الفهرس | Only 14 pages are availabe for public view |
Abstract In engineering and scientific applications of a natural phenomenon we usually find ourselves in front of one equation of three, differential equation, integral equation, or integro - differential equation. In fact the conversion of a scientific phenomenon to integral equation is the easy way to obtain numerical solutions, enable us to treat the singularities of the solution at the end - points, and enable us to prove fundamental results on the existence and uniqueness of the solution. In general, an integral equation is an equation where an unknown function occurs under an integral sign. Integral equations are classified according to three different dichotomies. If the limits of integration are both fixed then it is called Fredholm equations, if one limit is variable then it is called Volterra equations. If the Placement of the unknown function only inside the integral it is called of the first kind, and if the unknown function is in both inside and outside the integral it is called of the second kind. If the known function identically zero then it is called homogeneous, if not identically zero it is called inhomogeneous. Both Fredholm and Volterra equations are said to be linear integral equations, due to the linear behavior of the unknown function under the integral sign. Indeed, Integral equations are encountered in a variety of applications: in potential theory, geophysics, electricity and magnetism, radiation, and control systems. Many methods of solving Fredholm integral equations of the second kind have been developed in recent years, such as quadrature method, collocation method and Galerkin method, expansion method, product-integration method, deferred correction method, graded mesh method, and Petrov–Galerkin method. In addition, the iterated kernel method is a Traditional method for solving the second kind. However, this method also requires a huge size of calculations. The objective of this thesis is 5 to establish a promising algorithm that can be easily programmed in Matlab, and therefore, computational complexity can be considerably reduced and much computational time can be saved. The main idea of the thesis is given in chapter 3 where a modified Iterative Method via matrices for the solution of Fredholm Integral Equations of the second kind has been presented. The given method gives a very simple form for the iterated kernels via the well - known Hilbert matrix. Thus, the iterative solutions of an integral equation of the second kind can be reduced to the solution of a matrix equation, whereas only one coefficient matrix is required to be computed. Thus the computational complexity can be considerably reduced and much computational time can be saved. The new proposed approach needs a small number of iterations to provide an exact result that proofs the power of the given Method, and stimulates to find out the relation between the integral equations and Hilbert Matrix. The convergence of the obtained solution is studied and three conditions for the existence of the iterative solution are given. In the First chapter the definitions, concepts, types, and applications of integral equations of all types and kinds are presented. In chapter 2 all about Fredholm Integral Equations of the second kind and their different methods of solutions is presented (about 15 methods). In chapter 4 the matlab program for our method with approximate examples is given. The tables and figures of our obtained solutions are compared with the solutions by iterated kernel method is also given, showing that The new proposed approach needs a small number of iterations and much computational time can be saved to provide an exact result. |