Approximate Solution of the System of Nonlinear Fredholm Integral Equations of the Second Kind Using Spline Function

This study deals with introducing spline function to find the approximate solution of the system of nonlinear Fredholm integral equations of the second kind. The benefit of spline functions was demonstrated by presenting several examples. Introduction In this paper we consider the numerical solution of the systems of nonlinear Fredholm integral equation of the second kind (SNFIES),    b a m i i i dt t u t u t u t x k x f x u )) ( ..., ), ( , ) ( , , ( ) ( ) ( 2 1 , for i=1, 2,..., m, ... (1) Where ) (x ui , i=1, 2, ..., m are the unknown functions which are to be determined in the interval [a, b], the functions ) (x f i and the kernels )) ( ..., ), ( ), ( , , ( 2 1 t u t u t u t x k m i are given, by using spline function. Suppose that the system (1) has a unique solution. However, the necessary and sufficient conditions for existence and uniqueness of the solution of system (1) could be found in (Delves & Muhamed, 1985), (Jumaa, 2005).Spline function is a function that consists of polynomial pieces joined together with certain smoothness condition, which is used for data interpolation and can be used for finding derivatives and integrals of function. (Cheney & Kinciad, 2004) Splines can be any degree: linear splines are simply straight line segments connecting two points. Linear splines yield first-order approximation polynomials. The slopes (i.e., first derivatives) and curvature (i.e., second derivatives) are discontinuous at every data point. Quadratic splines yield second-order approximating polynomials. The slopes of the quadratic splines can be forced to be continuous at each data point, but the curvatures are still discontinuous. Cubic splines yield a third–degree polynomial connecting each pair of data points. The slopes and curvatures of the cubic splines can be forced to be continuous at each data point. (Hoffman, 2001) The name Spline comes from the thin flexible rod, called a spline,


Introduction
In this paper we consider the numerical solution of the systems of nonlinear Fredholm integral equation of the second kind (SNFIES), , for i=1, 2,…, m, … (1) Where ) (x u i , i=1, 2, …, m are the unknown functions which are to be determined in the interval [a, b] are given, by using spline function.Suppose that the system (1) has a unique solution.However, the necessary and sufficient conditions for existence and uniqueness of the solution of system (1) could be found in ( Delves & Muhamed, 1985), (Jumaa, 2005).Spline function is a function that consists of polynomial pieces joined together with certain smoothness condition, which is used for data interpolation and can be used for finding derivatives and integrals of function.(Cheney & Kinciad, 2004) Splines can be any degree: linear splines are simply straight line segments connecting two points.Linear splines yield first-order approximation polynomials.The slopes (i.e., first derivatives) and curvature (i.e., second derivatives) are discontinuous at every data point.Quadratic splines yield second-order approximating polynomials.The slopes of the quadratic splines can be forced to be continuous at each data point, but the curvatures are still discontinuous.Cubic splines yield a third-degree polynomial connecting each pair of data points.The slopes and curvatures of the cubic splines can be forced to be continuous at each data point.(Hoffman, 2001) The name Spline comes from the thin flexible rod, called a spline, use by draftsmen to draw smooth curves through a series of discrete points.The spline is placed over the points and either weighted or pinned at each point.Due to the flexure properties of a flexible rod (typically of rectangular cross section), the slope and curvature of the rod are continuous at each point.A smooth curve is then traced along the rod, yielding a spline curve.(Ahlberg, et. al, 1967)Some authors and researchers used spline functions to find the approximate solution of the integral equations, such as (Deputat, et. al, 2005;Saveljeva, 2006 andHoffman, 2001).Also, (Mustafa, 2004) and (Jumaa, 2005) used spline functions to solve a system of linear and nonlinear Volterra integral equations of the second kind respectively.In this paper, linear, quadratic and cubic spline functions are used for solving the system of non-linear Fredholm integral equation of the second kind.

are all continuous functions on [a, b],
3-There is a partitioning of the interval a=x 0 < x 1 < x 2 <…< x n =b, such that ٍS(x) is a polynomial of degree at most k on each subinterval [x i , x i+1 ], i=0, 1,…, n-1.Such a function is somewhat complicated to define in explicit terms, we are forced to write is a polynomial of degree at most k on each subinterval [x i , x i+1 ] for i=1, 2,…, n-1.In this section three types of spline functions have been considered, they are: 1-First-Degree Spline Function (LSF): A simple example for the spline functions is the spline of degree one, whose pieces are linear polynomials joined together to achieve continuity.The linear spline function L(x) consists of n functions which are a linear polynomial on each subinterval, [x i , x i+1 ].

