Successive approximation method (S.A.M.) for solving integral equation of the first kind with symmetric kernel

In this paper we used successive approximate method (S.A.M.) to solve Fredholm integral equation of the first kind (F.I.E.1 st . K.) with symmetric kernel. And also suggested an algorithm for this method the computer programming is given for the algorithm. The method and algorithm are tested on several numerical examples. After comparing the results with exact solution see tables (1, 2), it occurred that the results are good. (in this work Matlab programming used).


Abstract:
In this paper we used successive approximate method (S.A.M.) to solve Fredholm integral equation of the first kind (F.I.E.1 st . K.) with symmetric kernel. And also suggested an algorithm for this method the computer programming is given for the algorithm. The method and algorithm are tested on several numerical examples. After comparing the results with exact solution see tables (1, 2), it occurred that the results are good. (in this work Matlab programming used).

1-Introduction.
The mechanics problem of calculating the time a particle to side under gravity down a given smooth curve, from any point on the curve to it's lower end, leads to an exercise in integration. The time ) (τ f say, for the particle to descend from the height t is given by an expression of the form Where ) (t y embodies the shape of the given curve. The converse problem, in which the time of descent from height τ is given and the particular curve which produces this time has to be found is les straight for ward, as it entails the determination of the function φ form (1), . From this point of view (1) is called integral equation, this description expressing the fact that the function to be determined appears under an integral sign. The equation (1) is one of many integral equations which result directly from a physical problem.The notation adopted in this section, and through out by y or ) , and the function ) (s f called the free term, is also assumed known.[2, 9,12]. In general the kernel and free term will be complex -valued functions of real variables. A condition such as ( ) where λ is a numerical parameter, Generally complex, in practical applications λ is usually composed of physical quantities. To solve the equation (3) there are several methods, like Galerkin, collocation [1, 2, 4, 9,12]…etc. In this paper we try to solve type of equation (3)

Iterative kernels:-
The Fredholm Integral Equation has iterated kernels of the form. Let determined form formula (5) are called iterative kernels for them. [9].

Successive approximation method (S.A.M) to solve Fredholm integral equation of the first kind:-
The method of successive approximations consists in the following we have an integral equation The integral equation (6) Similarly it is established that generally. Step 2: Find ) ( 1 s u from equation (8).

Numerical examples and results:-
The following examples are solved by using S.A.M.

5-Conclusion.
After testing the SM and it's Algorithm on several Numerical examples the results are obtained are obvious in tables (1, 2), they are very good results, for instance (1) in table (1) for solving example (1) shows at s=0.314159, at n=17 iteration we obtain approximation solution with error 0.000000 at time 4.0150 second, and also for after values of s results are clear. In the table (2) for solving example (2) shows at s=0.2, at n=13 iteration error becomes 0.000001 and RT equal 1.797000 second, is also the excellent, results and it's clear for different values of s .
All of the numerical examples gave good results, but in this paper, we occurred only two examples. But results indicated the successive approximation method, was successively for solving (F.I.E.1 st . K.) with symmetric kernel.