we get:
(30)
(31)
where
Finally, we construct
nonlinear equations from (30) and (31) to find the unknown functions
. The resulting system is solved by using modified Newton-Raphson method.
5. Numerical Examples
We solve two examples by RKM and BBMat
,
,
and
.
In Tables 1-6: fExact®Exact solution, fR.K.®approximate solution of RKM, ER.K.®the absolute error of RKM, fB.B.®approximate solution of BBM, EB.B.®the absolute error of BBM.
Example 1
Consider:
Exact solution
Case 1:
Table 1. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.01, N = 20.
Table 2. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.1, N = 20.
Table 3. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.3, N = 20.
Case 2:
Table 4. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.01, N = 50.
Table 5. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.1, N = 50.
Table 6. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.3, N = 50.
6. Conclusions
From the previous discussions we conclude the following:
1) As N is increasing the errors are decreasing.
2) As x and t are increasing in
, the errors due to RKM and BBM are also increasing.
3) The errors due to the BBM are less than the errors due to RKM (i.e. BBM the better than RKM to solve NF-VIE with continuous kernel).