…(2)
] the Lagrange interpolation polynomial is used to represent this piecewise linear curve ( Cheney & Kinciad, 2004): 2‫ـ‬Quadratic Spline Function (QSF): A quadratic spline function is continuously differentiable piecewise quadratic function which consists of n separated polynomial over each subinterval [ is a piecewise quadratic function and continuously differentiable on the entire interval [x 0 , x n ] and which interpolates the table; that is Where This equation can be used to obtain the values of , starting with an arbitrary value for Property I states that ) (x S consists of n piecewise cubic polynomials.Property II states that the piecewise cubics interpolate the given set of data points.Property III and IV require that the piecewise cubics represent a smooth continuous function.Property V states that the second derivative of the resulting function is also continuous.There are 4n unknowns which are the a k , b k , c k and d k for k=0,1,…,n-1.By using the above properties we get We get for k=1, 2, …, n-1.Take .This makes the end cubics approach linearity at extremities.This condition, called a natural spline, matches precisely to the drafting device.This technique is used very frequently.

Solutions of a SNFIES Using Spline Functions
In this section, we use spline functions: linear, quadratic and natural cubic to find numerical solution of a SNFIES.
To apply the spline functions divide the intervals [a, b] into n subintervals as follows:

Note:
In this paper, evaluate the integrals directly if it's possible or numerically by using composite trapezoid method.

‫ـ1‬ Using Linear Spline Function (LSF):
To solve equations (1), suppose in the interval [ where Substituting equation ( 19) into equation ( 18) we get: In equation ( 20) replace x by n x x x ..., , , 1 0 for each i; we get V equations in the V unknowns where V=m(n+1).Solve the resulting nonlinear system by using modified Newton-Raphson method to find the unknowns.Substituted the values of in the equation ( 19), we get the approximation solution of (1).

The Algorithm LSF:
Step (1): Input n (number of subintervals).Set Step(3): Solve a nonlinear system in step (2) to find the unknowns by using modified Newton-Raphson method, and put the resulting values in equation ( 19) to get the approximation solution of (1).
‫ـ2‬ Using Quadratic Spline Functions (QSF): for i=1, 2, …, m; where for i=1, 2, …,m.Where  22) respectively for each i=1, 2, …, m we get a nonlinear system of V equations in the V unknowns Solve the resulting nonlinear system by using modified Newton-Raphson method to get the approximation solution of equation (1).

‫ـ3‬ Using Cubic Spline Functions (CSF):
In , and For finding the values of , rewrite the system (17) as follows where , then solve the system (25) .Substituting (24) into the equation ( 18) we get: by using modified Newton-Raphson method.Substitute the values in the equation ( 24), to get the approximation solution of (1).
Step(3): Solve a linear system (25) to find the values of in the equation ( 24) to get an approximation solution of (1).

Illustrative Examples
In this section, three examples presented for conforming the methods.In each example, we calculate the least-square error and the running times of the program for each method.Note:

Solution:
When applying the algorithms LSF, QSF and CSF, we get the following results: 2.
i=1, 2,…, m and k=1, 2, …, n is given by the equation (6).Replacing x by 2, …, n, and divide the interval [a, b] into n subintervals , n, and divide the interval [a, b] into n subintervals Solve the resulting nonlinear system in step (4) by using modified Newton-Raphson method, and then substitute the values of ,

Table ( 2
): shows a comparison between the exact solutions 2 2.

Table ( 3
): shows a comparison between the exact solution

Table ( 6
): shows a comparison between the exact solution 2

Table ( 7
) shows a comparison between the exact solution x and the numerical solution

Table ( 8
): shows a comparison between the solutions of Example 1 by using QSF with choosing two different initial values as follows:1.