Numerical Solution of Functional Integral and Integro-Differential Equations by Using B-Splines ()
1. Introduction
In recent years there has been a growing interest in the numerical treatment of the functional differential equations,
(1)
which is said to be of retarded type if
. It is said to be natural type
. If 
it is said to be advanced type. For more general functional equations see Arndt [1], In further details many authors such as, El-Gendi [2], Zennaro [3], Fox, et al. [4].
Rashed introduced new interpolation method for functional integral equations and functional integro-differential equations [5]. In this paper we approximate the numerical solution 
of the following functional integral equations and integro-differential equations:
(2)
(3)
(4)
(5)
(6)
where
For this approximation we first interpolate 
with following interpolation formula:
(7)
where
(8)
and
(9)
where
(10)
where 
is a well known function.
Then for computing 
we use B-Spline approximation that we present its details in the next section [6-9]. In the third section we give our method for functional integral equations. Also the fourth section is devoted to numerical solution of integro-differential equations. Of course for computing integrals both in the third and the fourth section we used Clenshaw-Curtis rule [10,11]. Finally in the latest section we give some applications of both functional integral equations and integro differential equations with numerical solutions. In addition, we compared our results with Rashed method [5,12]. We present some additional conclusions in Section 6.
2. Functional Linear Integral Equations of the Second Kind
In this paper we use spline function with Lagrange interpolation to compute the numerical solution 
of functional linear integral equations of the second kind:
(11)
(12)
(13)
(14)
In fact, We seek to find an approximation to 
which satisfies some interpolation property or variational principle.
In this functional integral Equations (2)-(5), we may use Lagrange interpolation of 
by
(15)
where
(16)
and
(17)
and also
(18)
where
(19)
and
(20)
where 
is well known function. Also
(21)
(22)
The integral part of each functional Equations (2)-(5) is given as follows: Integrating (1) w.r.t 
from 
to 
(23)
(24)
Integrating (10) w.r.t 
from 
to 
(25)
(26)
Integrating (10) w.r.t 
from 
to 
(27)
where
(28)
(29)
The integral in the relation (21) is approximated as:
(30)
where
Finally, the functional integral equation is approximated by system of 
linear equations. Also, the method is extended to treat the functional equations of advanced type (
in 
or
)
(31)
The last equation may not has analytically solution.
3. Functional Linear Integro-Differential Equations of the Second Kind
Consider the functional integro-differential equation of the second equation
(32)
where
With using Lagrange interpolation, the second derivative 
is given by
(33)
where
(34)
Integrating (35) w.r.t 
from 
to 
(35)
Integrating (37) w.r.t 
from 
to 
(36)
The integral is computed as (31). Also, integrating (36) w.r.t 
from 
to 
(37)
Substitution from (37) to (40) into (34) lead to system of 
linear equations
where
(38)
and
(39)
The integrals in (40) and (41) may be computed by Clenshaw-Curtis rule. It is obviously that the method may be extended to functional linear differential equations of the second order if 
in (34).
4. Numerical Examples
We compared our results with Rashed results [5]. We consider here the following examples on the functional integral equation, integro-differential and differential equations for comparison. The computed errors 
in these tables are defined to be
Example 1. Volterra integral equation of the second kind
Example 2. Fredholm integral equation of the second kind
Example 3. Volterra integral equation of the second kind
Example 4. Volterra integral equation of the second kind
Example 5. Volterra integro-differential equation of the second kind
5. Conclusions
1) The method give the approximate solution at the points
2) The method can be extended to the functional differential equation
3) The method may be used to treat boundary functional differential or integro-differential equations.
4) The small errors obtained shows that the method indeed successfully approximate the solution of problem.
Remark 1 The spline piecewise functions are very essential tools in approximation Theory. Then for this we applied B-Splines for approximation of unknown answer in integro-differential equations. From Tables, We see that for mesh points with
, we have small errors. Therefore the mentioned results for 
have high quality and application of B-Splines is acceptable (of course for getting convenient answer we must had
).
NOTES