الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
The thesis consists of four chapters: the first one is a comprehensive survey for a number of useful conventions and facts concerning about the numerical solution methods of the integral equations in general and Fredholm’s equation of the first and second kind in particular. Take for instance, encountering the numerical mathematics and the basic approximation theory. The second chapter deals with studying the numerical methods which are related to solving integral equations such as the repetition method which involves Newton method , reduction method , to the degenerate Kernel method, collection method , Bateman method ,and problem analysis in Fredholm’s integral equation of the second kind. In the third chapter , we deal with basic conventions concerning about the Neural net and especially the Back Propagation Neural Net which will be used in solving Fredholm’s equation of the first kind in the fourth chapter The fourth chapter deals with the possibility of analyzing Fredholm’s integral equation of the first kind by using the Back Propagation Neural net through being subjected to the net training and the architecture for the net method and applying and programming a computer program on Fredholm’s integral equation of the first kind. and This thesis cares about finding the numerical solution for the integral equation because of its several and important applications in many of the biological and natural sciences as sometimes the solution ways are not suitable because of the particularity of a special type of the integral equation in formation . This type is known as ill-posed problem as in Fredholm’s integral equation of the first kind ,and it is known as the ill-posed problem as the solution is presented in finding the value of function f(y) with the data processing of k(x,y),g(x) function. When Simpson’s Rule was used in solving this type , it failed as the matrix equation [g=Af ] necessitates the presence of the inverse matrix for matrix A to have the output (f) , [ f= A-1g ] . In this case the matrix A singular or non-singular but it has a high condition number and the solution is false . For this the Back Propagation Neural Net is used to overcome this problem . By using Fortran90 Language , a net training program was established when it contains three layers under the supervision of a teacher . Many several researches tackled the idea of using the Back Propagation Neural Net in finding the integral equation previously from a different respect despite the difficulty of training the net to find the integral equation . The following steps illustrate the solution idea : Consider the integral equation be the net training necessitates the following :- 1- Function f , g , w and the integration terms α1 , α2 must be known . 2- Function w(t), g are used as input data , function value f(x) is used as a net training target to have its values in the neural net . 3- The training process stops when the condition of continuity is satisfied . To explain the idea , the net was applied on the integral equation : as the exact solution f(x) = cos(x) and the output were 1-When the output layer contains one neuron and x=0.166 , t=0.9 in the input data ,the net output becomes 1.0612 and the exact is 0.9863 a) When x1=0.157 , t=0.9 the net output becomes 0.7113 and the exact solution is 0.9877 b)When x2=0.175 , t=0.9 the net output becomes 0.8960 and the exact solution is 0.9847 .