Romberg Algorithm to Solve a System of Nonlinear Fredholm Integral Equations of Second Kind

In this paper, we use Romberg algorithm, to find an approximation solution for a system of nonlinear Fredholm integral equations of the second kind. A comparison between this approximations and exact solutions for four numerical examples are given, depending on the least-square error and running computer time. Our numerical results are obtained by MATLAB 7.0 program and for a system of rank 2×2, 3×3 on Personal Computer P4.


Introduction:
Integral equations are used as mathematical models for many physical situations (Partial Differential Equations) and integral equations also occur as reformulations of other mathematical problems. Recently, many of the authors and co-workers are working in the system of nonlinear Fredholm integral equations of the second kind such as [1], [2] and [6].
Let us consider the following model for a system of nonlinear Fredholm integral equations of second kind are also given [8].
In this work, Romberg integration algorithm [4] has been used for solving the system of nonlinear Fredholm integral equations of the second kind defined in eq. 1. [3,4]:

Romberg Integration
where the error terms are: The pattern in (2) can be extended in the following sense. If an approximation rule is used with step sizes h and h 2 , then an algebraic manipulation of the two answers can be used to produce an improved answer. Each successive level of improvement increases the order of the error term from ) ( ) ( , this process called the Romberg integration [5].
The development of Romberg integration relies on the theoretical assumption that ) is smooth enough so that the error in the trapezoidal rule can be expanded in a series involving only even power of h , that is, The first column of this Similarly, ) 1 , 2 ( R is obtained with two trapezoids: . The second and successive column in the Romberg array are generated by extrapolation formula

Outlines of Romberg Algorithm to solve (eq. 1) [4]
First we find integrals of (eq.1) by Romberg algorithm and summarized by the following steps: and substituted in the (eq. 11) we get the approximate solution of (eq. 1).

Numerical Results:
In this part we present four numerical examples. The first and second are 2×2 system with integrable integrals, if we solve the system by another method such as weighted residual methods, spline function methods, or any approximation method, while the third example is 2×2 system the integrals are not integrable if we solve by another method and the fourth example is 3×3 system where m = number of columns in Romberg tables. It = number of iterations in Modified Newton-Raphson (Quadratically Convergent) method for solving nonlinear system or the number of rows in Romberg table. L.S.E=Least square error. R.T. =running time (CPU time).

Example 4.1: [2]
Consider the following system of nonlinear Fredholm integral equations of the second kind; with the exact solutions 2 ) (  ; approximation to the solution is: Some numerical values of this solution are presented in the following table: Solution: If we choose: ; approximation to the solution is: ; approximation to the solution is: Some numerical values of this solution are presented in the following table: Solution: If we choose: ; approximation to the solution is: ; approximation to the solution is: Some numerical values of this solution are presented in the following table: . By running the newly written program which is (in appendix) as: >>RombergforsolveSNFIESK3x3(0, 1,5,20,'-u1+(x/2)-(1/4)','x*ut1+t*ut2','-u2+x^2-(2/  x Exact value of Approximate value of

Discussions and Conclusions
Romberg integration algorithm has been known to be powerful device for evaluating integrals for many times. Here we use this algorithm for solving a system of nonlinear integral equations of the second kind because it's construction involved integrations. The real hard part of this algorithm is to construct a nonlinear system with respect to some unknowns. Furthermore, a new general program for implementing this algorithm and by increasing the number of rows in Romberg table or the number of iterations in the modified Newton Raphson method (if it is used instead of Romberg algorithm) can reach the desired accuracy. The present algorithm has successfully been applied to the system of nonlinear Fredholm integral equations of the second kind. Extensions of this algorithm for solving another system of nonlinear integro-differential equations offers an excellent opportunity for future research.
From our numerical results, tables (4.1, 4.2, 4.3 and 4.4) indicate that the approximation solution of ) ( 1 x u at m=5, it=25 is very successful even it requires more running time than the case of m=2, it=10. Also the approximation solution of ) ( 2 x u at m=4, it=30 is very successful even it requires an approximately the same running time than m=3, it=15.
Finally the appendix of this paper contains the main steps of the MATLAB program 7.0 to solve a system of rank 2×2 and 3×3.